# Geometry and Topology Seminar 2013

Current contact: Dave Futer or Matthew Stover

The Seminar usually takes place on Wednesdays at 2:45 PM in Room 617 on the sixth floor of Wachman Hall.

• Tuesday January 22, 2013 at 17:00, Wachman 617
GeoTop Seminar

Jason Behrstock, CUNY Lehman College, Divergence, thick groups, and Morse geodesics

In a metric space the divergence of a pair of rays is a way to measure how quickly they separate from each other. Understanding what divergence rates are possible in the presence of non-positive curvature was raised as a question by Gromov and then refined by Gersten. We will describe a construction of groups with several interesting properties, some of which shed light on the above question. (Joint work with Cornelia Drutu.)

• Thursday January 31, 2013 at 13:30, Wachman 527
GeoTop Seminar

-Note different day, location, and time-

Justin Malestein, Hebrew University of Jerusalem, Pseudo-Anosov density and dilatations in the Torelli groups

I will briefly discuss results/proofs relating to density of pseudo-Anosov mapping classes in the Torelli group. Then, I will discuss a method for estimating dilatations of pseudo-Anosovs from below. One result obtained via this method is an explicit lower bound for dilatations of a pseudo-Anosov in terms of its containment in a "higher Torelli" group. Specifically, an explicit function $f(k)$ will be exhibited such that the dilatation is at least $k$ if the mapping class acts trivially modulo the $k$-th step nilpotent quotient of the fundamental group of the surface.

• Thursday January 31, 2013 at 16:00, PATCH seminar, at Haverford College, KINSC H108
GeoTop Seminar

John Harer, Duke University, Topology, geometry and statistics: Merging methods for data analysis

Dimension reduction and shape description for scientifi c datasets are difficult problems, ones that continue to grow in importance within the statistical, mathematical and computer science communities. Powerful new methods of Topological Data Analysis (TDA) have emerged in the last 10 years, and these have added signi ficantly to the data analysis toolbox.

In this talk we will give an overview of these methods and describe some early efforts to make them work together with statistical approaches. In particular we will discuss how one can use topological priors in data analysis and how TDA applies to the study of shape in point clouds, dimension reduction, time varying data and finding quasi-periodic patterns in signals.

• Thursday January 31, 2013 at 17:30, PATCH seminar, at Haverford College, KINSC H108
GeoTop Seminar

Zoltan Szabo, Princeton University, Knot Floer homology and bordered algebras

In the talk, I will describe a new algebraic method that computes knot Floer homology for knots in the 3-sphere. This is joint work with Peter Ozsvath.

• Tuesday February 12, 2013 at 13:30, Wachman 527
GeoTop Seminar

-Note different place and time-

Kei Nakamura, Temple University, On Isosystolic inequalities and Z/2Z-homology

The systole $\mathrm{Sys}(M,g)$ of a Riemannian manifold $(M,g)$, is the length of the shortest geodesic loop. Given a smooth closed $n$-manifold $M$, an isosystolic inequality is a metric-independent inequality of the form $(\mathrm{Sys}(M,g))^n \leq C \mathrm{Vol}(M,g)$, where the constant $C$ is independent of Riemannian metric $g$ on $M$.

We show that, for any closed smooth $n$-manifold $M$ satisfying a certain homological/cohomological condition, the isosystolic inequality with constant $C=n!$ holds: for every Riemannian metric $g$ on $M$, $(\mathrm{Sys}(M,g))^n \leq n! \mathrm{Vol}(M,g)$. Our inequality can be regarded as a generalization of the inequality by Hebda and Burago, as well as a refinement of the inequality by Guth.

We show that the inequality readily applies to certain compact space forms and geometric 3-manifolds. We also generalize the inequality to some open manifolds, and derive the analogous inequality for all closed aspherical 3-manifolds.

• Tuesday February 19, 2013 at 17:00, Wachman 617
GeoTop Seminar

Mark Feighn, Rutgers University, The complex of free factors

Let $F_n$ be a free group of rank $n$. The complex of free factors $X$ of $F_n$ is the simplicial complex whose vertices are conjugacy classes $V$ of proper free factors of $F_n$ and whose simplices are determined by chains $V_1 < ...< V_k$. The outer automorphism group $Out(F_n)$ acts simplicially on $X$, and $X$ acts as an analogue of the curve complex of a compact surface with its action by the mapping class group. In seminal work, Masur and Minsky proved that the curve complex is hyperbolic.

I will discuss joint work with Mladen Bestvina showing that $X$ is hyperbolic. If time permits, I will also discuss further developments.

• Tuesday March 5, 2013 at 17:00, Wachman 617
GeoTop Seminar

Matthew Stover, University of Michigan, A whirlwind introduction to complex hyperbolic geometry

After hyperbolic 2-manifolds, which are quotients of the Poincare disk, a natural next step is the study of hyperbolic 3-manifolds, which received a great deal of attention in the past 30 years. Another natural generalization that retains the complex analytic structure so often useful in the study of hyperbolic 2-manifolds is the complex hyperbolic plane. After discussing the basics of complex hyperbolic space, I will draw several tantalizing parallels between the geometry and topology of complex hyperbolic 2-manifolds and hyperbolic 3-manifolds, then give some indication as to what progress has been made. This talk will be low on proofs and heavy on analogies.

• Tuesday March 26, 2013 at 17:00, Wachman 617
GeoTop Seminar

Tarik Aougab, Yale University, Effective results in curve graph geometry

In this talk, we are interested in studying how the geometry of the curve graph explicitly depends on the genus of the underlying surface. For example: how many times must a pair of curves intersect on the genus $g$ surface in order to be distance $k$ in the genus $g$ curve graph? We'll answer this question and (time permitting) we'll discuss how it is used to prove other effective curve graph results, such as:

1. All curve graphs are $k$-hyperbolic for some fixed $k$;
2. The disk set is a $O(g^{2})$-quasiconvex subset of the genus $g$ curve graph.

• Tuesday April 2, 2013 at 17:00, Wachman 617
GeoTop Seminar

Moira Chas, Stony Brook University, Normal distributions related curves on surfaces

In an orientable surface with boundary, free homotopy classes of closed, oriented curves on surfaces are in one to one correspondence with cyclic reduced words in a minimal set of generators of the fundamental group.

Given a cyclic reduced word, there are algorithms to compute the self-intersection of the corresponding free homotopy class (that is, the smallest number of self-crossings of a curve in the class, counted by multiplicity).

With the help of the computer, one can make a histogram of how many free homotopy classes of twenty letters have self-intersection 0, 1, 2,.... The obtained histogram is essentially Gaussian.

This experimental result led us to the following theorem, (joint with Steve Lalley): If a free homotopy class of curves is chosen at random from among all classes of L letters, then for large L the distribution of the self-intersection number approaches a Gaussian distribution.

The goal of this talk will be to discuss this theorem as well as related results and conjectures.

• Thursday April 4, 2013 at 16:30, PATCH seminar, at Penn, DRL room 4C8
GeoTop Seminar

Shea Vela-Vick, Louisiana State University, Transverse knots, branched covers and Heegaard Floer homology.

In recent years, Heegaard Floer theory has proven an invaluable tool for studying contact manifolds and the Legendrian and transverse knots they contain. After surveying a bit about the connections between transverse knot theory and branched coverings, I will discuss a method for defining a variant of Heegaard Floer theory for infinite cyclic covers of transverse knots in the standard contact 3-sphere. This invariant takes the form of a $Z[t,t^-1]$-module and generalizes one defined in joint work with Baldwin and Vertesi for transverse knots braided about open book decompositions. In this talk, I will discuss how our invariant is constructed and present some basic properties. This is joint work with Tye Lidman and Sucharit Sarkar.

• Thursday April 4, 2013 at 18:00, PATCH seminar, at Penn, DRL room 4C8
GeoTop Seminar

Bruce Kleiner, New York University, Mean convex mean curvature flow

In spite of much progress, our basic understanding of mean curvature flow is in some respects still lacking, apart from the case of curves in the plane. However, beautiful work of White and Huisken-Sinestrari in the last 10 years has shown that there is a far-reaching structure and regularity theory in the case of mean convex (i.e. positive mean curvature) mean curvature flow. After presenting some background, I will discuss joint work with Robert Haslhofer, which gives a new approach to mean convex flow that is substantially simpler and shorter than the original.

• Tuesday April 9, 2013 at 17:00, Wachman 617
GeoTop Seminar

Babak Modami, Yale University, Prescribing the behavior of Weil-Petersson geodesics

The Weil-Petersson (WP) metric is an incomplete Riemannian metric on the moduli space of Riemann surfaces with negative sectional curvatures which are not bounded away from 0. Brock, Masur and Minsky introduced a notion of "ending lamination" for WP geodesic rays which is an analogue of the vertical foliations of Teichmuller geodesics. In this talk we show that these laminations and the associated subsurface coefficients can be used to determine the itinerary of a class of WP geodesics in the moduli space. As a result we give examples of closed WP geodesics staying in the thin part of of the moduli space, geodesic rays recurrent to the thick part of the moduli space and diverging geodesic rays. These results can be considered as a kind of symbolic coding for WP geodesics.

• Tuesday April 23, 2013 at 17:00, Wachman 617
GeoTop Seminar

Abstract: Let $S$ be a surface with punctures, and let $f \in Mod(S)$ be a pseudo-Anosov mapping class. Associated to f is an attracting lamination $L$, which is the limit under the forward orbit of $f$ of any closed curve on $S$. We address the following question - is there a natural way to associate to $L$ some natural object in the homology of $S$? If so, can it be described using some limiting process? What would such an object tell us about $f$? We show that there is indeed such an object, and that it possesses a surprising amount of structure. For instance, if $f$ is in the Torelli group, then the homological lamination will be a convex polyhedron with rational vertices.

• Tuesday April 30, 2013 at 16:00, Wachman 617
GeoTop Seminar

PATCH seminar, joint with Bryn Mawr, Haverford, and Penn

Daniel Wise, McGill University, From riches to RAAGs: 3-manifolds, cubes, and right-angled Artin groups

Cube complexes have come to play an increasingly central role within geometric group theory, as their connection to right-angled Artin groups provides a powerful combinatorial bridge between geometry and algebra. This talk will describe the developments in this theory that have recently culminated in the resolution of the virtual Haken conjecture for 3-manifolds, and simultaneously dramatically extended our understanding of many infinite groups.

• Tuesday April 30, 2013 at 17:30, Wachman 617
GeoTop Seminar

PATCH seminar, joint with Bryn Mawr, Haverford, and Penn

Josh Greene, Boston College, Conway mutation and alternating links

I will discuss the proof, context, and consequences of the following result: a pair of reduced, alternating diagrams for a pair of links are mutants iff the Heegaard Floer homology of the links' branched double covers are isomorphic.

• Tuesday September 3, 2013 at 16:00, Wachman 617
GeoTop Seminar

Pre-talk at 3:30pm

Lisa Traynor, Bryn Mawr College, The geography of Lagrangian cobordisms

In topology, cobordisms define a fundamental equivalence relation on the set of compact manifolds: two compact, n-dimensional manifolds are cobordant if their disjoint union is the boundary of a (n+1)-dimensional manifold. I will discuss cobordisms that satisfy extra geometrical conditions imposed by symplectic and contact structures. Namely, I will discuss Lagrangian cobordisms between Legendrian manifolds. In contrast to the smooth setting, this cobordism relation no longer defines an equivalence relation on the set of Legendrian submanifolds. There are numerous interesting "geography" questions about the existence of Lagrangian cobordisms. I will discuss some obstructions to and constructions of Lagrangian cobordisms that give some geographic information.

• Tuesday September 10, 2013 at 16:00, Wachman 617
GeoTop Seminar

Pre-talk at 3:30pm

Daniel Studenmund, University of Chicago, Abstract commensurators of lattices in Lie groups

The abstract commensurator of a group G is the group of all isomorphisms between finite index subgroups of G up to a natural equivalence relation. Commensurators of lattices in semisimple Lie groups are well understood, using strong rigidity results of Mostow, Prasad, and Margulis. We will describe commensurators of lattices in solvable groups, where strong rigidity fails. If time permits, we will extend these results to lattices in certain groups that are neither solvable nor semisimple.

• Tuesday September 24, 2013 at 14:30, Wachman 617
GeoTop Seminar

-Note different time-

Jeff Danciger, UT Austin, Margulis spacetimes

Margulis found the first examples of complete affine manifolds with non-solvable fundamental group. Each of these manifolds, now called Margulis spacetimes, is equipped with a flat Lorentzian metric compatible with the affine structure. This talk will survey some recent work, joint with François Guéritaud and Fanny Kassel, which studies these flat spacetimes as limits of their negative curvature relatives, anti de Sitter (AdS) spacetimes. In particular, we prove the tameness conjecture for Margulis spacetimes and also give a parameterization of their moduli.

• Tuesday October 1, 2013 at 16:00, Wachman 617
GeoTop Seminar

Matthew Stover, Temple University, Moduli of flat tori I

This will be a multi-part introduction to how one parameterizes geometric structures, focusing on the space of flat n-tori. These talks will be aimed at a general audience, e.g., any graduate student with some very basic exposure to topology.

• Tuesday October 8, 2013 at 16:00, Wachman 617
GeoTop Seminar

Matthew Stover, Temple University, Moduli of flat tori II

This will be a multi-part introduction to how one parameterizes geometric structures, focusing on the space of flat n-tori. These talks will be aimed at a general audience, e.g., any graduate student with some very basic exposure to topology.

• Friday October 11, 2013 at 14:30, Wachman 617
GeoTop Seminar

-Note different day and time-

Ben McReynolds, Purdue University, Primitive lengths and arithmetic progression

Lengths of primitive closed geodesics on a closed negatively curved manifold and prime ideals in number fields share many common features. In this talk, I will discuss a few results, both old and new, that illustrates this connection. This talk is based on work joint with Jean Lafont.

• Tuesday October 15, 2013 at 16:00, Wachman 617
GeoTop Seminar

Matthew Stover, Temple University, Moduli of flat tori III

This will be a multi-part introduction to how one parameterizes geometric structures, focusing on the space of flat n-tori. These talks will be aimed at a general audience, e.g., any graduate student with some very basic exposure to topology.

• Tuesday October 29, 2013 at 16:00, Wachman 617
GeoTop Seminar

Jessica Purcell, BYU, Twisted checkerboard surfaces

Checkerboard surfaces in alternating knot complements have been used for many years to determine information about the knot. However, checkerboard surfaces become increasingly complicated as higher numbers of crossings are added to a knot diagram. When more and more crossings are added to a single twist of the diagram, the geometry of the knot complement begins to stabilize (it approaches a geometric limit), but the corresponding checkerboard surfaces continue to increase in complexity (area and genus). In this talk, we will discuss a generalization of checkerboard surfaces, called twisted checkerboard surfaces, which better reflect the geometric complexity of an alternating knot. We will construct the surfaces, discuss their geometric properties, and give some consequences. This is joint work with Marc Lackenby.

• Tuesday November 5, 2013 at 16:00, Wachman 617
GeoTop Seminar

Sam Taylor, University of Texas, Convex cocompactness in Mod(S) and generalizations to Out(Fn)

We discuss convex cocompactness in the mapping class group and focus on two well-studied open questions. We show how a potential approach to these problems involves right-angled Artin groups and explain how the Out(Fn) version of these questions may be more approachable. To do this, we describe some new tools to study the geometry of Out(Fn).

• Tuesday November 26, 2013 at 16:00, Wachman 617
GeoTop Seminar

Pre-talk at 3:30pm

Hongbin Sun, Princeton University, Virtual homological torsion of closed hyperbolic 3-manifolds.

We will use Kahn-Markovic's almost totally geodesic surfaces to construct certain $\pi_1$-injective 2-complexes in closed hyperbolic 3-manifolds. Such 2-complexes are locally almost totally geodesic except along a 1-dimensional subcomplex. Using Agol and Wise's result that fundamental groups of hyperbolic 3-manifolds are LERF and quasi-convex subgroups are virtual retracts, we will show that closed hyperbolic 3-manifolds virtually contain any prescribed homological torsion: For any finite abelian group A, and any closed hyperbolic 3-manifold M, there exists a finite cover N of M, such that A is a direct summand of $Tor(H_1(N; Z))$.

• Tuesday December 3, 2013 at 16:00, Wachman 617
GeoTop Seminar

Jonah Gaster, University of Illinois at Chicago, A non-injective skinning map with a critical point

Following Thurston, certain classes of 3-manifolds yield holomorphic maps on the Teichmuller spaces of their boundary components. Inspired by numerical evidence of Kent and Dumas, we present a negative result about these maps. Namely, we construct a path of deformations of a hyperbolic structure on a genus-2 handlebody with two rank-1 cusps. We exploit an orientation-reversing isometry to conclude that the skinning map sends a specied path to itself, and use estimates on extremal length functions to show non-monotonicity and the existence of a critical point. Time permitting, we will indicate some surprising unexplained symmetry that comes out of our calculations.