# Caltech GT Seminar

# 2020/2021

### Fridays 3pm - 4pm (Pacific time), online via Zoom

We meet at the Zoom room 963 2453 5627. The password is the value of π_{1}(S^{1}), repeated six times (for a total of six characters).

## 2020 Fall (via Zoom)

### Dec 4: Brieskorn spheres, homology cobordism and homology balls

Oguz Savk (Bogazici University)

Brieskorn spheres Σ(p,q,r) represent a special and important class of 3-manifolds which are defined to be links of singularities x^p+y^q+z^r=0. Over the years, they have been 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: TBA

### 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: TBA

### 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.