October 9th

Melissa Zhang (UC Davis)

Title: 

Decompositions and diagrams of symplectic surfaces in Weinstein domains 

Abstract:  

A closed 3-manifold can be diagrammatically encoded by a Heegaard diagram, which comes from a decomposition of the space into 1-handlebodies called a Heegaard splitting. Analogously, Gay and Kirby's trisections diagrammatically encode the decomposition of a smooth 4-manifold into three 1-handlebodies. More generally, Islambouli--Naylor multisections decompose 4-manifolds with boundary into n 1-handlebodies. Meier--Zupan's bridge trisections encode surfaces in these 4-manifolds. 

We may ask for decompositions that further encode geometric structure.  In 2023, Islambouli and Starkston defined "multisections with divides", which additionally capture the symplectic structure of a Weinstein domain. In joint work with Román Aranda, Patricia Cahn, and Agniva Roy, we introduce combinatorial and diagrammatic methods for representing properly embedded symplectic surfaces in 4-dimensional Weinstein domains. We show that positive ascending surfaces, which include complex curves in Stein domains and multisections of Lefschetz fibrations, can be placed in bridge position with respect to Islambouli--Starkston's bisection-with-divides structure on the Weinstein domain. In this talk, I will introduce the decompositions and diagrams described above, and then show how one can algorithmically relate various decompositions of such surfaces, including transverse banded unlink diagrams, quasipositive factorizations, bridge bisections with divides, shadow diagrams (curves on surfaces), and pointed monodromy factorizations. 

October 16th

Ka Ho Wong (UC Davis)

October 17th: in-person

N+13th Southern California Topology Colloquium at Caltech

Erica Flapan, (Pomona College) 

Qiuyu Ren, (Stanford University) 

Mayuko Yamashita, (Perimeter Institute for Theoretical Physics)

Tian Yang, (Texas A&M University)

See https://sites.google.com/view/socaltopology/home for details.

October 23th

TBA

October 30th

Tobias Ekholm (Uppsala University)

November 6th

TBA

November 13th

TBA

November 20th

TBA

November 27th: No seminar (Thanksgiving break)

December 4th

TBA

2026 Spring (in-person or online, as indicated)

April 3rd

First Talk (2pm-3pm), in-person:

Flexibility and degeneracy around a theorem of Thurston

Alex Nolte (IHES) 

We will describe some curious stability phenomena that appear when comparing lengths of curves on different hyperbolic surfaces. In particular, we will explain construct “noisy” families of geodesics for Thurston’s Lipschitz metric on Teichmüller space. We then explain a construction of n-ples of Fuchsian representations where an invariant built out of vectors of lengths of curves, called the limit cone, exhibits a surprising finite-sidedness behavior. If time allows, we will conclude by discussing the estimates on lengths of curves that we use to prove polyhedrality of our examples of limit cones.

Second Talk (3pm-4pm), in-person:

Classification of ancient cylindrical mean curvature flows and the Mean Convex Neighborhood Conjecture  

Yi Lai (UC Irvine)

We resolve the Mean Convex Neighborhood Conjecture for mean curvature flows in all dimensions and for all types of cylindrical singularities. Our proof relies on a complete classification of ancient, asymptotically cylindrical flows. We prove that any such flow is non-collapsed, convex, rotationally symmetric, and belongs to one of three canonical families: ancient ovals, the bowl soliton, or the flying wing translating solitons. This is joint work with Richard Bamler. 

April 17th

First Talk (2pm-3pm), in-person: 

Strict hyperbolization of flat manifolds 

Eduardo Reyes (Pontifical Catholic University of Chile)

Charney–Davis strict hyperbolization is a construction that takes a non-positively curved cubulated manifold (or cube complex) and turns it into a negatively curved space. In this talk, I will discuss two extra features of strict hyperbolization. First, it can be carried out in a way that preserves the stable tangent bundle of the input manifold. Second, when applied to a suitable class of flat manifolds, it produces closed hyperbolic manifolds with interesting topological properties. Applications include new examples of closed hyperbolic manifolds with nontrivial Pontryagin and Stiefel–Whitney classes, hyperbolic manifolds that occur as totally geodesic boundaries of other hyperbolic manifolds, and new exotic pairs of negatively curved manifolds. This is joint work with Mauricio Bustamante and Stefano Riolo. 

Second Talk (3pm-4pm), in-person:

Systoles of hyperbolic hybrids

Sami Douba (University of Bonn)

We explain a method for constructing in any dimension n>2 and for any positive integer m a collection of m pairwise incommensurable closed hyperbolic n-manifolds of the same volume each possessing a unique shortest closed geodesic of the same length less than 1/m. 

May 1st

First Talk (2pm-3pm), in-person:

A Steenrod Square for Knot Floer homology  

Yan Tao (UCLA)

We give a combinatorial description of link Floer homology. Then, we outline the Manolescu-Sarkar construction for the link Floer stable homotopy type and show its extension to the full grid. Finally, we find Steenrod squares for this homotopy type, which lets us compute the homotopy type for many examples of knots. 

Second Talk (3pm-4pm), in-person:

Orientations in Real Heegaard-Floer theory 

Ciprian Bonciocat (Stanford University)

Real Heegaard Floer homology is an invariant recently defined by Gary Guth and Ciprian Manolescu, which takes as input a 3-manifold equipped with a branched cover involution, and outputs a graded Z/2-vector space. It is defined in a way that parallels Jiakai Li's construction of Real Monopole Floer homology, by restricting attention only to Heegaard diagrams, intersection points, and pseudo-holomorphic curves that respect the involution. In this talk, we explore the question of orientability for the Floer moduli spaces, which would be required to lift the Guth-Manolescu theory to integer coefficients, and its consequences to being able to define a Heegard-Floer theoretic analogue of Miyazawa's invariant for surfaces in 4-manifolds (defined by counting real solutions to the Seiberg-Witten equations). In the case of roll-spun knots, this numerical invariant should be computable from the functorial maps of the 3-manifold invariant. Time permitting, we may also discuss the question of orientability in generalized cohomology theories, with the idea of defining Heegaard-Floer homotopy types analogous to the Seriberg-Witten story. 

May 8th

First Talk (2pm-3pm), in-person:

What the Heat Trace Hears at Curved Corners  

Sam Looi (Caltech)

What geometric information can one hear from the heat trace of a planar region? The classical McKean–Singer expansion recovers the area, boundary length, and Euler characteristic of a smoothly bounded domain. For domains with corners, each vertex contributes a term depending on the opening angle. These contributions are well understood when corners are "straight" (locally isometric to a Euclidean sector) but when the boundary curvature is nontrivial all the way down to a corner, much less is known.

In joint work with David Sher, we compute the t^{1/2} coefficient in the short-time heat trace expansion for planar curvilinear polygons, with both Dirichlet and Neumann boundary conditions. This coefficient splits into a boundary integral of the squared curvature of the boundary and a sum of local corner contributions, each depending only on the interior angle and the limiting one-sided curvatures of the adjacent arcs. Using a conformal model and a parametrix construction on the sector heat space, we characterize this contribution and compute it explicitly for right angles. As an application, we show that any admissible curvilinear polygon Dirichlet isospectral to a polygon must itself be a polygon, extending a result of Enciso and Gómez-Serrano beyond the straight-corner case.

Second Talk (3pm-4pm), in-person:

Curvature-homogeneous 4-manifolds  

Renato Ghini Bettiol (CUNY)

A Riemannian metric is curvature-homogeneous if its curvature operators at all points are the same, using linear isometries between tangent spaces at different points to compare them. Of course, locally homogeneous metrics are curvature-homogeneous, but, in dimensions > 2, these are far from the only examples. In this talk, I will discuss several new results about curvature-homogeneous manifolds of dimension 4, obtained in joint work with Robert Bryant.

May 15th

First Talk (2pm-3pm), in-person:

Isoperimetry by stretching 

Kobe Marshall-Stevens (Johns Hopkins University)

Isoperimetric boundaries minimise area for fixed an enclosed volume, with sharp regularity theory ensuring smoothness away from a closed singular set of codimension seven. I will discuss recent work, with G. Niu, which constructs isoperimetric regions from hypersurfaces in closed manifolds. As a direct application, we show that a wide variety of topological types and singular sets arise in isoperimetric boundaries. 

Second Talk (3pm-4pm),  in-person:

Z/2Z Eigenfunctions, Branched Minimal Surfaces and Thin Free Boundaries 

Federico Franceschini (Stanford University)

I will discuss a problem at the intersection of differential geometry and PDEs that has independently emerged in the work of three distinct research communities. The talk will emphasize intuition and accessible open questions. As an application, I will present a new and relatively elementary construction of branched singularities for minimal hypersurfaces, from joint work with Rafe Mazzeo and Paul Minter. 

May 22nd

3pm-4pm, in-person

Series Invariants for Plumbed 3-Manifolds 

Nicola Tarasca (Virginia Commonwealth University)

I will present a new series invariant for plumbed 3-manifolds recovering recent results of Gukov-Pei-Putrov-Vafa, Gukov-Manolescu, Park, and Ri. The series incorporates a root lattice as an input and applies more generally to 3-manifolds which are not necessarily negative definite. We show that our series are unique in an appropriate sense and verify certain gluing and splitting properties related to the corresponding operations on 3-manifolds. This is joint work with Allison Moore. 

May 29th

First Talk (1pm-2pm), in-person:

The Utility of Tangle Decompositions

Allison Moore (Virginia Commonwealth University)

A tangle decomposition along a Conway sphere breaks a knot or link into simpler pieces, each of which is a two-string tangle. Whether the goal is direct (investigate knot invariants under mutation), indirect (investigate unknotting number), or about something else entirely (Dehn surgery), tangle decompositions can be used to address classic problems in knot theory and three-manifold topology. Numerous instances illustrate the versatility of tangle decompositions: the Montesinos correspondence of Dehn surgery with rational tangle replacement, algebraic manifestations of tangles in knot homology theories, or tangle methods for analyzing enzymatic actions in biopolymers. These themes will be highlighted, and I will discuss recent applications of tangle-based methods in the study of spatial theta curves and three-manifolds arising in my own joint work with multiple collaborators.

Second Talk (2pm-3pm), in-person:

Knot complements, series invariants and Lie superalgebra 

John Chae (UC Davis)

Inspired by the categorification program for a numerical invariant of three-manifolds, series invariants for closed manifolds and for knot complements were introduced. This in turn motivated an extension of the series invariant of the former case to Lie superalgebras. It was recently generalized to knot complements. In this talk, we review the original series invariants and introduce the recent generalization and explore its properties and examples. 

Third Talk (3pm-4pm), in-person:

Deloopings of compact manifolds - a survey

Milo Moses (Caltech)  

Associated to every Lie group G there is a classifying space BG for principal BG-bundles. This classifying space BG is a "delooping" of G in the sense that its loop space is homotopy equivalent to G. In this talk I will survey the rich literature around deloopings of compact manifolds (this means BG as well as more exotic deloopings), which involves a beautiful mix of homotopy theory, Lie theory, and p-adic analysis.

2026 Winter (in-person or online, as indicated)

January 9th

First Talk (2pm-3pm), in-person: 

Second-to-Top Term of \hat{HFK} of Closed 3-Braids 

Zhaojun Chen (Caltech)

In this paper, we use the skein exact sequence and other techniques to compute the second-to-top term of HFK of closed 3-braids. We do it case-by-case according to Xu's classification. We also verify the rank inequality conjectured by Sivek. 

Second Talk (3pm-4pm), online: 

The Hasse Principle for Geometric Variational Problems: An Illustration via Area-minimizing Submanifolds 

Zhenhua Liu (Princeton University)

The Hasse principle in number theory states that information about integral solutions to Diophantine equations can be pieced together from real solutions and solutions modulo prime powers. We show that the Hasse principle holds for area-minimizing submanifolds: information about area-minimizing submanifolds in integral homology can be fully recovered from those in real homology and mod n homology for all n ∈ Z≥2. As a consequence we derive several surprising conclusions, including: area-minimizing submanifolds in mod n homology are asymptotically much smoother than expected, area-minimizing submanifolds are not generically calibrated, and products of area-minimizing submanifolds are not generically area-minimizing. We conjecture that the Hasse principle holds for all geometric variational problems that can be formulated on chain spaces over different coeffiicients, e.g., Almgren-Pitts min-max, mean curvature flow, Song’s spherical Plateau problem, minimizers of elliptic and other general functionals, etc. 

January 23rd

First Talk (2pm-3pm), in-person: 

Large-scale consequences of nonnegative Ollivier-Ricci curvature 

Tom Hutchcroft (Caltech)

The Ollivier-Ricci curvature is an analogue of the Ricci curvature that makes sense for discrete metric spaces like graphs. I will overview the basic theory of this notion and then discuss recent results on the volume growth of bounded degree graphs of non-negative Ollivier-Ricci curvature. Joint work with Isaac Lopez and Florentin Munch.

Second Talk (3pm-4pm), in-person: 

Concavity Properties of Dirichlet Eigenfunctions 

Malik Tuerkoen (UC Irvine)

On convex domains in R^n and S^n, the first Dirichlet eigenfunction is known to be log-concave, a fact that is crucial to estimate the spectral gap, which is the difference between the second and first Dirichlet eigenvalue. I will present power-concavity results for Dirichlet eigenfunctions: on convex domains in S^n and R^n, the first eigenfunction is p-concave (i.e., u^p  is concave; p = 0 recovers log-concavity) with an exponent p determined by the geometry of the domain. I will also discuss concavity estimates on horoconvex domains in hyperbolic space (which are domains whose boundaries second fundamental form is greater or equal than 1), which yield new spectral-gap bounds in H^n. This is based on joint work with Zhiqin Lu and on work with Gabriel Khan.

January 30th

3pm-4pm, in-person: 

Smoothly knotted surfaces in small closed 4-manifolds 

Dave Auckly (Kansas State University)

It has long been known that homeomorphic 4-manifolds may admit inequivalent smooth structures. Analogous behavior holds for embedded surfaces. Some surfaces are topologically isotopic without being smoothly isotopic. Such surfaces are said to be smoothly knotted. It turns out that it is easier to construct inequivalent smooth structures on larger 4-manifolds. Similarly, it is easier to construct closed, smoothly knotted surfaces in large 4-manifolds. In this talk, we will explain how to construct smoothly knotted surfaces in a small 4-manifold. This talk will have many pictures. This is joint work with Konno, Mukherjee, Ruberman, and Taniguchi. 

February 6th

First Talk (2pm-3pm), in-person: 

Quantum invariants and surfaces in knot complements 

Josef Svoboda (Caltech)

Murasugi sum is a natural gluing operation on surfaces in $S^3$. Rather surprisingly, some knot invariants behave multiplicatively under Murasugi sum along minimal genus Seifert surfaces, prominently the extremal knot Floer homology. I will present a conjecture in this direction concerning the invariant $F_K$ of Gukov--Manolescu and Park, together with some partial results in the case of fibered knots. If true, the conjecture would provide a powerful tool for the study of the Grothendieck group of links under Murasugi sum, introduced recently by Cheng, Hedden and Sarkar. This is joint work with Paul Orland, Lara San Martín Suárez and Toby Saunders-A'Court.

Second Talk (3pm-4pm), in-person: 

Khovanov Homology for Tangles in Connected Sums 

Alan Du (Caltech)

Khovanov homology is an invariant for links in the three sphere that categorizes the Jones polynomial. We extend Khovanov's construction to links in 3-manifolds that are connected sums of orientable interval bundles over surfaces. Cutting the 3-manifold along a separating sphere, we construct type D and type A structures that are invariants of tangles in the two halves following the work of Roberts. Gluing the type D and type A structures along the common boundary recovers the Khovanov homology of the link. 

February 20th

First Talk (2pm-3pm), in-person:

Widths, Index, Intersection, and Isospectrality 

Jared Marx-Kuo (Rice University)

In this talk, I will discuss a series of works on Gromov's p-widths, $\{\omega_p\}$, on surfaces. For ambient dimensions larger than $2$, $\omega_p$ morally realizes the area of an embedded minimal surface of index p. This characterization was historically used to prove the existence of infinitely minimal hypersurfaces in closed Riemannian manifolds. In ambient dimension $2$, $\omega_p$ realizes the length of a union of (potentially immersed) geodesics, and heuristically, $p$ is equal to the sum of the indices of the geodesics plus the number of points of self-intersection. Joint with Lorenzo Sarnataro and Douglas Stryker, we prove upper bounds on the index and vertices, making progress towards this heuristic. Along the way, we prove a generic regularity statement for immersed geodesics. If time allows, we will also discuss the isospectral problem for the p-widths and how surfaces provide a convenient setting to investigate this. 

Second Talk (3pm-4pm),  in-person:

Min-max constructions in noncompact manifolds 

Doug Stryker (Stanford University)

I will present a localization method in the min-max theory of minimal hypersurfaces. This tool enables a proof of the existence of minimal hypersurfaces, as well as hypersurfaces with prescribed mean curvature, in some noncompact manifolds.

March 6th

First Talk (2pm-3pm), in-person: 

3-symmetric spaces, their moduli, and Ricci solitons 

Tommy Murphy (Cal State Fullerton)

Riemannian 3-symmetric spaces are a natural generalization of the classical symmetric spaces of Cartan. Their classification has been open since the pioneering work of Gray and Wolf in the sixties, which focused on the semi-simple case. I will outline the full classification, stressing some of the more remarkable features including their rich moduli space structure. As a by-product, a very general construction of Ricci solitons is presented which is of independent interest.  

Second Talk (3pm-4pm), in-person:

New quantum invariants and knot fiberedness  

Lara San Martin Suarez (Caltech)

What do quantum invariants know about geometry? In this talk, I will discuss a relation between the quantum invariant $F_K$, defined by Gukov–Manolescu and Park, and knot fiberedness. As an application, we uncover a connection between the $\mathfrak{sl}(2)$ quantum invariants and the knot geometry, and obtain machinery for constructing new $q$-series associated to knots. I will also discuss an intriguing parallel between extremal knot Floer homology and the extremal part of $F_K$. This is based on joint work with Paul Orland, Toby Saunders-A'Court and Josef Svoboda. 

2025 Fall (in-person or online, as indicated)

October 3rd: in-person

Periodic Inscription Problems 

Ali Naseri Sadr (Boston College)

Periodic inscriptions are a variation of the Toeplitz square peg problem suggested by Tao in 2017. A periodic curve is an embedding of the real line in the plane that is invariant under translation by a fixed vector. We say that a pair of disjoint periodic curves inscribe a quadrilateral Q if there exist four points in the union of their images such that the quadrilateral they form is similar to Q. In this talk, I will present a proof of the existence of such inscriptions when Q is an isosceles trapezoid, using tools from symplectic geometry and Lagrangian Floer homology. If time permits, I will also discuss other modifications of the square peg problem in non-Euclidean geometries and explain how symplectic geometry can be used to establish analogous results.

October 10th: online

Torus surgeries on knot traces

Kai Nakamura (Stanford University)

This talk has a simple thesis statement: torus surgeries are a powerful tool to study 4-manifolds, we apply this technology to knot traces. The key insight is that annulus twisting a knot's Dehn surgery can be realized 4-dimensionally as a torus surgery on the knot's trace. We will then sketch a few applications of this idea.

Zoom link: https://caltech.zoom.us/j/89404595159

October 17th: in-person

Unoriented link Floer homology and Khovanov homology

Gheehyun Nahm (Princeton University)

We study a variant of the unoriented link Floer homology defined by Ozsváth, Stipsicz, and Szabó, and construct a spectral sequence from various versions of Khovanov homology to the corresponding versions of unoriented link Floer homology over the field with two elements. In particular, we obtain a spectral sequence from reduced Khovanov homology to knot Floer homology over the field with two elements. This is a Heegaard Floer analogue of Kronheimer and Mrowka’s spectral sequence from Khovanov homology to singular instanton Floer homology.

October 24th: in-person

Splitting spheres and barbells

Alison Tatsuoka (Princeton University)

In 2019, Budney and Gabai introduced barbell diffeomorphisms of four-manifolds. We will review their construction and use it to show the existence of infinitely many splitting 3-spheres for two unknotted, unlinked 2-spheres in S^4.  Time permitting, we will also discuss some joint work in progress with Seungwon Kim and Gheehyun Nahm, where we use barbells to construct more codimension-1 linking and knotting in S^4. 

October 31st: in-person

Curvature, macroscopic dimensions, and symmetric products of surfaces 

Ekansh Jauhari (University of Florida)

The objective of this talk is to explain how symmetric products, macroscopic dimensions, and scalar curvature help in the study of one another. We begin by discussing two distinct notions of macroscopic dimensions arising in the study of manifolds with positive scalar curvature. Then we show that symmetric products of surfaces sharply distinguish between these notions of macroscopic dimensions. Finally, we address versions of the classical Gromov—Lawson and Gromov conjectures that predict obstructions to Kähler metrics of positive scalar curvature on projective varieties in terms of their macroscopic dimensions. This talk is based on joint work with Luca Di Cerbo and Alexander Dranishnikov. 

November 7th: in-person

Singularities of Hamiltonian Stationary Lagrangian Surfaces 

Filippo Gaia (Stanford University)

I will discuss the existence and regularity of critical points of the area functional among Lagrangian surfaces in symplectic 4-manifolds, reviewing classical results by R. Schoen and J. Wolfson, as well as recent progress by A. Pigati and T. Rivière. I will then present a variational construction of Hamiltonian stationary Lagrangian immersions in C^2 with multiple singularities at prescribed points, based on the sharp Wente inequality and an asymptotic analysis near the singularities. This is based on joint work with G. Orriols and T. Rivière. 

November 14th: in-person

Beck-Fiala and Komlós Bounds Beyond Banaszczyk 

Haotian Jiang (University of Chicago)

The Beck-Fiala Conjecture asserts that any set system of n elements with degree k has combinatorial discrepancy O(\sqrt{k}). A substantial generalization is the Koml\'os Conjecture, which states that any m by n matrix with columns of unit Euclidean length has discrepancy O(1). 

In this talk, we describe an \tilde{O}(log^{1/4} n) bound for the Koml\'os problem, improving upon the O(log^{1/2} n) bound due to Banaszczyk from 1998. We will also see how these ideas can be used to resolve the Beck-Fiala Conjecture for k \geq \log^2 n, and give a \tilde{O}(k^{1/2} + \log^{1/2} n) bound for smaller k, which improves upon Banaszczyk's O(k^{1/2} \log^{1/2} n) bound. These results are based on a new technique of "Decoupling via Affine Spectral Independence" in designing rounding algorithms, which might also be useful in other contexts.

November 21st: in-person

Non-collapsing of Ricci shrinkers with bounded curvature 

Conghan Dong (Duke University)

In this talk, I will discuss a non-collapsing result for Ricci shrinkers. I will show that for any simply connected Ricci shrinker with finite second homotopy group, uniformly bounded curvature implies uniformly non-collapsing. The proof will be based on an analysis of collapsing structures. This is joint with Yu Li. 

2025 Spring (in-person or online, as indicated)

April 4: in-person

The L^2 Alexander torsion for 3-manifolds and sutured manifold theory

Jianru Duan (Peking  University)

The L^2-Alexander torsion is an invariant associated to a 3-manifold and an 1-cohomology class. This invariant is a real function with many properties similar to the classical Alexander polynomial. In this talk, I will first review the basics of L^2-theory of 3-manifolds (e.g. L^2-betti numbers, L^2-torsions), then discuss the "leading coefficient" of the L^2-Alexander torsion and show its connection with Gabai's sutured manifold theory and the guts theory recently developed by Agol-Zhang.

April 11: in-person

Bordered algebras and the wrapped Fukaya category

Isabella Khan (Princeton)

This talk will discuss an isomorphism between endomorphism algebras from the wrapped Fukaya category of a type of punctured surface, and a class of A-infinity algebras related to bordered knot Floer homology; see https://arxiv.org/abs/2501.06310 and https://arxiv.org/abs/2408.01564. After a brief outline of the constructions and algebraic methods used in the proof, this talk will focus on the model calculations with Fukaya categories that both motivate and help to verify this result.  

April 18: in-person

Tight contact structures on Seifert fibrations

Hyunki Min (UCLA)

In this talk, we will briefly review the history of tight contact structures on small Seifert fibered spaces. We will introduce a new classification strategy for tight contact structures on Seifert fibered spaces. As an example, we will classify tight structures on every surgery of every torus knot in the 3-spehre. As a consequence, we will disprove the conjecture of Ghiggini and Van Horn-Morris about symplectic fillings on a family of Brieskorn spheres.

April 25: in-person

Positive scalar curvature on trivial and nontrivial circle bundles

Aditya Kumar (John Hopkins University)

We will discuss two recent results on positive scalar curvature on circle bundles. In the context of trivial circle bundles over 4-manifolds, we consider two problems: Gromov's width inequality conjecture, and Rosenberg's S^1-stability conjecture. Both conjectures have counterexamples in dimension 4 based on Seiberg-Witten invariants. Nevertheless, we show that in the simply connected case both of these results are true upon considering 4-manifolds up to homeomorphism. We also obtain a result up to stabilization in the non-simply connected case. In the context of nontrivial bundles, we construct infinitely many examples of PSC circle bundles over enlargeable manifolds in all dimensions greater than 3. This answers a question of Gromov. We shall emphasize some analogies between symplectic geometry and positive scalar curvature that we encountered in the process. This is joint work with B. Sen.

May 2: online

Nearly geodesic surfaces are filling

Xiaolong Hans Han  (Shanghai Institute for Mathematics and Interdisciplinary Sciences and Fudan University)

A surface S in a manifold M is filling if S cuts M into contractible components. We prove for any closed hyperbolic 3-manifold M , there exists a K''> 0 such that every homotopy class of K-quasi-Fuchsian surfaces with 1<K ≤ K'' is filling. As a corollary, the set of embedded surfaces in M satisfies a dichotomy: it consists of at most finitely many totally geodesic surfaces and surfaces with a quasi-Fuchsian constant lower bound K''. Each of these nearly geodesic surfaces separates any pair of distinct points at the sphere of infinity. Crucial tools include the rigidity results of Mozes-Shah, Ratner, and Shah. This work is inspired by a question of Yunhui Wu and Yuhao Xue whether random geodesics on random hyperbolic surfaces are filling.

Zoom link: https://caltech.zoom.us/j/89994856119 

May 9: in-person

The Dehn twist on a connected sum of two homology tori

Haochen Qiu (Brandeis)

Kronheimer and Mrowka showed that the Dehn twist along a $3$-sphere in the neck of the connected sum of two $K3$ surfaces is not smoothly isotopic to the identity. The tool they used is the nonequivariant family Bauer-Furuta invariant, and their result requires that the manifolds are simply connected and the signature of one of them is $16 \mod 32$. We generalize the $S^1$-equivariant family Bauer-Furuta invariant to nonsimply connected manifolds, and construct a refinement of this invariant. We use it to show that if $X_1,X_2$ are two homology tori whose determinants $r_1,r_2$ are odd, then the Dehn twist along a $3$-sphere in the neck of $X_1\mathbin{\#} X_2$ is not smoothly isotopic to the identity.

May 16: in-person

On the Horowitz-Myers conjecture

Pei-Ken Hung (UIUC)

In 1998 Horowitz and Myers constructed a static manifold with a toroidal infinity, later referred as the Horowitz-Myers geon. They further conjectured that the Horowitz-Myers geon minimizes the mass among manifolds with same asymptotics and the scalar curvature lower bounds. In this talk, I will discuss joint works with Simon Brendle, in which we confirm this conjecture and the rigidity. 

May 23: in-person

From phase transitions to minimal submanifolds in low codimensions

Davide Parise (UC San Diego)

Proving existence of minimal submanifolds (i.e. critical points of the area functional) in various settings has been one of the driving forces of the development of modern calculus of variations, and geometric measure theory, with remarkable applications in differential geometry. In the 1960s Almgren developed a far-reaching technique to prove existence of (measure theoretic versions of) these objects. While effective, Almgren’s technique was notoriously complex which prompted the emergence in recent years of an alternative PDE-based method. The underlying idea is to construct minimal submanifolds as limits of nodal sets of critical points of functionals arising from the gradient theory of phase transitions and the theory of superconductors. After starting with a general overview, I will explain how the work of my collaborators and I fits into the broader picture. In particular, we will start with the Allen-Cahn functional and the codimension one theory discussing the construction of free boundary minimal surfaces, i.e. minimal submanifolds meeting the boundary orthogonally. We will then move to higher codimensions (specifically codimensions 2 and 3), where the theory is much less developed. We will introduce the Yang-Mills-Higgs functionals, in both the abelian and non-abelian settings. In particular, we will focus on what happens to their gradient flows, and their variational properties. I will highlight other successes of this theory, and point to some open problems along the way. The content of this talk is based on joint works with Martin Li, Lorenzo Sarnataro, Alessandro Pigati, and Daniel Stern.

May 30: in-person

On the topology of the manifolds with positive intermediate curvature

Xuan Yao (Cornell)

We formulate a conjecture relating the topology of a manifold's universal cover with the existence of metrics with positive m-intermediate curvature. We prove the result for manifolds of dimension n∈{3,4,5} and for most choices of m when n=6. As a corollary, we show that a closed, aspherical 6-manifold cannot admit a metric with positive 4-intermediate curvature.

June 6: in-person

A theory of anisotropic varifolds

Joshua Flynn (MIT)

In this talk I will introduce a varifold theory suitable for geometric settings which are locally "anisotropic." In such geometries, basic concepts like regularity, Lipschitzness and rectifiability have to be re-imagined. This varifold theory is built in such a way to respect these re-imaginations.

2025 Winter (in-person or online, as indicated)

January 17: Changed to on-line

Algebraic structure of knot Floer homology

David Popović (UCLA)

Knot Floer homology is a powerful link invariant. In its most general version, it is a bigraded chain complex over a polynomial ring F[U,V]. In this talk, I will describe the structure theorem of such objects - they are a direct sum of snake complexes and local systems - and explain what information each summand carries about the knot.

Zoom link: https://ucla.zoom.us/j/7036325556 

January 24: online

Weyl Law for the 1-cycles 

Bruno Staffa (University of Toronto)

Let $(M^{n},g)$ be an $n$-dimensional Riemannian manifold and $1\leq k\leq n-1$. In the 1980's, Gromov studied the asymptotic behavior of the volumes $\omega^{k}_{p}(M^{n},g)$ of certain (possibly singular) $k$-dimensional minimal submanifolds $N^{k}_{p}$ of $(M^{n},g)$ arising from a Morse Theory of the volume functional (the Almgren-Pitts Min-Max Theory). He conjectured that these volumes $\omega_{p}^{k}(M^{n},g)$ behave when $p\to\infty$ similarly to the eigenvalues $\lambda_{p}$ of the Laplacian in $(M^{n},g)$, obeying the so-called Weyl Law for the Volume Spectrum. In 2016, Liokumovich, Marques and Neves solved the conjecture for $k=n-1$ and $n$ arbitrary. More recently, Guth and Liokumovich proved it for $1$-cycles in $3$-manifolds ($k=1$ and $n=3$). I will discuss the main difficulties which arise when the codimension $n-k$ of the cycles is greater than $1$, which boils down to proving parametric versions of the isoperimetric and coarea inequalities; and the applications of the Weyl law for $1$-cycles on generic density and equidistrbution of stationary geodesic networks.

Zoom link: https://caltech.zoom.us/j/89155661233 

January 31: in-person

Manifolds with Ricci lower bounds in the spectral sense

Kai Xu (Duke)

A manifold is said to have Ricci curvature lower bound in the spectral sense, if the first eigenvalue of the elliptic operator -γΔ+Ric is bounded below. This condition is usually strictly weaker than pointwise Ricci lower bound. This talk concerns the geometry of manifolds satisfying this condition. In particular, we will discuss the analogue of the classical theorems (Bonnet-Myers, volume comparison, Cheeger-Gromoll splitting) under our main condition. This talk is based on joint works with Gioacchino Antonelli and Marco Pozzetta.

February  3:  (The Joint LA Topology Seminar at Caltech)

4pm-5pm

Exotic aspherical 4-manifolds

Danny Ruberman (Brandeis)

An aspherical space is one with vanishing higher homotopy groups; under mild assumptions this means that it is determined up to homotopy type by its fundamental group. A famous conjecture of Borel states that closed aspherical manifolds are in fact determined up to homeomorphism by the fundamental group. One could also ask (Borel famously didn’t) whether smooth aspherical manifolds are determined up to diffeomorphism by their fundamental group; this is known to hold in dimensions at most 3 and is false in dimension at least 5. We resolve the remaining case by exhibiting closed smooth aspherical manifolds that are homeomorphic but not diffeomorphic. This is joint with Davis, Hayden, Huang, and Sunukjian.

5pm-6pm

Irreducible 4-manifolds can admit exotic diffeomorphisms

Hokuto Konno (U Tokyo)

A diffeomorphism of a smooth manifold is called an exotic diffeomorphism if it is topologically isotopic to the identity but smoothly not. Since Ruberman's pioneering work in 1998, extensive research has been conducted to the detection of exotic diffeomorphisms of 4-manifolds, with particularly active developments in recent years. Nevertheless, the basic question of whether an irreducible 4-manifold can admit an exotic diffeomorphism has remained a major unresolved problem in the field. We resolve this open problem affirmatively by providing numerous examples of minimal complex surfaces that admit exotic diffeomorphisms. This is joint work with David Baraglia.

February  7: in-person

Annular links and categorified invariants

Fraser Binns (Princeton)

This talk will concern annular links i.e. links in the thickened annulus. It will come in two parts. In the first I will explore different notions of equivalence between annular links. In the second I will discuss categorified annular link invariants and their topological content. This talk is partially based on joint work with Román Aranda, Margaret Doig and, time permitting, Diana Hubbard.

February 14: in-person

Embedded Minimal Tori in the Three Sphere

Xingzhe Li (Cornell)

In this talk, we introduce the strong Morse inequalities for the area functional in the space of embedded tori and spheres in S^3. Applying this, we show the existence of at least nine embedded minimal tori in bumpy positively Ricci curved S^3. This talk is based on joint work with Zhichao Wang.

February 21: in-person

Hongjian Yang (Stanford)

Knot Detection Results in RP^3

Following Kronheimer and Mrowka's approach, we prove thta Khovanov homology detects the unknot and the standard RP^1 in RP^3. I'll explain the idea of the proof. Time permitting, I'll discuss some further detection results.

February 28: in-person

Jan Nienhaus (UCLA)

Einstein metrics on even-dimensional spheres

The first non-round Einstein metrics on spheres were described in 1973 by Jensen in dimensions 4n+3 (n > 0). For the next 25 years it remained an open problem whether the same could be done in even dimensions. This question was settled in 1998 when C. Böhm constructed infinite families of Einstein metrics on all Spheres of dimension between 5 and 9, in particular on S^6 and S^8. In the 25 years since then, all spheres of odd dimension (at least 5) have been shown to admit non-round Einstein metrics. However, there have been no new developments in even dimensions above 8, leaving open to speculation the question of whether, if the dimension is even, non-uniqueness of the round metric is a low-dimensional phenomenon or to be expected everywhere. I will give an overview of the methods used to construct such Einstein metrics, which we recently used to construct the first examples of non-round Einstein metrics on S^{10}. This is joint work with Matthias Wink.

March 7: in-person

Examples of invariants of 4-dimensional 2-handlebodies up to 2-equivalence

Maksymilian Manko (Universität Zürich)

It is not known whether two diffeomorphic 4-dimensional 2-handlebodies are always equivalent through handle moves of index smaller than 2 (i.e. 2-equivalent). A result of Beliakova, Bobtcheva, de Renzi and Piergallini implies that any so-called BP Hopf algebra in a braided monoidal category yields an invariant of 4-dimensional 2-handlebodies up to 2-equivalence. Unfortunately, it is hard to exhibit such algebras, since we want the category to be non-semisimple and non-factorizable. After explaining all of this in detail, I will discuss a series of such examples, both well-known and newly constructed. We proved some of them degenerate to the invariants boundary of the handle body, and some of them to the spine of the 2-handlebodies. (Joint work in progress with A. Beliakova and Q. Faes)

March 8: in-person

N+12th Southern California Topology Colloquium at Pitzer College of the Claremont Colleges

Dror Bar-Natan, (University of  Toronto) 

Patrick Orson, (California Polytechnic State University, San Luis Obispo) 

Rhea Palak Bakshi, (University of California, Santa Barbara)

Sucharit Sarkar, (University of California, Los Angeles)

See https://sites.google.com/view/socaltopology/home for details.

March 14: in-person

Gary Guth (Stanford)

Real Heegaard Floer homology

There has been a burst of interest in gauge theoretic invariants of 3- and 4-manifolds equipped with an involution, developed in various contexts by Tian-Wang, Nakamura, Konno-Miyazawa-Taniguchi, and Li. Notably, Miyazawa proved the existence of an infinite family of exotic RP^2-knots using real Seiberg-Witten theory. In joint work with Ciprian Manolescu, we construct an invariant of based 3-manifolds with an involution, called real Heegaard Floer homology. This is the  analogue of Li’s real monopole Floer homology.  Our construction is a particular case of a real version of Lagrangian Floer homology, which may be of independent interest to symplectic geometers. In the case of Heegaard Floer homology, the construction starts from a Heegaard diagram where the involution swaps the alpha and beta curves. We prove that real Heegaard Floer homology is indeed a topological invariant of the underlying pointed real 3-manifold. Further, we study the Euler characteristic of our theory, which is the Heegaard Floer analogue of Miyazawa’s invariant for twist-spun 2-knots. This quantity is algorithmically computable and, indeed, appears to agree with Miyazawa’s invariant.

2024 Fall (in-person or online, as indicated)

October 4: in-person

PDE analysis on stable minimal hypersurfaces: curvature estimates and sheeting

Costante Bellettini (University College London)

We consider properly immersed two-sided stable minimal hypersurfaces of dimension n. We illustrate the validity of curvature estimates for n \leq 6 (and associated Bernstein-type properties). For n \geq 7 we illustrate sheeting results around "flat points". The proof relies on PDE analysis. The results extend respectively the Schoen-Simon-Yau estimates (obtained for n \leq 5) and the Schoen-Simon sheeting theorem (valid for embeddings). 

October 11: in-person

Finiteness of Totally Geodesic Hypersurfaces in Negative Curvature 

Ben Lowe (University of Chicago)

There is a broad body of work devoted to proving theorems of the following form: spaces with infinitely many special sub-spaces are either nonexistent or rare.  Such finiteness statements are important in algebraic geometry, number theory, and the theory of moduli space and locally symmetric spaces. I will talk about joint work with Simion Filip and David Fisher proving a finiteness statement of this kind in a differential geometry setting.  Our main theorem is  that a closed negatively curved analytic Riemannian manifold with infinitely many totally geodesic hypersurfaces must be isometric to an arithmetic hyperbolic manifold. 

October 18: in-person

Towards isomorphisms among Floer homologies

Fan Ye (Harvard University)

Since Floer's work in 1988, various Floer homologies have been constructed for closed 3-manifolds, knots, and sutured manifolds.  In 2008, Kronheimer-Mrowka proposed a conjecture about isomorphisms among Floer homologies. In this talk, I will introduce an approach to proving the isomorphisms based on combinatorial version of Floer homology. This work is joint with Baldwin, Li, and Sivek.

October 21 (The Joint LA Topology Seminar at USC)

4pm-5pm

Integral counts of pseudo-holomorphic curves and applications

Shaoyun Bai (MIT)

Moduli spaces of pseudo-holomorphic curves in symplectic topology do not usually give rise to integral fundamental classes because of the existence of automorphisms. Nevertheless, in joint work with Guanbo Xu, following proposals of Fukaya-Ono and Parker, we constructed integer-valued counts from these moduli spaces in certain generality, including the setting of Gromov-Witten invariants and Hamiltonian Floer theory. In this talk, I will give an overview of our construction, and present concrete applications of these invariants including quantitative existence results of Hamiltonian periodic orbits and topology of Hamiltonian fibrations.

5pm-6pm

The exponential type conjecture for quantum connections

Zihong Chen (MIT)

The (small) quantum connection is one of the simplest objects built out of Gromov-Witten theory, yet it gives rise to a repertoire of rich and important questions such as the Gamma conjectures and the Dubrovin conjectures. There is a very basic question one can ask about this connection: what is its formal singularity type? People's (e.g. Kontsevich-Katzarkov-Pantev, Iritani) expectation for this is packaged into the so-called exponential type conjecture, and the goal of this talk is to discuss a proof in the case of closed monotone symplectic manifolds. My approach uses a reduction mod p argument, and I will start by introducing some basic ordinary differential equations (in particular in characteristic p) and Katz's local monodromy theorem. Then I will demonstrate the key idea of proof pretending we were working in a B-side mirror situation---matrix factorizations, where it is particularly simple. Finally, I will explain how to adapt the proof to the case of quantum connections using certain equivariant operations on quantum cohomology.

October 25: in-person

An iterable surgery formula for knot lattice homotopy

Seppo Niemi-Colvin (Indiana University Bloomington)

I compute the knot Floer complex for the regular fiber of \Sigma(2,3,7), and I show that its Seifert genus and genus in a self homology cobordism agree. The key step in this result was providing upgrades to the surgery formula for knot lattice homotopy. Ozsv\'ath, Stipsicz, and Szab\'o showed that knot lattice homology satisfies a surgery formula similar to the one relating knot Floer homology and Heegaard Floer homology. I provide an iterable version of this formula that given the doubly filtered knot lattice space with involutive maps for a generalized algebraic knot produces the corresponding data for the dual knot post-surgery.

November 1: in-person

The (fractional) Dehn twist coefficient and infinite-type surfaces

Hannah Turner (Stockton University)

The fractional Dehn twist coefficient (FDTC) is an invariant of a self-map of a surface which is some measure of how the map twists near a boundary component of the surface. It has mostly been studied for compact surfaces; in this setting the invariant is always a fraction. I will discuss work to extend this invariant to infinite-type surfaces and show that it has surprising properties in this setting. In particular, the invariant no longer needs to be a fraction - any real number amount of twisting can be achieved! I will also discuss a new set of examples of (tame) big mapping classes called wagon wheel maps which exhibit irrational twisting behavior. This is joint work in progress with Diana Hubbard and Peter Feller.

November 4 (The Joint LA Topology Seminar at UCLA, MS 6627)

4pm-5pm

Classifying vector bundles on complex projective spaces

Morgan Opie (UCLA)

A classical problem in topology is the following: given two vector bundles over a topological space, how can we determine whether or not they are isomorphic? There are certain standard approaches to such a question, but a full answer generally depends strongly on the base space. In this talk, I will survey results on this question for complex topological vector bundles on complex projective spaces. This includes my own work using "chromatic" generalized cohomology theories to approach the problem. This talk will be a gentle introduction to the more abstract tools involved.

5pm-6pm

Bohr-Sommerfeld surgeries on Lagrangian submanifolds

Soham Chanda (USC)

We introduce a class of surgery operations on Lagrangian submanifolds which switch between two different fillings of Bohr-Sommerfeld Legendrians. We will discuss the effect of such a surgery operation on the disk potential in the setting of monotone Lagrangians. As an application we will prove that such surgery operations yield exotic monotone Lagrangian tori in projective spaces of dimension 3 and higher.

November 8: in-person

The Space of Minimal Surfaces in S^3 and in B^3

Peter McGrath (North Carolina State University)

I will discuss recent progress, with Kapouleas and with Karpukhin-Kusner-Stern, on understanding the space of minimal surfaces in the round 3-sphere and in the euclidean 3-ball.

November 15: in-person

Minimal (hyper)surface doublings and their geometry

Nicos Kapouleas (Brown University)

I will first discuss briefly the general setting of minimal (hyper)surface doublings and the early stages of their development. Then I will discuss the introduction of the Linearized Doubling (LD) methodology and its applications so far and potential for the future. Finally I will concentrate on recent and ongoing work, including constructions in high dimensions, relation with classical Yau questions, questions on index, etc.

November 18 (The Joint LA Topology Seminar at Caltech)

Exploring non-left-orderability in 3-manifold groups

4pm-5pm

Zipei Nie (UIUC)

In this talk, I will introduce some computational techniques for establishing non-left-orderability in various groups, including the fundamental groups of the cyclic branched covers of the pretzel knot P(3, -3, -2k-1) and certain surgeries on chainmail links. I will also briefly discuss how a computational proof of non-left-orderability can provide insights into the construction of taut foliations.

5pm-6pm

L-space satellite operators and knot Floer homology

Daren Chen (Caltech)

A satellite operator on knots is called an L-space satellite operator if the 2-component link, consisting of the pattern and the meridian of the solid torus, is an L-space link. Examples of such operators include cabling, Whitehead double and Mazur patterns. We present an algorithm to compute the knot Floer homology of the resulting knot in terms of the knot Floer homology of the companion knot, as well as the Alexander polynomials of the 2-component link and the pattern. This algorithm is based on the link surgery formula of Manolescu and Ozsváth, together with a reinterpretation and a connected sum formula of it by Zemke. A key ingredient in this algorithm is a formality result for 2-component L-space links, which says we can determine the knot Floer chain complex of such a link from the Alexander polynomial of it and its sublinks. This is joint work with Ian Zemke and Hugo Zhou.

November 22: in-person

Framed real monopole Floer homology

Jiakai Li (Harvard University)

Seiberg-Witten theory has an analogue for 3- and 4-manifolds with involutions called real Seiberg-Witten theory. This theory can be used to construct invariants of links and embedded surfaces by passing to double branched covers. This talk will focus on a framed version of real Seiberg-Witten-Floer homology. It turns out this invariant of links has rather surprising properties not seen in ordinary Seiberg-Witten theory. I will explain why it is special and how it is related to some recent developments.

December 6: in-person

The Isoperimetric Problem in the Cube and Torus

Federico Glaudo (IAS)

A renowned conjecture—described by A. Ros as one of the nicest open problems in classical geometry—proposes that the sets minimizing the perimeter among subsets of the 3-dimensional torus R^3/Z^3 with a fixed volume v are spheres, cylinders, or stripes (depending on the value of v). An equivalent conjecture applies to isoperimetric sets in the 3-dimensional cube (0,1)^3. 

In this seminar, we will review the known results, primarily due to Ritorè, and discuss recent partial progress from an ongoing project with G. Antonelli.

2024 Spring (in-person or online, as indicated)

April 5: in-person

Polya's conjecture in spectral geometry

David Sher

Polya's conjecture in spectral geometry is a hypothesized uniform bound for eigenvalues of the Dirichlet Laplacian on a domain in Euclidean space. Although proven sixty years ago for domains which tile the plane, the conjecture remains completely open for non-tiling domains. Recently, we have proven Polya's conjecture for Euclidean balls. Our results make use of some novel bounds for zeroes of Bessel functions and their derivatives, as well as lattice point counting arguments. This is joint work with N. Filonov (St. Petersburg), M. Levitin (Reading), and I. Polterovich (Montreal).

April 12: on Zoom

Surgery Exact Triangles in Instanton Theory

Deeparaj Bhat

We prove an exact triangle relating the knot instanton homology to the instanton homology of surgeries along the knot. As the knot instanton homology is computable in many instances, this sheds some light on the instanton homology of closed 3-manifolds. We illustrate this with computations in the case of some surgeries on the trefoil. In particular, we show the Poincaré homology sphere is not an instanton L-space (with Z/2 coefficients), in contrast with Heegaard Floer and monopole Floer theories. Finally, we sketch the proof of the triangle inspired by the Atiyah-Floer conjecture and results from symplectic geometry.

April 19: on Zoom

Triple Grid Diagrams

Sarah Blackwell

The complex projective plane satisfies two special properties that set it apart from its peers: it admits (1) a symplectic structure and (2) a genus one trisection. Together these properties inspire the notion of a ``triple grid diagram,’’ the primary object of study in this talk.

Trisections, due to Gay and Kirby, are decompositions of (smooth, closed, connected, oriented) 4-manifolds into three nice pieces. Meier and Zupan showed that surfaces smoothly embedded in trisected 4-manifolds can inherit their own trisection, and can be represented with ``shadow diagrams.’’ Triple grid diagrams are specific shadow diagrams of surfaces in the complex projective plane that naturally arise as grid diagrams on the central surface of its standard (genus one) trisection. Diagrams satisfying one extra condition represent Lagrangian surfaces, thus capturing this geometric information combinatorially. This is joint work with David Gay and Peter Lambert-Cole.

April 26: in-person

Stable Minimal Hypersurfaces in \R^5

Paul Minter

I will discuss recent work (joint with Otis Chodosh, Chao Li, and Douglas Stryker) showing that every complete two-sided stable minimal hypersurface in \R^5 is flat. The ideas of the proof are motivated by those in the study of manifolds with positive curvature conditions in a suitable ‘weak’ sense, in our case bi-Ricci curvature.

May 3: in-person

Infinitesimal calculations in fundamental groups

Nir Gadish

How can you tell if some power of a word is an n-fold commutator?  How do cochains “see” the fundamental group? We give explicit answers to these questions, in joint work with A. Ozbek, D. Sinha and B. Walter. Thinking of a topological space as a presentation of its fundamental groups, I will introduce a collection of invariants for words in groups, starting from the 1-st cohomology of a space, using linking numbers of letters in words. These turn out to define a universal “finite type" invariant for arbitrary groups, a result which has not been achieved for other explicit functionals on words, namely those arising from Magnus expansion and Fox calculus. Moreover, the theory works over any PID, in particular, it relates mod-p Massey products and the p-central series of groups, and presents the rational Harrison complex as a dual to the Malcev Lie algebra.

May 10: in-person

Knots in circle bundles

Andrew Yarmola

In this talk we will be interested in knots in orientable circle bundles N over a genus g > 1 surfaces. Examples of such knots arise naturally when N is the unit tangent bundle of a surface and the knot is the set of tangents to a curve on the surface. We call such knots canonical or Legendrian. In joint work with Tommaso Cremaschi, we show that knots in N are determined by their complements, a special case of the oriented knot complement conjecture. In the setting of unit tangent bundles, this demonstrates that canonical knots have homeomorphic complements if and only if their shadows differ by Reidemeister moves, (de)stabilizations, loops/cusps added by transvections, and mapping classes of the surface.

May 17: in-person

Khovanov skein lasagna modules and exotica

Michael Willis

The Khovanov skein lasagna module S(X;L) is a smooth invariant of a 4-manifold X with link L in its boundary.  In this talk I will outline the construction of Khovanov skein lasagna modules, as well as new computations and applications including the detection of some exotic 4-manifolds.  This work is joint with Qiuyu Ren.

May 20 (The Joint LA Topology Seminar at Caltech)

4:00 - 5:00 pm

Sherry Gong

5:00 - 6:00 pm

Mike Hill

May 24: in-person

Area estimates and intersection properties for minimal hypersurfaces in space forms

Jonathan Zhu

We’ll discuss certain properties of minimal submanifolds in geodesic balls of space forms, including area estimates for minimal submanifolds either through a `prescribed point’, or which meet the boundary sphere orthogonally. Such area estimates are related to estimates of the width of the ambient space, and in turn related to stability of minimal submanifolds. In this latter direction, we will give some proofs of an improved Frankel property, whereby, in the appropriate settings, minimal submanifolds must intersect in every half-space. This is joint work with Keaton Naff.

May 31: on Zoom

Orbifold singularity formation along ancient and immortal Ricci flows

Tristan Ozuch

Dimension 4 is the next horizon for applications of Ricci flow to topology, where the main goal is to understand the topological operations that Ricci flow performs both at singular times, and in its long-term behavior. With Alix Deruelle, we explain how Ricci flow develops or resolves orbifold singularities by a notion of stability depending only on the curvature at the singular points. We construct a plethora of ancient and immortal Ricci flows spontaneously forming or desingularizing arbitrarily complicated orbifold singularities by bubbling-off Ricci-flat ALE metrics.

This unexpectedly predicts that singular spherical and cylindrical orbifolds should not appear as finite-time singularity models. On the other hand, (complex) hyperbolic orbifolds appear as limits of immortal “thick” 4-dimensional Ricci flows as $t\to+\infty$.

2024 Winter (in-person or online, as indicated)

January 5: No seminar

January 12: in-person

On a generalization of Geroch's conjecture

Sven Hirsch

The theorem of Bonnet-Myers implies that manifolds with topology $M^{n-1}\times S^1$ do not admit a metric of positive Ricci curvature, while the resolution of Geroch's conjecture shows that the torus $T^n$ does not admit a metric of positive scalar curvature. In this talk, I will introduce a new notion of curvature which interpolates between Ricci and scalar curvature (so-called $m$-intermediate curvature) and use stable weighted slicings to show that for $n\le7$ the manifolds $M^{n-m}\times T^m$ do not admit a metric of positive $m$-intermediate curvature. This is joint work with Simon Brendle and Florian Johne.

January 19: in-person

Kirby and the Skein Lasagna Module of $S^2 \times S^2$

Melissa Zhang

In 2018, Morrison, Walker, and Wedrich’s skein lasagna modules are 4-manifold invariants defined using Khovanov-Rozansky homology similarly to how skein modules for 3-manifolds are defined. In 2020, Manolescu and Neithalath developed a formula for computing this invariant for 2-handlebodies by defining an isomorphic object called cabled Khovanov-Rozansky homology; this is computed as a colimit of cables of the attaching link in the Kirby diagram of the 4-manifold.

One especially important conjecture states that the skein lasagna module of $S^2\times S^2$ is 0 (or infinite-dimensional). This is a necessary condition for the invariant to be able to detect exotic 4-manifold pairs. In this project, we lift the Manolescu-Neithalath construction to the level of Bar-Natan's tangles and cobordisms, and trade colimits of vector spaces for a homotopy colimit in Bar-Natan's category. This allows us to compute significant portions of the skein lasagna module of $S^2 \times S^2$, and relate the remainder to the Rozansky-Willis invariant of links in $S^2 \times S^1$. Our local techniques also allow for computations of the skein lasagna invariant for other 4-manifolds whose Kirby diagram contains a 0-framed unknot component.

This is joint upcoming work with Ian Sullivan (UC Davis).

January 26: in-person

Obstructing Knots from Being k-Slice

Qianhe Qin

A knot K in S^3 is k-slice in a 4-manifold X if it bounds a disk with self-intersection number -k in X removing the interior of a 4-ball. In this talk, we discuss how to obstruct a knot from being k-slice in nCP^2, using knot invariants from knot Floer homology and Khovanov homology. Moreover, we explore the use of k-RBG links to obstruct k-sliceness when these invariants for K are ineffective.

February 2: in-person

Heegaard Floer symplectic cohomology and generalized Viterbo's isomorphism

Roman Krutowski

I will define a novel invariant, called Heegaard Floer symplectic cohomology (HFSH), which serves as a closed string analog of the higher-dimensional Heegaard Floer homology. This invariant can also be regarded as a deformation of a k-th symmetric version of symplectic cohomology given by counting curves of higher genus. Alternatively, one may view it as a Floer invariant associated with a problem of Hamiltonian motion of multiple identical particles.  I will also introduce a multiloop Morse complex of a manifold, which is supposed to be a Morse-theoretic counterpart of HFSH. At last, I will show that HFSH of a cotangent bundle T^*M is isomorphic to the cohomology of the multiloop complex of M. This result generalizes Viterbo's isomorphism theorem to the case of multiple particles.

February 9: in-person

3- and 4-dimensional invariants of satellite knots with (1,1)-patterns

Holt Bodish

In this talk I will discuss computing knot Floer homology of satellites with arbitrary companions and patterns from a few families of (1,1)-patterns. I'll show how to compute $\tau$ and $\epsilon$ of satellites with these patterns in terms of $\tau$ and $\epsilon$ of the companion and show that there is an infinite subfamily of winding number 1 patterns (generalizing the Mazur pattern) that do not act surjectively on the smooth concordance group. I will also discuss determining the genus and fiberedness of these patterns (and their twisted relatives) in the solid torus. Some of this is based on joint work with Subhankar Dey.

February 16: on Zoom

Freedman's Link Packing Question

Elia Portnoy

Freedman recently posed a new question in quantitative topology about link packings. Given a link L, define the $\epsilon$-diagonal packing number $n_{L(\epsilon)}$ to be the number of copies of L that can be simultaneously embedded in $[0,1]^3$ so that (1) Each copy of $L$ is contained in a ball which is disjoint from the other copies. (2) Within each copy, the components are separated by a distance of at least $\epsilon$. We'll discuss a new construction for obtaining a lower bound on $n_{L(\epsilon)}$ and expand on Freedman's ideas to obtain an upper bound on $n_{L(\epsilon)}$ when $L$ has a non-trivial Milnor Invariant. At the end we'll mention several related open problems about link packings. This is joint work with Fedya Manin.

February 23: in-person

On the topology and index of minimal/Bryant framed surfaces

Franco Vargas Pallete

In this talk we'll discuss a 1-to-1 correspondence between Euclidean minimal and Bryant surfaces, known in the literature as Lawson's correspondence. Along this correspondence we will study framed surfaces, which is a class of Euclidean minimal and hyperbolic CMC-1 surfaces that generalize immersed Euclidean minimal surfaces and Bryant surfaces. For this class we prove a lower bound on the (unrestricted) Morse index by a linear function of the genus, number of ends and number of branch points (counting multiplicity), generalizing previous work in the literature. This is based on joint work with Davi Maximo.

March 1

From Gromov-Witten theory to dynamics

Julian Chaidez

A general philosophy due to Joel Fish and Helmut Hofer states that symplectic manifolds with sufficiently rich Gromov-Witten theory should contain hypersurfaces with rich dynamics. In this talk, we will discuss many recent results that represent specific instances of this philosophy. In low dimensions, where Gromov-Witten (and related Gromov-Taubes) invariants are more well understood, strong and general results like the smooth closing lemma have been proven. The higher dimensional setting is much more mysterious, and is the subject of ongoing research. This talk is based on joint work with Shira Tanny (and the work of others).

March 8: on Zoom

On complete Calabi-Yau manifolds asymptotic to cones

Junsheng Zhang

We proved a ‘’no semistability at infinity" result for complete Calabi-Yau metrics asymptotic to cones, by eliminating the possible appearance of an intermediate K-semistable cone in  Donaldson-Sun's 2-step degeneration theory. As a consequence, we establish a polynomial convergence rate result and a classification result for complete Calabi-Yau manifolds with Euclidean volume growth and quadratic curvature decay. Joint work with Song Sun.

2023 Fall (in-person or online, as indicated)

September 29: in-person

Lee filtration structure of torus links, adjunction inequality, and applications

Qiuyu Ren

We state our recent result determining the quantum filtration structure of the Lee homology of torus links. This implies a relative adjunction-type inequality for s-invariants of two links related by a link cobordism in kCP^2, originally conjectured by Manolescu-Marengon-Sarkar-Willis. As two applications of this inequality, we show the existence of knots with arbitrarily large CP^2 genus, and we sketch a potential approach to produce and detect exotic kCP^2's.

October 6: in-person

Geometric Boundary of Groups

Yulan Qing

Gromov boundary provides a useful compactification for all infinite-diameter Gromov hyperbolic spaces. It consists of all geodesic rays starting at a given base-point and it is an essential tool in the study of the coarse geometry of hyperbolic groups. In this talk we generalize the Gromov boundary. We first construct the sublinearly Morse boundaries and show that it is a QI-invariant, metrizable topological space. We show sublinearly Morse directions are generic both in the sense of Patterson-Sullivan and in the sense of random walk.  

The sublinearly Morse boundary is a subset of all directions with desired properties. In the second half we will truely consider the space of all directions and show that with some minimal assumptions on the space, the resulting boundary is a QI-invariant topology space in which many existing boundaries embeds. This talk is based on a series of work with Kasra Rafi and Giulio Tiozzo.

October 13: in-person

Labeled four cycles and the K(pi,1) problem for reflection arrangement complements

Jingyin Huang

The K(pi,1)-conjecture for reflection arrangement complements, due to Arnold, Brieskorn, Pham, and Thom, predicts that certain complexified hyperplane complements associated to infinite reflection groups are Eilenberg MacLane spaces. We establish a close connection between a very simple property in metric graph theory about 4-cycles and the K(pi,1)-conjecture, via elements of non-positively curvature geometry. We also propose a new approach for studying the K(pi,1)-conjecture. As a consequence, we deduce a large number of new cases of Artin groups which satisfies the K(pi,1)-conjecture.

October 20: in-person

The extremal structures of the Alexandrov-Fenchel inequality: convex polytopes and beyond

Yair Shenfeld

Solutions to classical geometric variational problems are often unique and nice: the ball is the only shape which minimizes surface area among shapes of equal volume. But the solutions to the variational problems of the Alexandrov-Fenchel inequalities are conjectured (by R. Schneider) to be non-unique and wild. I will talk about these variational problems, including their connections to areas beyond convexity such as analysis and combinatorics, and how we resolved some of these conjectures. Joint work with Ramon van Handel.

October 27: in-person

Genus one singularities in mean curvature flow

Adrian Chun-Pong Chu

We show that for certain one-parameter families of initial conditions in ℝ³, when we run mean curvature flow, a genus one singularity must appear in one of the flows. Moreover, such a singularity is robust under perturbation of the family of initial conditions. This contrasts sharply with the case of just a single flow. As an application, we construct an embedded, genus one self-shrinker with entropy lower than a shrinking doughnut.

October 30 (The Joint LA Topology Seminar at Caltech)

4:00 - 5:00 pm, Linde 310

Counting incompressible surfaces in hyperbolic 3-manifolds

Marc Lackenby

Incompressible embedded surfaces play a central role in 3-manifold theory. It is a natural and interesting question to ask how many such surfaces are contained in a given 3-manifold M, as a function of their genus g. I will present a new theorem that provides a surprisingly small upper bound. For any given g, there is a polynomial p_g with the following property. The number of closed incompressible surfaces of genus g in a hyperbolic 3-manifold M is at most p_g(vol(M)). This is joint work with Anastasiia Tsvietkova.

5:00 - 6:00 pm, Linde 310

PL-genus of surfaces in homology balls

Jennifer Hom

We consider manifold-knot pairs (Y, K) where Y is a homology 3-sphere that bounds a homology 4-ball. Adam Levine proved that there exists pairs (Y, K) such that K does not bound a PL-disk in any bounding homology ball. We show that the minimum genus of a PL surface S in any bounding homology ball can be arbitrarily large. The proof relies on Heegaard Floer homology. This is joint work with Matthew Stoffregen and Hugo Zhou.

November 3: in-person

Splitting spheres in S^4

Maggie Miller

We say that a 2-component codimension-2 link in S^n is split when its components lie in disjoint balls. I’ll talk about recent work proving that split 2-component links in S^4 can admit distinct non-isotopic splitting spheres, which contradicts classical intuition since any two splitting spheres in the complement of a 2-component split link in S^3 are isotopic. Specifically, I’ll talk about this classical motivation, some recent work on interesting embeddings of 2- and 3-manifolds in S^4, and some of my favorite classical theorems in 4-dimensional topology. This is joint work with Mark Hughes and Seungwon Kim.

November 10: on Zoom

Convergence of unitary representations and spectral gaps of manifolds

Michael Magee

Let M be a manifold. I'll discuss the notion of `strong convergence' of a sequence of finite dimensional unitary representations of the fundamental group of M. Once this convergence property is established for particular sequences of representations, it can be used to deduce information about the spectral gap of the Laplacian on covering spaces  of M, or on vector bundles over M. This has led to several recent advances. For example, it is now known that every compact hyperbolic surface has a sequence of covering spaces with asymptotically optimal relative spectral gap. I'll discuss what is known and conjectured for higher dimensional hyperbolic manifolds.

November 17: on Zoom

Log-Concavity of the Alexander Polynomial for Special Alternating Links

Elena Hafner

The central question of knot theory is that of distinguishing links up to isotopy. The first polynomial invariant of links devised to help answer this question was the Alexander polynomial (1928). Almost a century after its introduction, it still presents us with tantalizing questions, such as  Fox's conjecture (1962) that the absolute values of the coefficients of the Alexander polynomial $\Delta_L(t)$ of an alternating link $L$ are unimodal.  Fox's conjecture remains open in general with special cases settled by Hartley (1979) for two-bridged knots, by Murasugi (1985) for a family of alternating algebraic links, and by Ozsv\'ath and Szab\'o (2003) for the case of genus $2$ alternating knots, among others.  We settle Fox's conjecture for special alternating links. We do so by proving that a certain multivariate generalization of the Alexander polynomial of special alternating links is Lorentzian.  As a consequence, we obtain that the absolute values of the coefficients of $\Delta_L(t)$, where $L$ is a special alternating link, form a log-concave sequence with no internal zeros. In particular, they are unimodal.  This talk is based on joint work with Karola M\'esz\'aros and Alexander Vidinas.

November 24: No seminar (Thanksgiving break)

December 1: in-person

Optimal regularity for minimizers of the prescribed mean curvature functional over isotopies

Lorenzo Sarnataro

In this talk, I will describe the regularity theory for surfaces minimizing the prescribed mean curvature functional over isotopies in a closed Riemannian 3-manifold, which is a prescribed mean curvature counterpart of the celebrated regularity result of Meeks, Simon and Yau about minimizers of the area functional over isotopies. 

Whereas for the area functional minimizers over isotopies are smooth embedded minimal surfaces, minimizers of the prescribed mean curvature functional turn out to be C^{1,1} immersions which can have a large self-touching set where the mean curvature vanishes. 

Even though the proof broadly follows the same general strategy as in the case of the area functional, several new ideas are needed to deal with the lower regularity setting. This regularity theory plays an important role in Z. Wang-X. Zhou’s recent proof of the existence of 4 embedded minimal spheres in a generic metric on the 3-sphere.

The results in this talk are joint work with Douglas Stryker (Princeton). 

Previous Seminars:

2023 Spring (in-person or online, as indicated)

April 7: on Zoom

Area-minimizing subvarieties of general manifolds

Zhenhua Liu

Homologically area-minimizing currents are minimizers of area among homologous competitors. By Almgren-De Lellis-Spadaro's big regularity theorem, such currents are chains over smooth submanifolds whose closure adds a set of codimension at least two. Prominent examples include holomorphic subvarieties and special Lagrangians. For instance, a classification of area-minimizers for every homology class in a complex torus would immediately give a true or false for the Hodge conjecture of this variety. Thus, understanding the behavior of area-minimizing currents would have immense applications to many facets of differential geometry. In this talk, we will review some recent progress on the behavior of area-minimizing surfaces in general manifolds. We start with the simplest type of singularities, i.e., the self-intersection of area-minimizing immersions. We illustrate that both non-smoothable rigid singularities and malleable singularities with arbitrary, even fractal, deformations exist in abundance. This settles a conjecture of Almgren in the 1980s about the existence of area-minimizing currents with fractal singular sets. Then we go to more general types of singularities and show that the bordism rings give fairly general obstructions to the smoothing of singularities. Combining with the previous case, we show that area-minimizing currents in every homology class in general dimensions and codimensions can have non-smoothable singularities, thus essentially settling a conjecture of White in the 1980s about the generic smoothness of area-minimizing currents. We will also discuss some cases where we can fully determine the moduli space of area-minimizing currents.

April 14:  on Zoom

Scattering rigidity for analytic metrics

Malo Jézéquel

For analytic negatively curved compact connected Riemannian manifold with analytic strictly convex boundary, the scattering map for the geodesic flow determines the manifold up to isometry. After detailing this result, I will explain how it can be proved using a unique continuation principle. This requires to know that certain objects are real-analytic: I will give a hint on the method of real-analytic microlocal analysis that we used to prove it. This is a joint work with Yannick Guedes Bonthonneau and Colin Guillarmou.

April 21: in-person

Plateau's problem via the theory of phase transitions

Stephen Lynch

Plateau's problem asks whether every boundary curve in 3-space is spanned by an area minimizing surface. Various interpretations of this problem have been solved using eg. geometric measure theory. Froehlich and Struwe proposed another approach, in which the desired surface is produced using smooth sections of a twisted line bundle over the complement of the boundary curve. The idea is to consider sections which minimize an analogue of the Allen--Cahn functional (a classical model for phase transition phenomena) and show that these concentrate energy around a solution of Plateau's problem. After some background on the link between phase transition models and minimal surfaces, I will describe new work with Marco Guaraco in which we produce smooth solutions of Plateau's problem using the approach described above.

April 28: in-person

Colored sl(N) homology and SU(N) representations

Joshua Wang

The Khovanov homology of a rational knot or link happens to coincide with the cohomology of its space of SU(2) representations that send meridians to traceless matrices. This coincidence is closely related to the spectral sequence from Khovanov homology to an SU(2) instanton homology defined by Kronheimer and Mrowka. Motivated by a conjectural spectral sequence from colored sl(N) homology to a hypothetical colored SU(N) instanton homology, I'll explain how the colored sl(N) homology of the trefoil agrees with the cohomology of its space of SU(N) representations that send meridians to a conjugacy class associated to the color. This gives the first computation of colored sl(N) homology of a nontrivial knot.

May 5: in-person

Fixed point-free pseudo-Anosovs and the cinquefoil

Luya Wang

I will discuss joint work with Braeden Reinoso and Ethan Farber on showing that the pseudo-Anosov representative of any genus-two, hyperbolic, fibered knot in the 3-sphere with nonzero fractional Dehn twist coefficient has an interior fixed point. Combined with work of Baldwin-Hu-Sivek where they use the number of fixed points of appropriate area-preserving diffeomorphisms to constrain ranks of Floer homology, this shows that knot Floer homology detects the cinquefoil T(2,5). In particular, T(2,5) is the only genus-two L-space knot. I will discuss some tools from train tracks and train track maps, and outline the main ideas of our proof.

May 12: in-person

An $L_\infty$-module structure on annular Khovanov homology

Champ Davis

Khovanov homology is one of the most popular tools used to study links in $S^3$.  If the link is in a thickened annulus, there is an annular refinement of Khovanov homology that contains additional structure.  In particular, Grigsby-Licata-Wehrli showed that the annular Khovanov homology of $L$ is equipped with an action of $sl_2(\wedge)$, the exterior current algebra of the Lie algebra $sl_2$.  We will discuss how this structure can be understood in the setting of $L_\infty$-algebras and modules.  We show that $sl_2(\wedge)$ is an $L_\infty$-algebra and that the annular Khovanov homology of $L$ is an $L_\infty$-module over $sl_2(\wedge)$.  Up to $L_\infty$-quasi-isomorphism, this structure is invariant under Reidemeister moves, and the higher $L_\infty$-operations can be computed using explicit formulas.

May 19: on Zoom

On the stability of self-similar blowup for nonlinear wave equations

Po-Ning Chen

One topic of fundamental importance in studying nonlinear wave equations is the singularity development of the solutions. Within the context of energy supercritical wave equations, a typical way to investigate singularity development is through the self-similar blowup.

In this talk, we will discuss current work in progress towards establishing the asymptotic nonlinear stability of self-similar blowup in the strong-field Skyrme equation and the quadratic wave equation.

May 26: in-person

Statistical Bergman geometry

Gunhee Cho

In complex geometry, the Bergman metric plays a very important role as a canonical metric as a pullback metric of the Fubini-Study metric of complex projective ambient space. This work is trying to do something really new to find a whole new approach of studying hyperbolic complex geometry, especially for a bounded domain in C^n, we replace the infinite dimensional complex projective ambient space to the collection of probability distributions defined on a bounded domain. We prove that in this new framework, the Bergman metric is given as a pullback metric of the Fisher-Information metric considered in information geometry, and from this, a new perspective on the contraction property and biholomorphic invariance of the Bergman metric will be discussed. As an application of this framework, in the case of bounded hermitian symmetric domains, we will discuss about the existence of a sequence of i.i.d random variables in which the covariance matrix converges to a distribution sense with a normal distribution given by the Bergman metric, and if more time is left, we will talk about recent progresses on stochastic differential geometry.

June 2: in-person

SU(2) representations of small Dehn surgeries on knots

Fan Ye

Kronheimer-Mrowka proved that the fundamental group of r-surgery on a nontrivial knot for |r|<=2 always has an irreducible SU(2) representation, which answered the property P conjecture affirmly. They asked the case of r=3 and 4. The case r=4 was solved by Baldwin-Sivek and the case r=3 was solved by Baldwin-Li-Sivek-Ye. In this talk, I will describe the strategy of the proofs using instanton Floer homology.

2023 Winter (in-person or online, as indicated)

January 6: in-person

On embedding periodic maps of surfaces into those of $S^m$

Zhongzi Wang

It is known that in the smooth orientable category any periodic map of order $n$ on a closed surface of genus $g$ can extend periodically over some $m$-dimensional sphere with respect to an equivariant embedding. 

We will determine the smallest possible $m$ when $n\geq 3g$. We will also show that for each integer $k>1$ there exist infinitely many periodic maps such that the smallest possible $m$ is equal to $k$.

This is a joint work with Chao Wang. 

January 13: in-person

Second order elliptic operators  on triple junction surfaces

Gaoming Wang

In this talk, we will consider minimal triple junction surfaces, a special class of singular minimal surfaces whose boundaries are identified in a particular manner. Hence, it is quite natural to extend the classical theory of minimal surfaces to minimal triple junction surfaces. Indeed, we can show that the classical PDE theory holds on triple junction surfaces. As a consequence, we can prove a type of Generalized Bernstein Theorem and talk about the Morse index on minimal triple junction surfaces.

January 20: in-person

Extending periodic maps over the 4-sphere

Shicheng Wang

Let $F_g$ be the closed  orientable surface of genus $g$. We address the problem to extend torsion elements of the mapping class group ${\mathcal{M}}(F_g)$ over the 4-sphere $S^4$. Let  $w_g$  be a  torsion element of maximum order in ${\mathcal{M}}(F_g)$. Results including:

(1) For each $g$, $w_g$ is periodically  extendable over $S^4$ for some non-smooth embedding  $e: F_g\to S^4$, and  not periodically extendable over $S^4$ for any smooth embedding  $e: F_g\to S^4$.

(2) For each $g$, $w_g$  is extendable over $S^4$ for some smooth embedding  $e: F_g\to S^4$ if and only if $g=4k, 4k+3$.

(3) For infinitely many primes,  each periodic map of order $p$ on $F_g$ is extendable over $S^4$ for some smooth embedding  $e: F_g\to S^4$. 

This is a joint work with Zhongzi Wang. 

January 27: in-person

Khovanov-type homology of null homologous links in RP^3

Daren Chen

Khovanov homology is originally defined for links in S^3, and it was extended for links in I-bundles over surface by Asaeda, Przytycki and Sikora. In this talk, we will exhibit some generalization of their construction for null homologous links in RP^3. On the other side of the story, Ozsvath and Szabo defined a spectral sequence relating the Heegaard Floer homology of the branched double cover of S^3 over L to the Khovanov homology of L. We will extend this construction for null homologous links in RP^3 as well, relating the Heegaard Floer homology of the branched double cover of RP^3 over L to our Khovanov-type homology of L.

February 3: in-person

Fibrations, depth 1 foliations, and branched surfaces

Chi Cheuk Tsang

A depth 1 foliation on a 3-manifold is a foliation having finitely many compact leaves and with all other leaves spiraling into the compact leaves. These are a natural extension of fibrations of 3-manifolds over $S^1$, and as shown in work of Cantwell, Conlon, and Fenley, a lot of that theory carries over. In this talk, we will first recall how the recent theory of veering branched surfaces offers a neat package of much that is known about fibrations, and explain how these can be generalized to apply to depth 1 foliations, providing a new way of studying finite depth foliations and their associated big mapping classes. This is joint work with Michael Landry.

February 10: in-person

Finite Bowen-Margulis-Sullivan measures in higher rank

Mikolaj Fraczyk

Let M be a negatively curved closed manifold. The Bowen Margulis Sullivan (BMS) measure on M can be understood as the unique probability measure on the unit tangent bundle of M which maximizes the entropy of the geodesic flow. The construction of BMS measures can be extended to locally symmetric spaces. where they proved to be a useful tool in the study of discrete subgroups of semi-simple Lie groups. I will talk about a joint work with Min Ju Lee where we show that a higher rank locally symmetric space admits a finite BMS measure if and only if it is finite volume. This leads to a new criterion detecting lattices among the discrete subgroups of higher rank Lie groups. 

February 17: on Zoom

Boundedness problems in conformal dynamics

Yusheng Luo

In 1980s, Thurston’s formulated the geometrization conjecture for 3-manifolds, and proved the hyperbolization theorem. The keys to Thurston’s proof are two bounded results for certain deformation spaces of Kleinian groups. In early 1990s, motivated by Thurston’s boundedness theorem and the Sullivan dictionary, McMullen conjectured that certain hyperbolic components of rational maps are bounded.

In this talk, I will start with a historical discussion on a general strategy of the proof of Thurston’s boundedness theorem. I will then explain how a similar strategy could work for rational maps, and discuss some recent breakthrough towards McMullen's boundedness conjecture.

February 24: in-person

Higgs bundles and SYZ geometry

Charles Ouyang

Special Lagrangian 3-torus fibrations over a 3 dimensional base play an important role in mirror symmetry and the SYZ conjecture. In this talk, we discuss the construction via Higgs bundles of an infinite family of semi-flat Calabi-Yau metrics on special Lagrangian torus bundles over an open ball in R^3 with a Y-vertex deleted. This is joint work with S. Heller and F. Pedit.

March 3: in-person

Exotic codim-1 submanifolds in 4-manifolds

Anubhav Mukherjee

One of the key topics of study in smooth 4-manifold theory is to understand how submanifolds are embedded in 4-manifolds. In this talk we will discuss about exotic codim-1 embeddings, i.e. those which are topologically isotopic but not smoothly in 4-manifolds. I will try to propose a few motivating questions and explain some new ideas on how to find such interesting embeddings in 4-manifolds with small Betti numbers. This is a joint work with Hokuto Konno and Masaki Taniguchi. 

March 10: in-person

Large genus asymptotics in flat surfaces

Amol Aggarwal

In this talk we will describe the behaviors of flat surfaces, as their genera tend to infinity. We first discuss enumerative results that count the number of such surfaces (which can be viewed as volumes of particular moduli spaces) in the large genus limit, and then we will explain how a randomly sampled such object looks. Prior work of Delecroix-Goujard-Zograf-Zorich expresses these topological counts in terms of more algebraic quantities, namely, certain intersection numbers. Thus, the large genus limits of these counts will rely on a new asymptotic result on the behaviors of these intersection numbers at high genus, which might be of independent interest.

2022 Fall (in-person or online, as indicated)

September 30: in-person
A  filtered mapping cone formula for cables of the knot meridian
Hugo Zhou
We construct a filtered mapping cone formula that computes the knot Floer complex of the (n,1)-cable of the knot meridian in any rational surgery, generalizing Truong's result about the (n,1)-cable of the knot meridian in large surgery and Hedden-Levine's filtered mapping cone formula. As an application, we show that there exist knots in integer homology spheres with arbitrary $\varphi_{i,j}$ values for any i>j>0, where $\varphi_{i,j}$ are the concordance homomorphisms defined by Dai-Hom-Stoffregen-Truong.

October 7: in-person
Blowing up and down with knot traces
Kai  Nakamura
Manolescu and Piccirillo recently proposed potential constructions of an exotic 4-sphere. These came in the form of knots K that if slice, then an exotic 4-sphere exists. The key property these knots have is that they share a zero surgery with a knot K' that has non-vanishing s-invariant. Here we show that the Manolescu-Piccirillo knots are not slice and rule out this exciting possibility. To do this, we show that the zero traces of K and K' become diffeomorphic after blowing up. This allows us to stably relate their slice properties and use the s-invariant of K′ to show K is not slice. Despite the success of our proof, a closer examination reveals a strange coincidence among the Manolescu-Piccirillo knots that allow our proof to work. Explaining this coincidence allows us to strongly generalize our proof from the original five knots to the infinite family of zero surgery homeomorphisms that Manolescu and Piccirillo considered.

October 14: in-person
The Kervaire conjecture and the minimal complexity of surfaces
Lvzhou Chen
We use topological methods to solve special cases of a fundamental problem in group theory, the Kervaire conjecture, which has connection to various problems in topology. The conjecture asserts that, for any nontrivial group G and any element w in the free product G*Z, the quotient (G*Z)/<<w>> is still nontrivial. We interpret this as a problem of estimating the minimal complexity (in terms of Euler characteristic) of surface maps to certain spaces. This gives a conceptually simple proof of Klyachko's theorem that confirms the Kervaire conjecture for any G torsion-free. We also obtain new results concerning injectivity of the map G->(G*Z)/<<w>> when w is a proper power.

October 21: in-person
Geography problem in  4 dimensional topology and Mahowald invariants
Zhouli Xu
The geography problem in 4 dimensional topology asks which closed simply connected 4-manifolds admit a smooth structure. After the celebrated work of Kirby-Siebenmann, Freedman, and Donaldson, the last uncharted territory of this geography question is the "11/8-Conjecture'', which states that for any smooth spin 4-manifold, the ratio of its second-Betti number and signature is least 11/8.

In this talk, I will discuss some recent progress on the "11/8-Conjecture'' by studying a problem in Pin(2)-equivariant stable homotopy theory using Mahowald invariants. This is a joint work with Mike Hopkins, Jianfeng Lin and XiaoLin Danny Shi.

Oct 28: in-person

The (2,1)-cable of the figure-eight knot is not smoothly slice
Sungkyung Kang

We prove that the (2,1)-cable of the figure-eight knot is not smoothly slice, thereby solving a question posed by Kawauchi in 1980. While doing so, we will review the involutive and equivariant actions in Heegaard Floer homology, and how one can use those symmetries to tackle various knot slicing problems in the smooth category. This is a joint work with Irving Dai, Abhishek Mallick, JungHwan Park, and Matthew Stoffregen.

Nov 11: in-person
Disk-like surfaces of section and symplectic embeddings
Oliver Edtmair

Symplectic embedding problems, i.e. the question whether one symplectic manifold embeds into another, are of central importance in symplectic geometry. Such problems are intimately related to Hamiltonian dynamics and this relationship has been used to construct a plethora of obstructions to symplectic embeddings. Going in the opposite direction, I will discuss how disk-like global surfaces of section, a concept from dynamics, can be used to construct symplectic embeddings. This yields partial progress towards Viterbo’s conjecture on symplectic capacities of convex domains: The cylindrical embedding capacity agrees with the minimal action of an unknotted Reeb orbit.

Nov 18: on Zoom
Does the Jones polynomial of a knot detect the unknot? A novel approach via braid group representations and class numbers of number fields.
Amitesh Datta
How good of an invariant is the Jones polynomial? The question is closely tied to studying braid group representations since the Jones polynomial can be defined as a (normalized) trace of a braid group representation.
In this talk, I will present my work developing a new theory to precisely characterize the entries of classical braid group representations, which leads to a generic faithfulness result for the Burau representation of B_4 (the faithfulness is a longstanding question since the 1930s). In forthcoming work, I use this theory to furthermore explicitly characterize the Jones polynomial of all 3-braid closures and generic 4-braid closures. I will also describe my work which uses the class numbers of quadratic number fields to show that the Jones polynomial detects the unknot for 3-braid links - this work also answers (in a strong form) a question of Vaughan Jones.
I will discuss all of the relevant background from scratch and illustrate my techniques through simple examples.

Nov 25: no seminar (Thanksgiving break)

Dec 2: in-person at 3pm (not 4pm)
Homology growth, fibering, and aspherical manifolds
Kevin Schreve
It is now known that every closed, hyperbolic 3-manifold has a finite cover which fibers over the circle. There has been recent interest in generalizing this result in various directions. In a geometric direction, one can ask whether higher odd dimensional hyperbolic manifolds virtually fiber. In an algebraic direction one can ask which groups have algebraic analogues of a virtual fibering. In all of this, there is a curious dearth of odd-dimensional examples which do not virtually fiber. We partially correct this by constructing closed, aspherical, odd-dimensional manifolds with word hyperbolic fundamental group that do not virtually fiber over the circle. This is joint work with Grigori Avramidi and Boris Okun.

2022 Spring (In person OR online as indicated)

June 10: In person at 1pm

On Steklov Eigenspaces for Free Boundary Minimal Surfaces

Peter McGrath

It has been conjectured that the first nontrivial eigenvalue of the Dirichlet-to-Neumann map on an embedded free boundary minimal surface in the unit 3-ball is one.  I will discuss recent work with R. Kusner which provides sufficient criteria for the first eigenvalue on such a surface to be equal to one, and moreover that the corresponding eigenspace is spanned by the coordinate functions.  A consequence of this work is that an embedded antipodally invariant free boundary minimal annulus in the unit ball is congruent to the critical catenoid.

May 20: Online

The alternation number and the Upsilon-invariant at 1 of positive 3-braid knots

Paula Trüol

The alternation number of a knot is the minimal number of crossing changes needed to deform the knot into an alternating knot, i.e. a knot with a diagram where the crossings alternate between over- and under-crossings as one travels around the knot. The tau- and the Upsilon-invariant from knot Floer homology give a lower bound on the alternation number of any knot. We use this lower bound and an upper bound by Abe and Kishimoto to determine the alternation number of all positive 3-braid knots.

The key tool and a result of independent interest is an explicit calculation of the Upsilon invariant at 1 of all 3-braid knots. We determine this integer-valued (concordance) invariant - which was defined by Ozsváth, Stipsicz and Szabó - by constructing cobordisms between 3-braid knots and (connected sums of) torus knots. In particular, we will only work with properties of Upsilon and not its definition, so no background in knot Floer homology will be assumed.

May 13: In person

A nonlinear spectrum on closed manifolds

Christos Mantoulidis

The p-widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of its Laplace--Beltrami operator, which was defined by Gromov in the 1980s and corresponds to areas of a certain min-max sequence of hypersurfaces. By a recent theorem of Liokumovich--Marques--Neves, the p-widths obey a Weyl law, just like the eigenvalues do. However, even though eigenvalues are explicitly computable for many manifolds, there had previously not been any >= 2-dimensional manifold for which all the p-widths are known. In recent joint work with Otis Chodosh, we found all p-widths on the round 2-sphere and thus the previously unknown Liokumovich--Marques--Neves Weyl law constant in dimension 2.

April 8: In person

Chromatic invariants of vector bundles on projective spaces

Morgan Opie

In this talk, I will discuss my ongoing work on complex rank 3 topological vector bundles on CP^5. I will describe a classification of such bundles using twisted, topological modular form-valued invariants, and the subtleties involved in actually computing this invariant. As time allows, I will outline future chromatic directions suggested by this result and by prior work of Atiyah and Rees. 

2022 Winter

March 4: In person

Quantization and non-quantization of energy along higher-dimensional Ginzburg-Landau vortices

Daniel Stern

Abstract: The complex Ginzburg-Landau equations are a family of geometric pdes arising in the study of harmonic maps, as well as simple models of superconductivity and superfluids. Around 20 years ago, it was observed that families of solutions satisfying natural energy bounds exhibit energy concentration along certain generalized minimal submanifolds of codimension two. Since then, it has been an open question whether energy is quantized along these concentration sets, in the sense that the limiting energy measure has integer multiplicity almost everywhere. In this talk, I'll describe joint work with Alessandro Pigati showing that this quantization does not hold in general, but that quantization does hold where energy density is <2.

February 25: Online

Reeb flows transverse to foliations

Jonathan Zung

Eliashberg and Thurston established an important link between the theory of foliations and the theory of contact structures on 3-manifolds: they showed that taut foliations on 3-manifolds can be approximated by tight contact structures. I will explain a new approach to this theorem which allows one to control the resulting Reeb flow and hence produce many hypertight contact structures. Along the way, I will explain how harmonic transverse measures may be used to understand the holonomy of foliations.

February 18: Online

Eigenvalue extremal metrics and n-harmonic maps

Antoine Métras

In this talk, I will discuss a correspondence between extremal problems from spectral geometry and geometric object such as n-harmonic maps. On a Riemannian manifold M, one can study the eigenvalues of the Laplacian or of the Steklov problem as functionals of the metric. Of particular interest are their critical points, which we call extremal metrics. If is known that for the Laplace eigenvalues, extremal metrics correspond to minimal immersions to a sphere; while for the Steklov eigenvalues on a surface, extremal metrics give rise to free-boundary minimal surface in a ball. I will discuss higher dimensional generalization of this Steklov result and how one obtains n-harmonic maps when considering extremal metrics in a conformal class. This is joint work with Mikhail Karpukhin.

February 11: In person

Waists, widths and symplectic embeddings

Jonathan Zhu

Waists and widths measure the size of a manifold with respect to measures of families of submanifolds. We’ll discuss related area estimates for minimal submanifolds, as well as applications to quantitative symplectic camels.

January 21: Online

From 3-manifolds to modular data

Yang Qiu

The progress of TQFT has revealed connections between the algebraic world of tensor categories and the topological world of 3-manifolds, such as Reshetikhin-Turaev and Turaev-Viro theories. Motivated by M-theory in physics, Cho-Gang-Kim recently proposed another relation by outlining a program to construct modular data from certain classes of closed oriented 3-manifolds. In this talk, I will talk about our mathematical exploration of this program. This talk is based on the joint works: [Cui-Qiu-Wang, arXiv: 2101.01674], [Cui-Gustafson-Qiu-Zhang, arXiv: 2106.01959].

January 14: Online

Smallest non-cyclic quotients of braid groups

Sudipta Kolay

We will discuss a new approach to determine the minimal quotients of groups of geometric interest, primarily focussing on the Artin braid groups. There were conjectures due to Zimmermann, Mecchia-Zimmermann, and Margalit, for the cases of mapping class groups, outer automorphism groups of free groups, and braid groups respectively. The first two conjectures were proved by Kielak-Pierro, and Baumeister-Kielak-Pierro respectively, and several recent results show support for the conjecture of Margalit. We will outline these approaches, and then describe how the novel inductive orbit-stabilizer method can be used to resolve these conjectures.

2021 Fall

November 19: In Person

Floer Homology and quasipositive surfaces

Lev Tovstopyat-Nelip

Ozsvath and Szabo have shown that knot Floer homology detects knot genus - the largest Alexander grading of a non-trivial homology class is equal to the genus.

We give a new contact geometric interpretation of this fact by realizing such a class via the transverse knot invariant introduced by Lisca, Ozsvath, Stipsicz and Szabo. Our approach relies on the "convex decomposition theory" of Honda, Kazez and Matic - a contact geometric interpretation of Gabai's sutured hierarchies. 

We use this new interpretation to study the "next-to-top" summand of knot Floer homology, and to show that Heegaard Floer homology detects quasi-positive Seifert surfaces. Some of this talk represents joint work with Matthew Hedden.

November 12: In Person

Singular behavior of min-max minimal hypersurfaces in dimension 8

Luca Spolaor

In this talk I will discuss an estimate on the index and number of "non-minimizing" singular points for one parameter min-max minimal hypersurfaces in 8-dimensional compact manifolds. As a consequence I will discuss the generic existence of smooth embedded minimal  hypersurfaces in 8-dimensional compact manifolds.

This joint work with Otis Chodosh (Stanford) and Yevgeny Liokumovich (University of Toronto).

November 5: In Person

Volume estimates of complements of lifts of geodesics in projective tangent bundles

Tommaso Cremaschi

In this talk I will discuss some results about volume estimates of knot complements in Seifert-fibered spaces with particular care of tangent lifts in projective tangent bundles of hyperbolic surfaces. We will of give various estimates depending on topological and geometric quantities associated to the projection of the knot to the surface. Finally inspired by some experimental data I will discuss recent work in the setting of random geodesics.

October 29: In Person

Quintic differentials and Penrose tilings

Philip Engel

I will discuss work-in-progress with Peter Smillie on the construction of rhombus-tiled surfaces in moduli spaces of quintic (and higher) differentials. There are some intriguing connections to abelian varieties, quasicrystals, Hurwitz theory, modular forms, and Masur-Veech volumes. 

October 22: Online

Coloured Jones and Alexander polynomials unified through Lagrangian intersections in configuration spaces

Cristina Anghel (Geneva)

The theory of quantum invariants started with the Jones polynomial and continued with the Reshetikhin-Turaev algebraic construction of link invariants. In this context, the quantum group Uq(sl(2)) leads to the sequence of coloured Jones polynomials, which contains the original Jones polynomial. Dually, the quantum group at roots of unity gives the sequence of coloured Alexander polynomials. We construct a unified topological model for these two sequences of quantum invariants. More specifically, we define certain homology classes given by Lagrangian submanifolds in configuration spaces. Then, we prove that the Nth coloured Jones and Nth coloured Alexander invariants come as different specialisations of a state sum (defined over 3 variables) of Lagrangian intersections in configuration spaces. As a particular case, we see both Jones and Alexander polynomials from the same intersection pairing in a configuration space.

October 15:  Online

A large surgery formula for instanton Floer homology

Fan Ye (Cambridge)

In Heegaard Floer homology, Oszváth-Szabó and Rasmussen introduced a large surgery formula computing HF^\hat(S^3_m(K)) for any knot K and large integer m by bent complexes from CFK^-(K). In this talk, I'll introduce a similar formula for instanton Floer homology. More precisely, I construct two differentials on the instanton knot homology KHI(K) and use them to compute the framed instanton homology I^#(S^3_m(K)) for any large integer m. As an application, I show that if the coefficients of the Alexander polynomial of K are not ±1, then there exists an irreducible SU(2) representation of the fundamental group of S^3_r(K)) for all but finitely many rational r. In particular, all hyperbolic alternating knots satisfy this condition. Also by this large surgery formula, I show KHI(K)=HFK^\hat(K) for any Berge knot and I^#(S^3_r(K))=HF^\hat(S^3_r(K)) for any genus-one alternating knot.

October 8: In person

Connor Mooney (UC Irvine)

Solutions to the Monge-Ampere equation with polyhedral and Y-shaped singularities

The Monge-Ampere equation det(D^2u) = 1 arises in prescribed curvature problems and in optimal transport. An interesting feature of the equation is that it admits singular solutions. We will discuss new examples of convex functions on R^n that solve the Monge-Ampere equation away from finitely many points, but contain polyhedral and Y-shaped singular structures. Along the way we will discuss geometric motivations for constructing such examples, as well as their connection to a certain obstacle problem.

October 1: In person

Reflection in algebra and topology

Aaron Mazel-Gee (Caltech)

In this talk, I will discuss a new duality that was recently discovered in joint work with David Ayala and Nick Rozenblyum, which we refer to as reflection.

In essence, reflection amounts to two dual methods for reconstructing objects, based on a stratification of the category that they live in. As a basic example, an abelian group can be reconstructed on the one hand in terms of its p-completions and its rationalization, or on the other (reflected) hand in terms of its p-torsion components and its corationalization; and these both come from a certain "closed-open decomposition" of the category of abelian groups.

Examples and applications of reflection are abundant, because stratifications are abundant. In algebra, reflection recovers the derived equivalences of quivers coming from BGP reflection functors (hence the terminology "reflection"). In topology, reflection is closely related to Verdier duality, a generalization of Poincaré duality that applies to singular spaces. Moreover, an explicit description of reflection leads to a categorification of the classical Möbius inversion formula, a Fourier inversion theorem for functions on posets.

2021 Spring (via Zoom)

June 4: Ancient solutions in geometric flows

Natasa Sesum (Rutgers)

We will talk about classification of ancient solutions in geometric flows. In particular, we will show the only closed ancient noncollapsed Ricci flow solutions are the shrinking spheres and Perelman's solution. We will talk about the higher dimensional analogue of this result under suitable curvature assumptions as well. These are joint works with Brendle, Daskalopoulos and Naff.

May 28: Lens space surgeries, lattices, and the Poincaré homology sphere

Jacob Caudell (BC)

Moser's classification of Dehn surgeries on torus knots (1971) inspired a now fifty-years-old project to classify "exceptional" Dehn surgeries on knots in the three-sphere. A prominent component of this project seeks to classify which knots admit surgeries to the "simplest" non-trivial 3-manifolds--lens spaces. By combining data from Floer homology and the theory of integer lattices into the notion of a changemaker lattice, Greene (2010) solved the lens space realization problem: every lens space which may be realized as surgery on a knot in the three-sphere may be realized by a knot already known to surger to that lens space (i.e. a torus knot or a Berge knot). In this talk, we present a survey of techniques in Dehn surgery and their applications, introduce a generalization of Greene's changemaker lattices, and discuss applications to surgeries on knots in the Poincaré homology sphere.

May 21: Ancient solutions to mean curvature flow

Theodora Bourni (UT Knoxville)

Mean curvature flow (MCF) is the gradient flow of the area functional; it moves the surface in the direction of steepest decrease of area.  An important motivation for the study of MCF comes from its potential geometric applications, such as classification theorems and geometric inequalities. MCF develops “singularities” (curvature blow-up), which obstruct the flow from existing for all times and therefore understanding these high curvature regions is of great interest.  This is done by studying ancient solutions, solutions that have existed for all times in the past, and which model singularities. In this talk we will discuss their importance and ways of constructing and classifying such solutions. In particular, we will focus on “collapsed” solutions and construct, in all dimensions n>=2, a large family of new examples, including both symmetric and asymmetric examples, as well as many eternal examples that do not evolve by translation. Moreover,  we will show that collapsed solutions decompose “backwards in time” into a canonical configuration of Grim hyperplanes which satisfies certain necessary conditions. This is joint work with Mat Langford and Giuseppe Tinaglia.

May 14: Khovanov homology via Floer theory of the 4-punctured sphere

Artem Kotelskiy (Indiana)

Consider a Conway two-sphere S intersecting a knot K in 4 points, and thus decomposing the knot into two 4-ended tangles, T and T’. We will first interpret Khovanov homology Kh(K) as Lagrangian Floer homology of a pair of specifically constructed immersed curves C(T) and C'(T’) on the dividing 4-punctured sphere S. Next, motivated by several tangle-replacement questions in knot theory, we will describe a recently obtained structural result concerning the curve invariant C(T), which severely restricts the types of curves that may appear as tangle invariants. The proof relies on the matrix factorization framework of Khovanov-Rozansky, as well as the homological mirror symmetry statement for the 3-punctured sphere. This is joint work with Liam Watson and Claudius Zibrowius.

May 7: Definite surfaces, plumbing, and Tait's conjectures

Thomas Kindred (UN Lincoln)

In 1898, P.G. Tait asserted several properties of alternating link diagrams, which remained unproven until the discovery of the Jones polynomial in 1985. By 1993, the Jones polynomial had led to proofs of all of Tait’s conjectures, but the geometric content of these new results remained mysterious.

In 2017, Howie and Greene independently gave the first geometric characterizations of alternating links; as a corollary, Greene obtained the first purely geometric proof of part of Tait’s conjectures. Recently, I used these characterizations and "replumbing" moves, among other techniques, to give the first entirely geometric proof of Tait’s flyping conjecture, first proven in 1993 by Menasco and Thistlethwaite.

I will describe these recent developments, focusing in particular on the fundamentals of plumbing (also called Murasugi sum), and definite surfaces (which characterize alternating links a la Greene). As an aside, I will also sketch a (partly new, simplified) proof of the classical result of Murasugi and Crowell that the genus of an alternating knot equals half the degree of its Alexander polynomial.  The talk will be broadly accessible. Expect lots of pictures!

April 30: Algebraic fibrations of surface-by-surface groups

Stefano Vidussi (UC Riverside)

An algebraic fibration of a group G is an epimorphism to the integers with a finitely generated kernel. This notion has been studied at least since the '60s, and has recently attracted renewed attention. Among other things, we will study it in the context of fundamental groups of surface bundles over a surface, where it has some interesting relations with some classical problems about the mapping class group. This is based on joint work with S. Friedl, and with R. Kropholler and G. Walsh.

April 23: The failure of the 4D light bulb theorem with dual spheres of non-zero square

Hannah Schwartz (Princeton)

Examples of surfaces embedded in a 4-manifold that are homotopic but not isotopic are neither rare nor surprising. It is then quite amazing that, in settings such as the recent 4D light bulb theorems of both Gabai and Schneiderman-Teichner, the existence of an embedded sphere of square zero intersecting a surface transversally in a single point has the power to "upgrade" a homotopy of that surface into a smooth isotopy. We will discuss the limitations of this phenonemon, using contractible 4-manifolds called corks to produce homotopic spheres in a 4-manifold with a common dual of non-zero square that are not smoothly isotopic.

April 16: Existence of static vacuum extensions

Lan-Hsuan Huang (UConn)

The study of static vacuum Riemannian metrics arises naturally in general relativity and differential geometry. A static vacuum metric produces a static spacetime by a warped product, and it is related to scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static vacuum metric with black hole boundary must belong to the Schwarzschild family. In contrast to the rigidity phenomenon, R. Bartnik conjectured that there are asymptotically flat, static vacuum metric realizing certain arbitrarily specified boundary data. I will discuss recent progress toward this conjecture. It is based on joint work with Zhongshan An.

April 9: Amphichiral knots with large 4-genera

Allison Miller (Rice)

An oriented knot is called negative amphichiral if it is isotopic to the reverse of its mirror image. Such knots have order at most two in the concordance group, and many modern concordance invariants vanish on them. Nevertheless, we will see that there are negative amphichiral knots with arbitrarily large 4-genera (i.e. which are highly 4-dimensionally complex), using Casson-Gordon signature invariants as a primary tool.

April 2: Skein lasagna modules of 2-handlebodies

Ikshu Neithalath (UCLA)

Morrison, Walker and Wedrich recently defined a generalization of Khovanov-Rozansky homology to links in the boundary of a 4-manifold. We will discuss recent joint work with Ciprian Manolescu on computing the "skein lasagna module," a basic part of MWW's invariant, for a certain class of 4-manifolds.

2021 Winter (via Zoom)

March 5: Stable and unstable homology of graph braid groups

Ben Knudsen (Northeastern)

The homology of the configuration spaces of a graph forms a finitely generated module over the polynomial ring generated by its edges; in particular, each Betti number is eventually equal to a polynomial in the number of particles, an analogue of classical homological stability. The degree of this polynomial is captured by a connectivity invariant of the graph, and its leading coefficient may be computed explicitly in terms of cut counts and vertex valences. This "stable" (asymptotic) homology is generated entirely by the fundamental classes of certain tori of geometric origin, but exotic non-toric classes abound unstably. These mysterious classes are intimately tied to questions about generation and torsion whose answers remain elusive except in a few special cases. This talk represents joint work with Byung Hee An and Gabriel Drummond-Cole.

Feb 26: Harmonic forms and norms on cohomology of non-compact hyperbolic 3-manifolds

Xiaolong Hans Han (UIUC)

We will talk about generalizations of an inequality of Brock-Dunfield to the non-compact case, with tools from Hodge theory for non-compact hyperbolic manifolds and recent developments in the theory of minimal surfaces. We also prove that their inequality is not sharp, using holomorphic quadratic differentials and recent ideas of Wolf and Wu on minimal geometric foliations. If time permits, we will also describe a partial generalization to the infinite volume case.

Feb 19: Knot Floer homology and relative adjunction inequalities

Katherine Raoux (Michigan State)

In this talk, we present a relative adjunction inequality for 4-manifolds with boundary. We begin by constructing generalized Heegaard Floer tau-invariants associated to a knot in a 3-manifold and a nontrivial Floer class. Given a 4-manifold with boundary, the tau-invariant associated to a Floer class provides a lower bound for the genus of a properly embedded surface, provided that the Floer class is in the image of the cobordism map induced by the 4-manifold. We will also discuss some applications to links and contact manifolds. This is joint work with Matthew Hedden.

Feb 12: Constructing minimal submanifolds via gauge theory

Daniel Stern (UChicago)

The self-dual Yang-Mills-Higgs (or Ginzburg-Landau) functionals are a natural family of energies associated to sections and metric connections of Hermitian line bundles, whose critical points (particularly those satisfying a first-order system known as the "vortex equations" in the Kahler setting) have long been studied as a basic model problem in gauge theory. In this talk, we will discuss joint work with Alessandro Pigati characterizing the behavior of critical points over manifolds of arbitrary dimension. We show in particular that critical points give rise to minimal submanifolds of codimension two in certain natural scaling limits, and use this information to provide new constructions of codimension-two minimal varieties in arbitrary Riemannian manifolds. We will also discuss recent work with Davide Parise and Alessandro Pigati developing the associated Gamma-convergence machinery, and describe some geometric applications.

Feb 5: Harmonic branched coverings and uniformization of CAT(k) spheres

Christine Breiner (Fordham)

Consider a metric space (S,d) with an upper curvature bound in the sense of Alexandrov (i.e. via triangle comparison).  We show that if (S,d) is homeomorphically equivalent to the 2-sphere S^2, then it is conformally equivalent to S^2.  The method of proof is through harmonic maps, and we show that the conformal equivalence is achieved by an almost conformal harmonic map.  The proof  relies on the analysis of the local behavior of harmonic maps between surfaces, and the key step is to show that an  almost conformal harmonic map from  a compact surface  onto a surface with an upper curvature bound is a branched covering. This work is joint with Chikako Mese.

Jan 29: Choosing points on cubic plane curves

Weiyan Chen (Tsinghua)

It is a classical topic to study structures of certain special points on complex smooth cubic plane curves, for example, the 9 flex points and the 27 sextactic points. We consider the following topological question asked by Farb: Is it true that the known algebraic structures give all the possible ways to continuously choose n distinct points on every smooth cubic plane curve, for each given positive integer n? This work is joint with Ishan Banerjee.

Jan 22: A family of 3d steady gradient solitons that are flying wings

Yi Lai (UC Berkeley)

We find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at infinity. The 3d flying wings are collapsed. For dimension n ≥ 4, we find a family of Z2 × O(n − 1)-symmetric but non-rotationally symmetric n-dimensional steady gradient solitons with positive curvature operator. We show that these solitons are non-collapsed.

Jan 15: Links all of whose branched cyclic covers are L-spaces

Hannah Turner (UT Austin)

Given an oriented link in the three-sphere and a fixed positive integer n, there is a unique 3-manifold called its branched cyclic cover of index n. It is not well understood when these manifolds are L-spaces - that is, when their Heegaard Floer homology is as simple as possible. In this talk I'll describe new examples of links whose cyclic branched covers are L-spaces for any index n. The proof uses a symmetry argument and a generalization of alternating links due to Scaduto-Stoffregen. This is joint work with Ahmad Issa.

Jan 8: Orthogonal group and higher categorical adjoints

David Ayala (Montana State)

In this talk I will articulate and contextualize the following sequence of results.

• The Bruhat decomposition of the general linear group defines a stratification of the orthogonal group.

• Matrix multiplication defines an algebra structure on its exit-path category in a certain Morita category of categories.

• In this Morita category, this algebra acts on the categeory of n-categories -- this action is given by adjoining adjoints to n-categories.

This result is extracted from a larger program -- entirely joint with John Francis, some parts joint with Nick Rozenblyum -- which proves the cobordism hypothesis.

2020 Fall (via Zoom)

Dec 4: Brieskorn spheres, homology cobordism and homology balls

Oguz Savk (Bogazici University)

A classical question in low-dimensional topology asks which homology 3-spheres bound homology 4-balls. This question is fairly addressed to Brieskorn spheres Σ(p,q,r), which are defined to be links of singularities x^p+y^q+z^r=0. Over the years, Brieskorn spheres also have been the main objects for the understanding of the algebraic structure of the integral homology cobordism group.

In this talk, we will present several families of Brieskorn spheres which do or do not bound integral and rational homology balls via Ozsváth-Szabó d-invariant, involutive Heegaard Floer homology, and Kirby calculus. Also, we will investigate their positions in both integral and rational homology cobordism groups.

November 27: no seminar

November 20: Pre-Calabi-Yau categories and dualizability in 2d

Alex Takeda (IHES)

In this talk I will describe some joint work with Maxim Kontsevich on the study of pre-Calabi Yau categories. I will discuss the action of a PROP on the Hochschild invariants of such a category and explain how this notion and more familiar notions of Calabi-Yau objects relate to different dualizability conditions of 2d TQFTs. Time allowing I will present some motivating examples from symplectic geometry.

November 13: Compactness and partial regularity theory of Ricci flows in higher dimensions

Richard Bamler (UC Berkeley)

We present a new compactness theory of Ricci flows. This theory states that any sequence of Ricci flows that is pointed in an appropriate sense, subsequentially converges to a synthetic flow. Under a natural non-collapsing condition, this limiting flow is smooth on the complement of a singular set of parabolic codimension at least 4. We furthermore obtain a stratification of the singular set with optimal dimensional bounds depending on the symmetries of the tangent flows. Our methods also imply the corresponding quantitative stratification result and the expected L^p-curvature bounds.

As an application we obtain a description of the singularity formation at the first singular time and a long-time characterization of immortal flows, which generalizes the thick-thin decomposition in dimension 3. We also obtain a backwards pseudolocality theorem and discuss several other applications.

November 6: 3-manifold invariants, G-equivariant TQFT, and complexity

Eric Samperton (UIUC)

Let G be a finite group.  G-equivariant TQFTs have received attention from both mathematicians and physicists, motivated in part by the search for new topological phases that can be used as the hardware for a universal quantum computer.  Our goal will be to convey two complexity-theoretic lessons.  First, when G is sufficiently complicated (nonabelian simple), 3-manifold invariants derived from G-equivariant TQFTs are very difficult to compute (#P-hard), even on a quantum computer.  Second, no matter what finite group G one uses, a 3-dimensional G-equivariant TQFT can not be used for universal topological quantum computation if the underlying non-equivariant theory is not already universal.  This talk is based on joint works with Greg Kuperberg and Colleen Delaney.

October 30: Generalized soap bubbles and the topology of manifolds with positive scalar curvature

Chao Li (Princeton)

It has been a classical question which manifolds admit Riemannian metrics with positive scalar curvature. I will present some recent progress on this question, ruling out positive scalar curvature on closed aspherical manifolds of dimensions 4 and 5 (as conjectured by Schoen-Yau and by Gromov), as well as complete metrics of positive scalar curvature on an arbitrary manifold connect sum with a torus. Applications include a Schoen-Yau Liouville theorem for all locally conformally flat manifolds. The proofs of these results are based on analyzing generalized soap bubbles - surfaces that are stable solutions to the prescribed mean curvature problem. This talk is based on joint work with O. Chodosh.

October 23: Several detection results of Khovanov homology on links

Boyu Zhang (Princeton)

The Khovanov homology is a combinatorially defined invariant for knots and links. I will present several new detection results of Khovanov homology on links. In particular, we show that if L is an n-component link with Khovanov homology of rank 2^n, then it is given by the connected sums and disjoint unions of unknots and Hopf links. This result gives a positive answer to a question asked by Batson-Seed, and it generalizes the unlink detection theorem by Hedden-Ni and Batson-Seed. The proof relies on a new excision formula for the singular instanton Floer homology. This is joint work with Yi Xie.

October 16: Kasteleyn operators from mirror symmetry

Harold Williams (USC)

In this talk we explain an interpretation of the Kasteleyn operator of a doubly-periodic bipartite graph from the perspective of homological mirror symmetry. Specifically, given a consistent bipartite graph G in T^2 with a complex-valued edge weighting E we show the following two constructions are the same. The first is to form the Kasteleyn operator of (G,E) and pass to its spectral transform, a coherent sheaf supported on a spectral curve in (C*)^2. The second is to take a certain Lagrangian surface L in T^* T^2 canonically associated to G, equip it with a brane structure prescribed by E, and pass to its homologically mirror coherent sheaf. This lives on a toric compactification of (C*)^2 determined by the Legendrian link which lifts the zig-zag paths of G (and to which the noncompact Lagrangian L is asymptotic). As a corollary, we obtain a complementary geometric perspective on certain features of algebraic integrable systems associated to lattice polygons, studied for example by Goncharov-Kenyon and Fock-Marshakov. This is joint work with David Treumann and Eric Zaslow. 

October 9: Symmetric knots and the equivariant 4-ball genus

Ahmad Issa (UBC)

Given a knot K in the 3-sphere, the 4-genus of K is the minimal genus of an orientable surface embedded in the 4-ball with boundary K. If the knot K has a symmetry (e.g. K is periodic or strongly invertible), one can define the equivariant 4-genus by only minimising the genus over those surfaces in the 4-ball which respect the symmetry of the knot. I'll discuss some ongoing work with Keegan Boyle on trying to understanding the equivariant 4-genus.

October 2: Concordance invariants from the E(-1) spectral sequence

William Ballinger (Princeton)

Many recent concordance invariants of knots come from perturbing the differential on a knot homology theory to get a complex with trivial homology but an interesting filtration. I describe the invariant coming from Rasmussen's E(-1) spectral sequence from Khovanov homology in this way, and show that it gives a bound on the nonorientable slice genus.

2020 Spring (via Zoom)

May 22: Surgery for gauge theoretical invariants of integral homology S1 × S3

Langte Ma (Brandeis)

Abstract: Given an integral homology S 1 × S 3 , one can define the Furuta-Ohta invariant λF O using Yang-Mills theory and Casson-Seiberg-Witten invariant λSW using Seiberg-Witten theory. In this talk I will discuss a torus surgery formula for those two invariants. I will also discuss some computations resulting from the surgery formula, which provides us with more evidence on the equivalence of these invariants.

May 15: Isometric submersions of Teichmueller spaces

Mark Greenfield (Michigan)

Abstract: We study holomorphic and isometric submersions between Teichmueller spaces of finite-type surfaces. Our main result is that (possibly excepting low-genus phenomena) any such map is a forgetful map obtained by filling in punctures. This generalizes Royden's theorem which states that the isometry group of a Teichmueller space is the mapping class group of the underlying surface (again excepting low-genus phenomena). In this talk, after discussing some background and related results, we will give an overview of the proof, including how we adapt and utilize analytic methods originally developed by Markovic for generalizing Royden's theorem to infinite-type surfaces. This is joint work with Dmitri Gekhtman.

May 8: Characterizing handle-ribbon knots

Maggie Miller (Princeton)

The stable Kauffman conjecture posits that a knot in the 3-sphere is slice if and only if it admits a slice derivative. In joint work with Alexander Zupan, we prove a related statement: A knot is handle-ribbon (also called strongly homotopy-ribbon) in a homotopy 4-ball B if and only if it admits an R-link derivative. We also show that K bounds a handle-ribbon disk D in B if and only the 3-manifold obtained by zero-surgery on K admits a singular fibration that extends over handlebodies to the complement of D, generalizing a classical theorem of Casson and Gordon stated for fibered knots. I will discuss the background (e.g. what is a knot derivative?) and the motivation (e.g. which theorem of Casson and Gordon?) behind this result, and sketch the techniques used in the proof. All of this work is joint with Alexander Zupan (University of Nebraska-Lincoln), and will appear on the arXiv in the near future.

April 24: Scalable Spaces

Fedya Manin (UCSB)

Abstract: Scalable spaces are simply connected compact Riemannian manifolds with lots of geometrically "efficient" self-maps.  For just one example, this is true for $(\mathbb{CP}^2)^{\#3}$ but not $(\mathbb{CP}^2)^{\#4}$.  This property, though defined geometrically, turns out to be a topological and indeed a rational homotopy invariant with several equivalent formulations.  It can be thought of as a metric refinement of Sullivan's notion of formality; there are also analogies with work of Wenger on asymptotic properties of nilpotent groups.  The work I will be discussing is joint with Sasha Berdnikov.

April 17: Minimal Surfaces in Hyperbolic 3-Manifolds

Baris Coskunuzer (UT Dallas)

Abstract: In this talk, we will show the existence of smoothly embedded closed minimal surfaces in infinite volume hyperbolic 3-manifolds. The talk will be non-technical, and accessible to graduate students.

2020 Winter

March 6: SL(3) foam evaluation and its relation to the Kronheimer-Mrowka homology theory of graphs.

Mikhail Khovanov (Columbia)

Abstract: In this talk we'll introduce SL(3) foams and their evaluation to symmetric functions in three variables. This construction gives rise to a functorial homology theory for plane trivalent graphs G. A naive conjecture is that the rank of this homology group assigned to G is the number of Tait colorings of G. A Tait coloring is the coloring of edges of a trivalent graph in three colors such that at each vertex the three edges have different colors. We'll discuss relation of this story to the Kronheimer-Mrowka homology theory and to the four-color theorem.

February 21: Generalizing Rasmussen's s-invariant, and applications

Michael Willis (UCLA)

Abstract: I will discuss a method to define Khovanov and Lee homology for links in connected sums of copies of S1 times S2.  From here we can define an s-invariant that gives genus bounds on oriented cobordisms between links.  I will discuss some applications to surfaces in certain 4-manifolds, including a proof that the s-invariant cannot detect exotic 4-balls coming from Gluck twists of the standard 4-ball.  If time allows, I will also discuss our new combinatorial proof of the slice Bennequin inequality in S1 times S2.  All of this is joint work with Ciprian Manolescu, Marco Marengon, and Sucharit Sarkar.

February 14: Decomposing sutured Instanton Floer homology

Zhenkun Li (MIT)

Abstract: Sutured Instanton Floer homology was introduced by Kronheimer and Mrowka. Though it has many important applications to the study of 3-dimensional topology, many basic aspects of the theory remain undeveloped. In this talk I will explain how to decompose sutured Instanton Floer homology with respect to properly embedded surfaces inside the sutured manifold, and present some applications of this decomposition to the development of the theory: performing  some computations, bounding the depth of taut sutured manifolds, detecting the Thurston norm on link complements, and constructing some invariants for knots and links.

February 7: Surface bundles, monodromy, and arithmetic groups

Bena Tshishiku (Brown)

Abstract: In the 1960s Atiyah and Kodaira constructed surface bundles over surfaces with many interesting properties. The topology of such a bundle is completely encoded by its monodromy representation (a homomorphism to a mapping class group), and it is a basic problem to understand precisely how the topology of the bundle is reflected in algebraic properties of the monodromy. The main result of this talk is that the Atiyah-Kodaira bundles have arithmetic monodromy groups. A corollary of this result is that Atiyah-Kodaira bundles fiber in exactly two ways. This is joint work with Nick Salter.

January 31: Stable commutator length in groups acting on trees

Lvzhou Chen (Chicago)

Abstract: The stable commutator length (scl) is a relative version of the Gromov-Thurston norm. It is a group invariant sensitive to geometric and dynamical properties. Scl can be used to understand homomorphism rigidity and to find surface subgroups. The latter problem is related to the rationality of scl, and the former is about lower bounds.

     In this talk, we will discuss the computation and rationality of scl in certain groups acting on trees by linear programming. In the special case of Baumslag-Solitar groups, we will see a convergence to scl in free groups resembling the convergence of metrics under hyperbolic Dehn surgeries. If time permits, I will also briefly explain lower bounds of scl in 3-manifold groups and right-angled Artin groups, which is joint work with Nicolaus Heuer.

January 24: Harmonic maps and anti-de Sitter 3-manifolds

Nathaniel Sagman (Caltech)

Abstract: Anti-de Sitter n-space is a complete Lorentzian manifold of constant sectional curvature -1 across all non-degenerate 2-planes. In dimension 3, anti-de Sitter space identifies with PSL_2(R), equipped with (a constant multiple of) its Killing metric, and the Lorentzian isometry group is (up to finite index) PSL_2(R) x PSL_2(R) acting via left and right multiplication. Recently attention has been devoted to understanding which subgroups of the isometry group admit properly discontinous actions, thus yielding what are called complete AdS 3-manifolds. I will describe aspects of this program and outline a construction of a new class of non-compact AdS 3-manifolds. The main technical tool in this case is the theory of equivariant harmonic maps.

January 17: Large scale geometry of large mapping class groups

Kathryn Mann (Cornell)

Abstract: The mapping class groups of infinite type surfaces aren't finitely generated or locally compact, so don't fall within the normal scope of geometric group theory.  Nevertheless, there has been much recent work (starting with that of Calegari and J. Bavard) on essentially geometric questions about these groups: distortion of elements, hyperbolicity of associated curve graphs, etc.    In new work with Kasra Rafi, we attack these kinds of questions by showing many large mapping class groups do have a well-defined large scale geometry, using Christian Rosendal's framework for geometric group theory of non-locally compact groups.    In this talk I'll explain our classification theorem, some of the tools involved in its proof, and advertise some next steps.  

January 10: Geodesic planes in hyperbolic 3-manifolds 

Amir Mohammadi (UCSD)

Abstract: Let M be a hyperbolic 3-manifold, a geodesic plane in M is a totally geodesic immersion of the hyperbolic plane into M. In this talk we will give an overview of some results which highlight how geometric, topological, and arithmetic properties of M are related to behavior of geodesic planes in M. This talk is based on joint projects with McMullen and Oh, and in another direction with Margulis. 

2019 Fall

December 6: The topology of representation varieties

Maxime Bergeron (Chicago)

Abstract: Let \Gamma be a finitely generated group and let G be a complex reductive algebraic group such as a special linear group. I will discuss various aspects of the topology of the space Hom(\Gamma,G) of representations of \Gamma in G.

November 15: GPV invariants and knot complements

Ciprian Manolescu (Stanford)

Abstract: Gukov, Putrov and Vafa predicted (from physics) the existence of some 3-manifold invariants that take the form of power series with integer coefficients, converging in the unit disk. Their radial limits at the roots of unity should recover the Witten-Reshetikhin-Turaev invariants. Further, they should admit a categorification, in the spirit of Khovanov homology. Although a mathematical definition of the GPV invariants is lacking, they can be computed in many cases. In this talk I will discuss what is known about the GPV invariants, and their behavior with respect to Dehn surgery. The surgery formula involves associating to a knot a two-variable series, obtained by parametric resurgence from the asymptotic expansion of the colored Jones polynomial. This is based on joint work with Sergei Gukov.

November 8: Comparing complexities of bounded area minimal hypersurfaces

Antoine Song (Berkeley)

Abstract: For a closed minimal surface with area less than A in a Riemannian 3-manifold, there are two natural measures of complexity: its Morse index as a critical point of the area functional, and its genus. How do these two relate? After giving some context, we will prove that they are actually comparable up to a constant factor depending only on the ambient manifold and the area bound A. As we will see, this result generalizes to higher dimensional minimal hypersurfaces with area less than A and with small singular sets in the following way: the index dominates the total Betti number in dimensions 3 to 7, or the size of the singular set in dimensions 8 and above. The proof’s arguments are essentially combinatorial.

November 1 **2:30-4:30**: Generic Multiplicity One Singularities of Mean Curvature Flow of Surfaces

Ao Sun (MIT)

Abstract: One of the central topics in mean curvature flow is understanding the singularities. In 1995, Ilmanen conjectured that the first singularity appeared in a smooth mean curvature flow of surfaces must have multiplicity one. Following the theory of generic mean curvature flow developed by Colding-Minicozzi, we prove that a closed singularity with high multiplicity is not generic, in the sense that we may perturb the flow so that this singularity with high multiplicity can never show up. One of the main techniques is the local entropy, which is an extension of the entropy used by Colding-Minicozzi to study the generic mean curvature flow.

October 25: Ricci Flow of Doubly-warped Product Metrics

Maxwell Stolarski (Arizona State)

Abstract: The Ricci flow of rotationally symmetric metrics has been a source of interesting dynamics for the flow that include the formation of slow blow-up degenerate neckpinch singularities and the forward continuation of the flow through neckpinch singularities. A natural next source of examples is then the Ricci flow of doubly-warped product metrics. This structure allows for a potentially larger collection of singularity models compared to the rotationally symmetric case. Indeed, formal matched asymptotic expansions suggest a non-generic set of initial metrics on a closed manifold form finite-time, type II singularities modeled on a Ricci-flat cone at parabolic scales. I will outline the formal matched asymptotics of this singularity formation and discuss the applications of such solutions to questions regarding the possible rates of singularity formation and the blow-up of scalar curvature. In the second half, we will examine in detail the topological argument used to prove the existence of Ricci flow solutions with these dynamics.

October 18: Harmonic maps with polynomial growth

Andrea Tamburelli (Rice) 

Abstract: To a conformal harmonic map from the complex plane to the symmetric space SL(n,R)/SO(n) one can associate holomorphic differentials q_k of degree k=3, ..., n. We say that a harmonic map has polynomial growth if all such differentials are polynomials and cyclic if only q_n in non-zero . In this talk, we will describe the asymptotic geometry of the minimal surface associated to cyclic harmonic maps with polynomial growth.

Monday, October 14, 3-5, Room 255 ***Special Date and Location****: 

Quantitative recurrence and hyperbolicity of the Teichmüller flow

Ian Frankel (Queen's University)

Abstract: In hyperbolic space, the unit tangent bundle has a foliation by horospheres, which are exponentially contracted by the geodesic flow as a function of time. We show that the same behavior holds in the universal cover of the moduli space of Riemann surfaces, endowed with the Teichmüller metric, except possibly when geodesics spend too much time in the cusps of moduli space. (It is known that this behavior can fail in the cusps.)

October 4: Harmonic maps for Hitchin representations

 Qiongling Li (Chern Institute, Nankai)

Abstract: Hitchin representations are an important class of representations of fundamental groups of closed hyperbolic surfaces into PSL(n,R), at the heart of higher Teichmüller theory. Given such a representation j, there is a unique j-equivariant harmonic map from the universal cover of the hyperbolic surface to the symmetric space of PSL(n,R). We show that its energy density is at least 1 and that rigidity holds. In particular, we show that given a Hitchin representation, every equivariant minimal immersion from the hyperbolic plane into the symmetric space of PSL(n,R) is distance-increasing. Moreover, equality holds at one point if and only if it holds everywhere and the given Hitchin representation j is an n-Fuchsian representation.

2019 Winter:

1. Katie Mann (Brown) (2019/1/12)

2. Du Pei (Caltech) (2019/1/25)

3. Peter Smillie (Caltech) (2019/2/1)

4. Matt Durham (Riverside) (2019/2/8)

5. Nick Salter (Coblumbia) (2019/2/15)

6. Yusheng Luo(Harvard) (2019/3/1)

7. Philip Engel (Georgia) (2019/3/8)

Philip Engel (Georgia) (2019/3/8) A Monstrous(?) Moduli Space

Abstract: Let B = CH^13 be 13-dimensional complex hyperbolic space (a complex ball). There is an arithmetic group Γ in PU(13) acting on B generated by order 3 Hermitian isometries s_i called triflections. Basak and Allcock have studied the geometry of X = Γ \ B in detail; it is intimately connected with the finite projective plane P^2F_3. A conjecture of Allcock states that if one replaces relations s_i^3=1 in Γ with s_i^2=1, the resulting group is the Bimonster---the wreath product of the monster with Z_2. A resolution of this conjecture likely leads to a resolution of the "Hirzebruch prize question": The existence of a compact 12-complex dimensional manifold with certain topological invariants and an action of the monster. Such a manifold would lead in a known way to a new, geometric proof of monstrous moonshine.

I will discuss three moduli spaces, which might (depending on the status of computations at the time of the talk) be isomorphic to ball quotients of dimensions 13, 7, 4 relating to the projective planes P^2F_3, P^2F_2, "P^2F_1" = {3 points}. The corresponding finite groups, gotten by replacing 3, 4, 6 with 2, should be the bimonster, an orthogonal group O_8(2) of a quadratic form on F_2^8, and the symmetric group S_6 respectively. Should these examples work out, they will produce many surprising things: For instance, a formula for the order of the monster group in terms of Hurwitz numbers. This talk is highly speculative and represents joint with Peter Smillie and Francois Greer.

Nick Salter (Columbia) (2019/2/15) Linear-central filtrations and representations of the braid group

Abstract: In 1994, Bass and Lubotzky introduced the notion of a “linear-central filtration” which is a structure present in many situations in geometric group theory, e.g. in the Johnson filtration of the mapping class group of a surface. We survey here some recent work investigating the Burau representation of the braid group using these tools. Our main result is a complete and simple description of the image of the Burau representation, answering a 1974 question of Birman. We are also able to obtain a new abelian quotient of the so-called braid Torelli group. Portions of this work are joint with Kevin Kordek.

Matt Durham (Riverside) (2019/2/8) Stable cubulations in mapping class groups

As with much of geometric group theory, the study of the coarse geometry of the mapping class group has recently seen an influx of ideas coming from the world of CAT(0) cubical complexes.  Perhaps most remarkably, Behrstock-Hagen-Sisto recently proved that the coarse convex hulls of finite sets of points in the mapping class group are coarsely modeled by cube complexes.

Using work of Bestvina-Bromberg-Fujiwara-Sisto, we improve their construction to build modeling cube complexes which remain coarsely stable under perturbation of the relevant data.  As initial applications, we build a bicombing of the mapping class group and prove that finite sets admit coarse barycenters.

This is joint work with Yair Minsky and Alessandro Sisto.

Peter Smillie (Caltech) (2019/2/1) Hyperbolic surfaces in Minkowski 3-space

I'll first give a characterization of all hyperbolic surfaces properly isometrically embedded in Minkowski 3-space in terms of their asymptotics. I'll then discuss the problem of reading off properties of these surfaces from these asymptotics: first, completeness of the intrinsic metric, and second, conformal type of a closely related surface. In the second hour, I'll give some details of the proof of the characterization. This is joint work with Francesco Bonsante and Andrea Seppi.

Katie Mann (2019/1/12): Rigidity and geometricity of surface group actions

Abstract:  An action of a finitely generated group G on a manifold M is called "geometric" if it comes from an embedding of G as a lattice in a Lie group acting transitively on M.   In this talk, I'll explain some past work on rigidity of geometric actions on the circle, and newer joint work with Maxime Wolff that effectively characterizes geometric actions of surface groups on the circle by dynamical rigidity.

Du Pei(2019/1/25): Modular tensor categories from the Coulomb branch 

Abstract: We propose a new link between geometry of moduli spaces and quantum topology. The construction goes through a class of four-dimensional supersymmetric quantum field theories. Each such theory gives rise to a family of modular tensor categories, whose algebraic structures are intimately related to the geometry of the Coulomb branch. This is based on joint work with Mykola Dedushenko, Sergei Gukov, Hiraku Nakajima and Ke Ye. .