Caltech Geometry & Topology Seminar
2023/2024
Fridays 4pm - 5pm (Pacific time)
in-person (Linde Hall, Room 187) or online
Online meetings are conducted in the Zoom room 882 2905 2667. Its password is the value of π1(S1), repeated six times (for a total of six characters).
2024 Spring (in-person or online, as indicated)
April 5: in-person
Polya's conjecture in spectral geometry
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
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
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
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
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
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
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
5:00 - 6:00 pm
May 24: in-person
Area estimates and intersection properties for minimal hypersurfaces in space forms
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
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
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 $S^2 \times S^2$
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
2023 Winter (in-person or online, as indicated)
January 6: in-person
On embedding periodic maps of surfaces into those of $S^m$
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
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
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
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
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
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
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
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
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
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)
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
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
May 20: Online
The alternation number and the Upsilon-invariant at 1 of positive 3-braid knots
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
Previous Terms
2019 Spring:
Michael Landry (Yale) (2019/4/5)
Oishee Banerjee (Chicago) (2019/4/10, Wednesday 4-5)
Sam Nariman (Northwestern) (2019/4/12)
Marco Marengon (UCLA) (2019/4/19)
Jonathan Zhu (Princeton) (2019/4/26)
Ken Bromberg (Utah) (2019/5/3)
Andy Putman (Notre Dame) (2019/5/8)
Aaron Mazel-Gee (USC) (2019/5/17)
Tengren Zhang (NUS, Singapore) (2019/5/24)
Tengren Zhang (2019/5/24): Affine actions with Hitchin linear part
Abstract: We prove that if a surface group acts properly on R^d via affine transformations, then its linear part is not the lift of a PSL(d,R)-Hitchin representation. To do this, we proved two theorems that are of independent interest. First, we showed that PSO(n,n)-Hitchin representations, when viewed as representations into PSL(2n,R), are never Anosov with respect to the stabilizer of the n-plane. Following Danciger-Gueritaud-Kassel, we also view affine actions on R^{n,n-1} as a geometric limit of isometric actions on H^{n,n-1}. The second theorem we prove is a criterion for when an affine action on R^{n,n-1} is proper, in terms of the isometric actions on H^{n,n-1} that converge to it. This is joint work with Jeff Danciger, with some overlap with independent work by Sourav Ghosh.
Aaron Mazel-Gee (2019/5/17): The geometry of the cyclotomic trace
Abstract: K-theory is a means of probing geometric objects by studying their vector bundles. Algebraic K-theory, the version applying to varieties and schemes, is a particularly deep and far-reaching invariant, but it is notoriously difficult to compute. The primary means of computing it is through its "cyclotomic trace" map K→TC to another theory called topological cyclic homology. However, despite the enormous computational success of these so-called "trace methods" in algebraic K-theory computations, the algebro-geometric nature of the cyclotomic trace has remained mysterious. In the first talk, I will describe a new construction of TC that affords a precise interpretation of the cyclotomic trace at the level of derived algebraic geometry, which is joint work with David Ayala and Nick Rozenblyum. By the end of the talk, you will be able to take home with you a very nice and down-to-earth fact about traces of matrices. No prior knowledge of algebraic K-theory or derived algebraic geometry will be assumed.
In the second talk, I will explain our theory of stratified noncommutative geometry. This provides a means of decomposing and reconstructing categories; it generalizes the theory of recollements, which e.g. from a closed-open decomposition of a scheme (or of a topological space) gives a decomposition of its category of quasicoherent (resp. constructible) sheaves. It encompasses a version of adelic reconstruction (after Beilinson, Parshin, and Tate), which is itself an elaboration of the classical "arithmetic fracture square" that reconstructs an abelian group from its p-completions and its rationalization. It also applies to chain complexes or spectra with "genuine" G-action, which are the relevant objects for equivariant Poincare duality. We use this theory to deduce a universal mapping-in property for TC (generalizing recent work of Nikolaus--Scholze), which gives rise to the cyclotomic trace from algebraic K-theory.
Andy Putman (Notre Dame) (2019/5/8): The stable cohomology of the moduli space of curves with level structures
Abstract: We will prove that in a stable range, the rational cohomology of the
moduli space of curves with level structures is the same as that of
the ordinary moduli space of curves: a polynomial ring in the
Miller-Morita-Mumford classes.
Ken Bromberg (Utah) (2019/5/3): Renormalized volume of hyperbolic 3-manifolds
Abstract: Renormalized volume is a way of assigning a finite volume to a hyperbolic 3-manifold that has infinite volume in the usual sense. While the definition was motivated by ideas from physics, it has a number of interesting properties that make it a natural quantity to study from a purely mathematical perspective. I will begin with some basic background on renormalized volume and then describe how it can be used to give bounds on the volume of convex cores of convex co-compact hyperbolic 3-manifolds. This is joint work with M. Bridgeman and J. Brock.
Michael Landry (Yale) (2019/4/5): Homology directions and veering triangulations
Abstract: The cone over a fibered face of a hyperbolic 3-manifold has a nice characterization as the dual of the so-called cone of homology directions of a certain pseudo-Anosov flow. We give a new characterization of this cone of homology directions in terms of Agol’s veering triangulations.
Oishee Banerjee (Chicago) (2019/4/10, Wednesday 4-5) Title: Cohomology of the space of polynomial morphisms on with prescribed ramifications
Abstract: In this talk we will discuss the moduli spaces of degree morphisms with "ramification length " over an algebraically closed field . For each , the moduli space is a Zariski open subset of the space of degree polynomials over up to . It is, in a way, orthogonal to the many papers about polynomials with prescribed zeroes- here we are prescribing, instead, the ramification data. We will also see why and how our results align, in spirit, with the long standing open problem of understanding the topology of the Hurwitz space.
Sam Nariman (Northwestern) (2019/4/12): Dynamical and cohomological obstruction to extending group actions
Abstract: For any 3-manifold $M$ with torus boundary, we find finitely generated subgroups of $\Diff_0(\partial M)$ whose actions do not extend to actions on $M$; in many cases, there is even no action by homeomorphisms. The obstructions are both dynamical and cohomological in nature. If time permits, we also discuss cohomological obstruction to extending $SO(3)$ action on $S^2$ to certain $3$-manifolds that bound $S^2$. This is joint work with Kathryn Mann.
Marco Marengon (UCLA) (2019/4/19) Strands algebras and Ozsváth-Szabó's Kauffman states functor
Abstract: Ozsváth and Szabó introduced in 2016 a knot invariant, which they announced to be isomorphic to the usual knot Floer homology. Their construction is reminiscent of bordered Floer homology: for example, their invariant is defined by tensoring bimodules over certain algebras.
During the talk I will introduce a more geometric construction, closer in spirit to bordered sutured Floer homology, based on strands on a particular class of generalised arc diagrams. The resulting strands algebras are quasi-isomorphic to the Ozsváth-Szabó algebras, suggesting that Ozsváth and Szabó's theory may be part of a hypothetical generalisation of bordered sutured Floer homology. This is a joint work with Mike Willis and Andy Manion.
2019 Winter:
Katie Mann (Brown) (2019/1/12)
Du Pei (Caltech) (2019/1/25)
Peter Smillie (Caltech) (2019/2/1)
Matt Durham (Riverside) (2019/2/8)
Nick Salter (Coblumbia) (2019/2/15)
Yusheng Luo(Harvard) (2019/3/1)
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.
Yusheng Luo(Harvard) (2019/3/1) Classification of hyperbolic component with bounded escape
Abstract: A hyperbolic component is said to have bounded escape if there is a sequence of rational maps which is degenerating as conjugacy classes, but for any period $p$, the multipliers of periodic points of period $p$ remain bounded. A hyperbolic component is said to have nested Julia set if the Julia set is a Cantor set of nested continuum.
In this talk, we will study the barycentric extensions of rational maps on hyperbolic $3$ space and its geometric limit as branched coverings on a $\R$-tree.
We will use them to show that a hyperbolic component has bounded escape if and only if it has nested Julia set.
We remark that either phenomenon cannot happen for a finitely generated Kleinian group.
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. .
2018 Fall: Titles
Ahmad Issa (UT Austin) (10.5)
Andy Manion (USC) (10.12)
Eylem Zeliha Yildiz (Michigan State) (10.19)
Beibei Liu (UC Davis) (11.2)
Sherry Gong (UCLA) (11.9)
Sebastian Hurtado (UChicago) (11.16)
Mikhail Khovanov (Columbia) (Monday 11.19)
Andrew Zimmer (LSU) (11.30)
Paul Apisa (Yale) (12.7)
Andrew Zimmer (LSU) (11.30)
Title: The geometry of domains with negatively pinched Kahler metrics
Abstract: Every bounded pseudoconvex domain in C^n has a natural complete Kahler metric: the Kahler-Einstein metric constructed by Cheng-Yau. In this talk I will describe how the curvature of this metric restricts the CR-geometry of the boundary. In particular, I will sketch the proofs of the following two theorems: First, if a smoothly bounded convex domain has a complete Kahler metric with pinched negatively curved bisectional curvature, then the boundary of the domain has finite type in the sense of D'Angelo. Second, if a smoothly bounded convex domain has a complete Kahler metric with sufficiently tight pinched negatively curved holomorphic sectional curvature, then the boundary of the domain is strongly pseudoconvex. The proofs use recent results of Wu-Yau, classical results of Shi on the Ricci flow, and ideas from Benoist's work in real projective geometry. This is joint work with F. Bracci and H. Gaussier.
Paul Apisa (Yale) (12.7)
Title: Using flat geometry to understand the dynamics of every point - Hausdorff dimension, divergence, and Teichmuller geodesic flow!
Abstract: The moduli space of Riemann surfaces admits a Kobayashi hyperbolic metric called the Teichmuller metric. The geodesic flow in this metric can be concretely understand in terms of a linear action on flat surfaces represented as polygons in the plane. In this talk, we will study the dynamics of this geodesic flow using the geometry of flat surfaces.
Given such a flat surface there is a circle of directions in which one might travel along Teichmuller geodesics. We will describe work showing that for every (not just almost every!) flat surface the set of directions in which Teichmuller geodesic flow diverges on average - i.e. spends asymptotically zero percent of its time in any compact set - is 1/2.
In the first part of the talk, we will recall work of Masur, which connects divergence of Teichmuller geodesic flow with the dynamics of straight line flow on flat surfaces.
In the second part of the talk, we will describe the lower bound (joint with H. Masur) and how it uses flat geometry to prove a quantitative recurrence result for Teichmuller geodesic flow.
In the third and final part of the talk, we will describe the upper bound (joint with H. al-Saqban, A. Erchenko, O. Khalil, S. Mirzadeh, and C. Uyanik), which adapts the work of Kadyrov, Kleinbock, Lindenstrauss, and Margulis to the Teichmuller geodesic flow setting using Margulis functions.
Sebastian Hurtado (UChicago) (11.16)
Topic: Burnside problem and Zimmer program: which group can act on a manifold?
Mikhail Khovanov (Columbia) (Monday 11.19)
Title: Foam evaluation and Kronheimer-Mrowka theories
Abstract: Foams are two-dimensional complexes with generic singularities, usually embedded in 3-space. The talk will revolve around the formula by Robert and Wagner assigning a symmetric polynomial to a foam via suitable colorings of facets of the foam by a finite set of colors. Euler characteristics of subsurfaces of the foam associated to colorings appear prominently in the formula. We'll explain an application of this construction and its conjectural relation to Kronheimer-Mrowka homology theories that come from a gauge theory for orbifolds.
Ahmad Issa (UT Austin) (10.5)
Title: Embedding Seifert fibered spaces in the 4-sphere
Abstract: Which 3-manifolds smoothly embed in the 4-sphere? This seemingly simple question turns out to be rather subtle. Using Donaldson's theorem, we derive strong restrictions to embedding a Seifert fibered space over an orientable base surface, which in particular gives a complete classification when e > k/2, where k is the number of exceptional fibers and e is the normalized central weight. Our results point towards an interesting conjecture which I'll discuss. This is joint work with Duncan McCoy.
Andy Manion (USC) (10.12)
Title: Heegaard Floer homology and higher representations
Abstract: I will discuss joint work with Rouquier (in preparation) relating certain tensor products of 2-representations to constructions of Douglas-Manolescu in Heegaard Floer homology.
Eylem Zeliha Yildiz (Michigan State) (10.19)
Title: Knot concordance in 3-manifolds with an application
Abstract: I will discuss PL and smooth knot concordances in 3-manifolds. In particular I will show that all the knots in the free homotopy class of $S^1 \times pt$ in $S^1 \times S^2$ are smoothly concordant to each other. I will also discuss an application of this concordances to constructing exotic 4-manifolds.
Beibei Liu (UC Davis) (11.2)
Title: Some geometric applications of Heegaard Floer homology
Abstract: For oriented links in the three sphere, there are two geometric questions: determining Thurston polytopes of the link complements and 4-genera of links with vanishing pairwise linking numbers. I will explain how to use the Heegaard Floer homology introduced by Ozsvath and Szabo to determine the Thurston polytope, and give some bounds on the 4-genus in terms of the so-called d-invariants. In particular, for 2-component L-space links, d-invariants of integral surgeries along the link can be computed, generalizing Ni-Wu’s formula for knot surgeries, and Thurston polytopes for such links are determined by Alexander polynomials explicitly. I will also show some examples for both questions.
Sherry Gong (UCLA) (11.9)
Title: Results on Spectral Sequences for Singular Instanton Floer Homology
Abstract: We introduce a version of Khovanov homology for alternating links with marking data, $\omega$, inspired by instanton theory. We show that the analogue of the spectral sequence from Khovanov homology to singular instanton homology (Kronheimer and Mrowka, \textit{Khovanov homology is an unknot-detector}) collapses on the $E_2$ page for alternating links. We moreover show that the Khovanov homology we introduce for alternating links does not depend on $\omega$; thus, the instanton homology also does not depend on $\omega$ for alternating links.