# Geometry and Topology Seminar 2017

Current contact: Dave Futer or Matthew Stover

The Seminar usually takes place on Wednesdays at 2:30 PM in Room 617 on the sixth floor of Wachman Hall.

• Wednesday January 25, 2017 at 14:30, Wachman 617
Divergence of CAT(0) cube complexes and right-angled Coxeter groups

Abstract: The divergence function of a metric space, a quasi-isometry invariant, roughly measures the rate that pairs of geodesic rays stray apart. We will present new results regarding divergence functions of CAT(0) cube complexes. Right-angled Coxeter groups, in particular, exhibit a rich spectrum of possible divergence functions, and we will give special focus to applications of our results to these groups. Applications to the theory of random right-angled Coxeter groups will also be briefly discussed.

• Wednesday February 1, 2017 at 14:30, Wachman 617
Universal acylindrical actions

Carolyn Abbott, University of Wisconsin

Abstract: Given a finitely generated group, one can look for an acylindrical action on a hyperbolic space in which all elements that are loxodromic for some acylindrical action of the group are loxodromic for this particular action. Such an action is called a universal acylindrical action and, for acylindrically hyperbolic groups, tends to give a lot of information about the group. I will discuss recent results in the search for universal acylindrical actions, describing a class of groups for which it is always possible to construct such an action as well as an example of a group for which no such action exists.

• Thursday February 9, 2017 at 16:30, David Rittenhouse Labs, UPenn
Stability and vanishing in the unstable cohomology of SL_n(Z)

Thomas Church, Stanford/IAS

PATCH seminar (organized by Bryn Mawr, Haverford, Penn, and Temple)

Abstract: Borel proved that in low dimensions, the cohomology of a locally symmetric space can be represented not just by harmonic forms but by invariant forms. This implies that the $k$-th rational cohomology of $SL_n(Z)$ is independent of $n$ in a linear range $n \geq c k$, and tells us exactly what this "stable cohomology" is. In contrast, very little is known about the unstable cohomology, in higher dimensions outside this range.

In this talk I will explain a conjecture on a new kind of stability in the unstable cohomology of arithmetic groups like $SL_n(Z)$. These conjectures deal with the "codimension-k" cohomology near the top dimension (the virtual cohomological dimension), and for $SL_n(Z)$ they imply the cohomology vanishes there. Although the full conjecture is still open, I will explain how we proved it for codimension-0 and codimension-1. The key ingredient is a version of Poincare duality for these groups based on the algebra of modular symbols, and a new presentation for modular symbols. Joint work with Benson Farb and Andrew Putman.

• Thursday February 9, 2017 at 17:45, David Rittenhouse Labs, UPenn
Lagrangian tori, mutations and toric degenerations

Denis Auroux, UC Berkeley/IAS

PATCH seminar (organized by Bryn Mawr, Haverford, Penn, and Temple)

Abstract: A basic open problem in symplectic topology is to classify Lagrangian submanifolds (up to, say, Hamiltonian isotopy) in a given symplectic manifold. In recent years, ideas from mirror symmetry have led to the realization that even the simplest symplectic manifolds (eg. vector spaces or complex projective spaces) contain many more Lagrangian tori than previously thought. We will present some of the recent developments on this problem, and discuss some of the connections (established and conjectural) between Lagrangian tori, cluster mutations, and toric degenerations, that arise out of this story.

• Wednesday February 15, 2017 at 14:30, Wachman 617
Experimental statistics of veering triangulations

William Worden, Temple University

Abstract: Certain fibered hyperbolic 3-manifolds admit a layered veering triangulation, which can be constructed algorithmically given the stable lamination of the monodromy. These triangulations were introduced by Agol in 2011, and have been further studied by several others in the years since. We present experimental results which shed light on the combinatorial structure of veering triangulations, and its relation to certain topological invariants of the underlying manifold. We will begin by discussing essential background material, including hyperbolic manifolds and ideal triangulations, and more particularly fibered hyperbolic manifolds and the construction of the veering triangulation.

• Wednesday February 22, 2017 at 14:30, Wachman 617
Minimal surfaces with bounded index

Davi Maximo, University of Pennsylvania

Abstract: In this talk we will give a precise picture on how a sequence of minimal surfaces with bounded index might degenerate on a given closed three-manifold. As an application, we prove several compactness results.

• Wednesday March 1, 2017 at 14:30, Wachman 617
The dynamics of classifying geometric structures

William Goldman, University of Maryland

The general theory of locally homogeneous geometric structures (flat Cartan connections) originated with Ehresmann's 1936 paper Sur les espaces localement homogènes''. Their classification leads to interesting dynamical systems.

For example, classifying Euclidean geometries on the torus leads to the usual action of the SL(2,Z) on the upper half­plane. This action is dynamically trivial, with a quotient space the familiar modular curve. In contrast, the classification of other simple geometries on the torus leads to the standard linear action of SL(2,Z) on R^2, with chaotic dynamics and a pathological quotient space. This talk describes such dynamical systems, where the moduli space is described by the nonlinear symmetries of cubic equations like Markoff’s equation x^2 + y^2 + z^2 = x y z.

Both trivial and chaotic dynamics arise simultaneously, relating to possibly singular hyperbolic metrics on surfaces of Euler characteristic ­1.

• Friday March 3, 2017 at 14:30, Wachman 617
Polyhedra inscribed in quadrics and their geometry

Sara Maloni, University of Virginia

PATCH seminar (organized by Bryn Mawr, Haverford, Penn, and Temple)

In 1832 Steiner asked for a characterization of polyhedra which can be inscribed in quadrics. In 1992 Rivin answered in the case of the sphere, using hyperbolic geometry. In this talk, I will describe the complete answer to Steiner's question, which involves the study of interesting analogues of hyperbolic geometry including anti de Sitter geometry. Time permitting, we will also discuss future directions in the study of convex hyperbolic and anti de Sitter manifolds. This is joint work with J. Danciger and J.-M. Schlenker.

In the morning talk (at 9:30am), I will recall the idea of a geometric structure and the definitions of hyperbolic and anti de Sitter geometry. I will also explain hyperbolic quasi-Fuchsian manifolds and their AdS analogues.

• Friday March 3, 2017 at 16:00, Wachman 617
Manipulating singularities of Weinstein skeleta

Laura Starkston, Stanford University

PATCH seminar (organized by Bryn Mawr, Haverford, Penn, and Temple)

Weinstein manifolds are an important class of symplectic manifolds with convex ends/boundary. These 2n dimensional manifolds come with a retraction onto a core n-dimensional stratified complex called the skeleton, which generally has singularities. The topology of the skeleton does not generally determine the smooth or symplectic structure of the 2n dimensional Weinstein manifold. However, if the singularities fall into a simple enough class (Nadler’s arboreal singularities), the whole Weinstein manifold can be recovered just from the data of the n-dimensional complex. We discuss work in progress showing that every Weinstein manifold can be homotoped to have a skeleton with only arboreal singularities (focusing in low-dimensions). This has significance for combinatorially computing deep invariants of symplectic manifolds like the Fukaya category.

In the morning background talk (at 11:00), I will discuss the original example of a symplectic manifold: the cotangent bundle $T^*M$ of any smooth manifold $M$.

• Wednesday March 29, 2017 at 14:30, Wachman 617
Counting loxodromics for hyperbolic actions

Samuel Taylor, Yale University

Abstract: Consider a nonelementary action by isometries of a hyperbolic group $G$ on a hyperbolic metric space $X$. Besides the action of $G$ on its Cayley graph, some examples to bear in mind are actions of $G$ on trees and quasi-trees, actions on nonelementary hyperbolic quotients of $G$, or examples arising from naturally associated spaces, like subgroups of the mapping class group acting on the curve graph.

We show that the set of elements of $G$ which act as loxodromic isometries of $X$ (i.e those with sink-source dynamics) is generic. That is, for any finite generating set of $G$, the proportion of $X$-loxodromics in the ball of radius n about the identity in $G$ approaches 1 as n goes to infinity. We also establish several results about the behavior in $X$ of the images of typical geodesic rays in $G$; for example, we prove that they make linear progress in $X$ and converge to the boundary of $X$. Our techniques make use of the automatic structure of $G$, Patterson-Sullivan measure, and the ergodic theory of randoms walks for groups acting on hyperbolic spaces. This is joint work with I. Gekhtman and G. Tiozzo.

• Friday March 31, 2017 at 15:00, Haverford College, room KINSC H108
Lagrangian handlebodies in R^6

David Treumann, Boston College

PATCH Seminar (organized by Bryn Mawr, Haverford, Penn, and Temple)

Abstract: I will discuss an approach using microlocal sheaf theory to study Legendrian surfaces in $S^5$ and their Lagrangian fillings in $R^6$. This talk is based on joint work with Eric Zaslow and Linhui Shen about open Gromov-Witten invariants in $R^6$.

MORNING BACKGROUND TALK: The first talk will explain some basic notions about sheaves, Legendrians, and Lagrangians. The background talk takes place at 9:30 AM in in the Science Library (KINSC H305C),

• Friday March 31, 2017 at 16:15, Haverford College, room KINSC H108

Anastasiia Tsvietkova, Rutgers University Newark

PATCH Seminar (organized by Bryn Mawr, Haverford, Penn, and Temple)

Abstract: Hyperbolic volume is a powerful invariant of hyperbolic 3-manifolds. For 3-manifolds that are not hyperbolic, simplicial volume, that is closely related to Gromov norm, can be seen as a generalization of hyperbolic volume. The hyperbolic volume of a link complement in a 3-sphere is known to be unchanged when a half-twist is added to a link diagram, and a suitable 3-punctured sphere is present in the complement. We generalize this to the simplicial volume of link complements by analyzing the corresponding toroidal decompositions. We then use it to prove a refined upper bound for the (simplicial and hyperbolic) volume in terms of twists of various lengths in a link diagram. The bound found an application in the work relating coefficients of the colored Jones polynomial to volume, in the spirit of the Volume Conjecture. This is a joint work with Oliver Dasbach.

MORNING BACKGROUND TALK: In this background talk, I will discuss incompressible surfaces in 3-manifolds, decomposing 3-manifolds along spheres and tori, and hyperbolic and simplicial volume. The background talk takes place at 11:30 AM in in the Science Library (KINSC H305C),

• Wednesday April 5, 2017 at 14:30, Wachman 617
Counting and dynamics on the Markoff-Hurwitz variety

Michael Magee, Yale University

Abstract: I'll discuss some recent results on the Markoff-Hurwitz equation. I'll give some explanation about the fundamental relationship between this equation and geometry. We recently obtained a true asymptotic formula for the number of integer points of bounded height on the Markoff-Hurwitz variety in at least 4 variables. The previous best result here was by Baragar (1998) that gives a rough polylogarithmic rate of growth with a mysterious exponent of growth that is not in general an integer. As a consequence of our work we obtain an asymptotic formula for the number of one sided simple closed curves of given length on a certain hyperbolic thrice punctured projective plane. This is joint work with Gamburd and Ronan. If time permits I'll also report on recent work on the dynamics of pseudo-Anosov automorphisms of the Markoff surface over finite fields. This is joint work with undergraduate students Cerbu, Gunther and Peilen. I'll also try to point out some interesting open questions.

• Wednesday April 12, 2017 at 14:30, Wachman 617
Azumaya algebras and hyperbolic knots

Matthew Stover, Temple

I will talk about arithmetic geometry of SL(2,C) character varieties of hyperbolic knots. A simple criterion on roots of the Alexander polynomial determines whether or not a natural construction extends to determine a so-called Azumaya algebra on the so-called canonical component of the character variety, and I'll then explain how this forces significant restrictions on arithmetic invariants of Dehn surgeries on the knot. This is joint work with Ted Chinburg and Alan Reid.

• Wednesday April 19, 2017 at 14:30, Wachman 617
Weitzenböck Formulae and Sectional Curvature

Renato Bettiol, University of Pennsylvania

Classical geometric applications of Weitzenböck formulae establish that manifolds with positive Ricci curvature have vanishing first Betti number, while manifolds with negative Ricci curvature have no nontrivial Killing vector fields. In this talk, I will describe a framework to produce more general Weitzenböck formulae due to Hitchin, and derive two geometric applications that regard sectional curvature. The first implies a certain geometric restriction on 4-manifolds with positive sectional curvature and indefinite intersection form; the second provides a characterization of nonnegative sectional curvature in terms of Weitzenböck formulae for symmetric tensors. These methods potentially yield applications to negatively curved manifolds as well. This is joint work with R. Mendes (WWU Münster).

• Wednesday May 3, 2017 at 14:30, Wachman 1036
Deligne-Mostow lattices and cone metrics on the sphere

Irene Pasquinelli, Durham

Finding lattices in $PU(n,1)$ has been one of the major challenges of the last decades. One way of constructing lattices is to give a fundamental domain for its action on the complex hyperbolic space.

One approach, successful for some lattices, consists of seeing the complex hyperbolic space as the configuration space of cone metrics on the sphere and of studying the action of some maps exchanging the cone points with same cone angle.

In this talk we will see how this construction of fundamental polyhedra can be extended to almost all Deligne-Mostow lattices with 3-fold symmetry. Time permitting, we will see how this can be extended to Deligne-Mostow lattices with 2-fold symmetry (work in progress).

Please note the change of location this week.

• Wednesday May 3, 2017 at 16:00, Wachman 1036
The Poisson boundary for WPD actions

Giulio Tiozzo, University of Toronto

Abstract: Let $G$ be a group of isometries of a hyperbolic space $X$. If $X$ is not proper (e.g., a locally infinite graph), a weak form of properness is given by the WPD (weak proper discontinuity) condition, as defined by Bestvina-Bromberg-Fujiwara.

We consider random walks on groups which act weakly properly discontinuously on a hyperbolic space, and prove that the topological (Gromov) boundary is a model for the measure-theoretic (Poisson) boundary.

This provides as a corollary an identification of the Poisson boundary of $Out(F_n)$ without using the theory of outer space. Joint work with J. Maher.

• Wednesday September 6, 2017 at 14:45, Wachman 617
A McCarthy-type theorem for linearly growing outer automorphisms of F_n

Edgar Bering, Temple University

Abstract: In his proof of the Tits alternative for the mapping class group of a surface, McCarthy also proved that given any two mapping classes $\sigma$ and $\tau$, there exists an integer $N$ such that the group generated by $\sigma^N, \tau^N$ is either free of rank two or abelian. An analogous statement for two-generator subgroups of a linear group is false, due to the presence of the Heisenberg group. In the setting of $Out(F_n)$, whether or not such a statement is true remains open, though there are many partial results. In this talk I will give an overview of the problem in the context of the analogy among the three families of groups, survey previous work, and give some of the ideas in my proof of a McCarthy-type theorem for linearly growing outer automorphisms.

• Wednesday September 13, 2017 at 14:45, Wachman 617
Geometrically finite amalgamations of hyperbolic 3-manifold groups are not LERF

Hongbin Sun, Rutgers University

Abstract: We will show that, for any two finite volume hyperbolic 3-manifolds, the amalgamations of their fundamental groups along nontrivial geometrically finite subgroups are always not LERF. A consequence of this result is: all arithmetic hyperbolic manifolds with dimension at least 4, with possible exceptions in 7-dimensional manifolds defined by the octonion, their fundamental groups are not LERF.

• Wednesday September 27, 2017 at 14:45, Wachman 617
Free products and diffeomorphisms of compact manifolds

Thomas Koberda, University of Virginia

Abstract: It is a well-known fact that if $G$ and $H$ are groups of homeomorphisms of the interval or of the circle, then the free product $G*H$ is also a group of homeomorphisms of the interval or of the circle, respectively. I will discuss higher regularity of group actions, showing that if $G$ and $H$ are groups of $C^{\infty}$ diffeomorphisms of the interval or of the circle, then $G*H$ may fail to act by even $C^2$ diffeomorphisms on any compact one-manifold. As a corollary, we can classify the right-angled Artin groups which admit faithful $C^2$ actions on the circle, and recover a joint result with H. Baik and S. Kim. This is joint work with S. Kim.

• Friday October 6, 2017 at 15:00, Wachman 617
Introducing symplectic billiards

Sergei Tabachnikov, Penn State University

PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)

Abstract: I shall introduce a simple dynamical system called symplectic billiards. As opposed to the usual (Birkhoff) billiards, where length is the generating function, for symplectic billiards the symplectic area is the generating function. I shall explore basic properties and exhibit several similarities, but also differences, of symplectic billiards to Birkhoff billiards. Symplectic billiards can be defined not only in the plane, but also in linear symplectic spaces. In this multi-dimensional setting, I shall discuss the existence of periodic trajectories and describe the integrable dynamics of symplectic billiards in ellipsoids.

In the morning background talk (at 10am), I shall survey the conventional and outer billiards to provide context for my afternoon talk.

• Friday October 6, 2017 at 16:30, Wachman 617
Symplectic and exotic 4-manifolds via positive factorizations

Inanc Baykur, UMass

PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)

Abstract: We will discuss new ideas and techniques for producing positive Dehn twist factorizations of surface mapping classes which yield novel constructions of interesting symplectic and smooth 4-manifolds, such as small symplectic Calabi-Yau surfaces and exotic rational surfaces, via Lefschetz fibrations and pencils.

In the morning talk (11:30am), I will provide background on these topics.

• Wednesday October 11, 2017 at 14:45, Wachman 617
Deviation inequalities, Martin boundary, and equidistribution for random walks on relatively hyperbolic groups

Ilya Gekhtman, Yale University

Abstract: Consider a random walk $\mu$ on a finitely generated group $G$. The associated Green's metric is defined as minus log of the probability that a random trajectory starting at the first point ever reaches the second. The horofunction boundary of the Green metric is called the Martin boundary of $(G, \mu)$. Identifying the Martin boundary with some geometric boundary of $G$ is a difficult question with many dynamical applications. We show that the Martin boundary of a relatively hyperbolic group admits an equivariant surjection to the Bowditch boundary, with the preimage of conical points being a singleton. When the relatively hyperbolic group acts properly and cocompactly on a CAT(0) space, we show the Martin boundary coincides with the boundary of the CAT(0) space. The key technical result is that a random path between two points in a relatively hyperbolic group (e.g. a nonuniform lattice in hyperbolic space) has a uniformly high probability of passing any point on a word metric geodesic between them that is not inside a long subsegment close to a translate of a parabolic subgroup.

We derive some dynamical consequences:

-For a geometrically finite action with parabolics on a Gromov hyperbolic space the Patterson-Sullivan and harmonic measure are singular.

-For a geometrically finite action on a negatively curved manifold, the axes of loxodromic elements defined by random walk trajectories equidistribute with respect to a flow invariant measure on the unit tangent bundle (which when there are parabolics is singular to the measure of maximal entropy).

• Wednesday October 25, 2017 at 14:45, Wachman 617
Price inequality and Betti numbers of manifolds without conjugate points

Luca Di Cerbo, Stony Brook

In this talk, I will present a Price type inequality for harmonic forms on manifolds without conjugate points and negative Ricci curvature. The techniques employed in the proof work particularly well for manifolds of non-positive sectional curvature, and in this case one can prove a strengthened result. Equipped with these Price type inequalities, I then study the asymptotic behavior of Betti numbers along infinite towers of regular coverings. If time permits, I will discuss the case of hyperbolic manifolds in some detail. This is joint work with M. Stern.

• Wednesday November 1, 2017 at 14:45, Wachman 617
Recent results about Kauffman bracket skein algebras

Helen Wong, Institute for Advanced Study

Abstract: The definition of the Kauffman bracket skein algebra of an oriented surface was originally motivated by the Jones polynomial invariant of knots and links in space, and more precisely by Witten's topological quantum field theory interpretation of the Jones invariant. But the skein algebra is also closely related to the $SL_2 \mathbb C$ -character variety of the surface. We'll describe two seemingly different methods for constructing finite-dimensional representations of the skein algebra --- one uses combinatorial skein theory whereas the other comes from the quantum Teichmuller space. Very recently, Frohman, Le and Kania-Bartoszynska show that for generic representations, the two methods yield exactly the same representations. We'll discuss implications of this result and some of the many questions that remain.

• Wednesday November 8, 2017 at 14:45, Wachman 617
The geometry of outer automorphism groups of universal right-angled Coxeter groups

Charles Cunningham, Haverford College

Abstract: Abstract: We investigate the combinatorial and geometric properties of automorphism groups of universal right-angled Coxeter groups. McCullough-Miller space is a polyhedral complex which is virtually a geometric model for the outer automorphism group of a universal right-angled Coxeter group, $Out(W_n)$. As it is currently an open question as to whether or not $Out(W_n)$ is CAT(0) or not, it would be helpful to know whether McCullough-Miller space can always be equipped with an $Out(W_n)$-equivariant CAT(0) metric. We show that the answer is in the negative. This is particularly interesting as there are very few non-trivial examples of proving that a space of independent interest is not CAT(0).

• Wednesday November 15, 2017 at 14:45, Wachman 617
Lower bounds on cubical dimension of C(6) groups

Kasia Jankiewicz, McGill University

Abstract: I will discuss a construction which for each n gives an example of a finitely presented C(6) small cancellation group that does not act properly on any n-dimensional CAT(0) cube complex.

• Friday November 17, 2017 at 15:00, Sharpless 113, Haverford College
Exact Lagrangian cobordisms the Augmentation category

Yu Pan, MIT

PATCH seminar (joint with Bryn Mawr, Haverford, and Penn). There will be a background talk and an afternoon research talk.

Abstract (background): I will give a brief introduction of the Legendrian contact homology, which is an invariant of Legendrian knots $\Lambda$ defined in the spirit of Symplectic Field Theory. With the similar idea applied to a 2-copy of a Lagrangian filling of $\Lambda), the wrapped Floer homology gives an isomorphism between the linearized contact homology of \(\Lambda$ and the singular homology of the Lagrangian filling. At the end, I would like to mention an on-going project with Dan Rutherford about the wrapped Floer theory for immersed exact Lagrangian fillings.

Abstract (research): To a Legendrian knot, one can associate an $A_{\infty}$ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study the functor and establish a long exact sequence relating the corresponding cohomology of morphisms of the two ends. As applications, we prove that the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to the existence of exact Lagrangian cobordisms in terms of linearized contact homology and ruling polynomials.

• Friday November 17, 2017 at 16:30, Room Sharpless 113, Haverford College
Duality and semiduality in cohomology of arithmetic groups

Daniel Studenmund, Notre Dame University

PATCH seminar (joint with Bryn Mawr, Haverford, and Penn). There will be a morning background talk and an afternoon research talk.

Abstract (background): An arithmetic group acts naturally on a product of symmetric spaces and Euclidean buildings. We will discuss the examples of SL(2, Z) acting on the hyperbolic plane, SL(2, Z[sqrt(2)]) acting on a product of hyperbolic planes, and SL(2, F_p[t]) acting on a tree.

Abstract (research talk): A duality group has a pairing exhibiting isomorphisms between its homology and cohomology groups, analogous to Poincare duality for manifolds. Arithmetic groups over number fields form a large class of examples of duality groups, by work of Borel and Serre. Many naturally occurring groups fail to be duality groups, but are morally very close. In this talk we make this precise with the notion of a semiduality group, and sketch a proof that certain arithmetic groups in positive characteristic are semiduality groups, building on the result of Borel--Serre. This talk covers work joint with Kevin Wortman.

• Wednesday December 6, 2017 at 14:45, Wachman 617
TBA

Ben Bakker, UGA

• Wednesday December 13, 2017 at 14:45, Wachman 617
Tessellations from long geodesics on surfaces

Jenya Sapir, Binghamton University

Abstract: I will talk about a recent result of Athreya, Lalley, Wroten and myself. Given a hyperbolic surface S, a typical long geodesic arc will divide the surface into many polygons. We give statistics for the geometry of this tessellation. Along the way, we look at how very long geodesic arcs behave in very small balls on the surface.