# Geometry and Topology Seminar 2016

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 20, 2016 at 14:30, Wachman 617
Abundant quasifuchsian surfaces in cusped hyperbolic 3-manifolds

Dave Futer, Temple University

I will discuss a proof that a cusped hyperbolic 3-manifold M contains an abundant collection of immersed, quasifuchsian surfaces. These surfaces are abundant in the sense that their boundaries separate any pair of points on the sphere at infinity. As a corollary, we recover Wise's theorem that the fundamental group of M is cubulated. This is joint work with Daryl Cooper.

• Wednesday January 27, 2016 at 14:30, Wachman 617
Topological constructions of manifolds with geometric structures

Matthew Stover, Temple University

Classical uniformization implies that the existence of a complete hyperbolic metric on a Riemann surface depends only on its topological type. In dimension 3, Thurston's geometrization program also gives a necessary and sufficient topological condition. I will discuss topological methods for proving existence of a metric of constant holomorphic sectional curvature -1 on the complement of curves in a smooth complex projective surface. I will mainly focus on an interesting example due to Hirzebruch, and hopefully turn to some applications of these topological constructions, e.g., to questions about betti number growth. This is mostly joint with Luca Di Cerbo.

• Wednesday February 3, 2016 at 14:30, Wachman 617
Spacious knots

Richard Kent, University of Wisconsin

Brock and Dunfield showed that there are integral homology spheres whose thick parts are very thick and take up most of the volume. Precisely, they show that, given $R$ big and $r$ small, there is an integral homology 3-sphere whose $R$-thick part has volume $(1-r) vol(M)$. Purcell and I find knots in the 3-sphere with this property, answering a question of Brock and Dunfield.

• Wednesday February 10, 2016 at 14:30, Wachman 617
The topology of local commensurability graphs

Khalid Bou-Rabee, City College of New York

The p-local commensurability graph (p-local graph) of a group has vertices consisting of all finite-index subgroups, where an edge is drawn between two subgroups if their commensurability index is a power of p. Sitting at the interface between intersection graphs, containment graphs, and commensurability, these p-local graphs give insights to Lubotzky-Segal's subgroup growth functions. In this talk, we connect topological properties of p-local graphs to nilpotence, solvability, and largeness (containing a free subgroup of finite index) of the target group. This talk covers joint work with Daniel Studenmund and Chen Shi.

• Wednesday February 24, 2016 at 14:30, Wachman 617
Compactifying spaces of Riemannian manifolds, with applications

Ian Biringer, Boston College

We will describe how to compactify sets of Riemannian manifolds with constrained geometry (e.g. locally symmetric spaces), where the added limit points are transverse measures on some universal foliated space. As an application, we study the ratio of the $k$-th Betti number of a manifold to its volume, and give a strong convergence result for higher rank locally symmetric spaces.

• Friday February 26, 2016 at 15:00, Wachman 617
Knot contact homology and string topology

Lenny Ng, Duke University [PATCH seminar, joint with Bryn Mawr, Haverford, and Penn]

Symplectic geometry has recently emerged as a key tool in the study of low-dimensional topology. One approach, championed by Arnol'd, is to examine the topology of a smooth manifold through the symplectic geometry of its cotangent bundle, building on the familiar concept of phase space from classical mechanics. I'll describe a way to use this approach, combined with the modern theory of Legendrian contact homology (which I'll also introduce), to construct a rather powerful invariant of knots called "knot contact homology".

• Friday February 26, 2016 at 16:30, Wachman 617

Josh Greene, Boston College [PATCH seminar, joint with Bryn Mawr, Haverford, and Penn]

I will describe a characterization of alternating links in terms intrinsic to the link exterior and use it to derive some properties of these links, including algorithmic detection and new proofs of some of Tait's conjectures.

• Wednesday March 9, 2016 at 14:30, Wachman 617
$k$-geodesics and lifting curves simply

Tarik Aougab, Brown University

Let $\gamma$ be a closed curve on a surface $S$ with negative Euler characteristic, and suppose gamma has at most $k$ self-intersections. We construct a hyperbolic metric with respect to which $\gamma$ has length (on the order of) $\sqrt{k}$, and whose injectivity radius is bounded below by $1/\sqrt{k}$; these results are optimal. As an application, we give sharp upper bounds on the minimum degree of a cover for which gamma lifts to a simple closed curve. This is joint work with Jonah Gaster, Priyam Patel, and Jenya Sapir.

• Friday March 18, 2016 at 13:30, Wachman 617
Counting curves on hyperbolic surfaces

-Note different day and time-

Viveka Erlandsson, Aalto University

In this talk I will discuss the growth of the number of closed geodesic of bounded length, and the length grows. More precisely, let $c$ be a closed curve on a hyperbolic surface $S=S(g,n)$ and let $N_c(L)$ denote the number of curves in the mapping class orbit of $c$ with length bounded by $L$. Mirzakhani showed that when $c$ is simple, this number is asymptotic to $L^{6g-6+2n}$. Here we consider the case when $c$ is an arbitrary closed curve, i.e. not necessarily simple. This is joint work with Juan Souto.

• Friday March 25, 2016 at 14:00, PATCH seminar, at Bryn Mawr, Park Science Building room 328

Genevieve Walsh, Tufts University

• Friday March 25, 2016 at 16:00, PATCH seminar, at Bryn Mawr, Park Science Building room 328

John Etnyre, Georgia Tech

• Wednesday April 6, 2016 at 14:30, Wachman 617
Veering Dehn surgery

Saul Schleimer, University of Warwick

It is a theorem of Moise that every three-manifold admits a triangulation, and thus infinitely many. Thus, it can be difficult to learn anything really interesting about the three-manifold from any given triangulation. Thurston introduced ideal triangulations'' for studying manifolds with torus boundary; Lackenby introduced taut ideal triangulations'' for studying the Thurston norm ball; Agol introduced veering triangulations'' for studying punctured surface bundles over the circle. Veering triangulations are very rigid; one current conjecture is that any fixed three-manifold admits only finitely many veering triangulations.

After giving an overview of these ideas, we will introduce veering Dehn surgery''. We use this to give the first infinite families of veering triangulations with various interesting properties. This is joint work with Henry Segerman.

• Thursday April 14, 2016 at 16:30, PATCH seminar, at Penn, DRL room 4C8
Controlling Ray Bundles with Reflectors

Andrew Hicks, Drexel University

• Thursday April 14, 2016 at 17:30, PATCH seminar, at Penn, DRL room 4C8
Loop products, index growth, and closed geodesics

Nancy Hingston, The College of New Jersey

• Wednesday April 20, 2016 at 14:30, Wachman 617
Unsmoothable group actions on one-manifolds

Thomas Koberda, University of Virginia

I will discuss virtual mapping class group actions on compact one-manifolds. The main result will be that there exists no faithful $C^2$ action of a finite index subgroup of the mapping class group on the circle, which generalizes results of Farb-Franks and establishes a higher rank phenomenon for the mapping class group, mirroring a result of Ghys and Burger-Monod.

• Wednesday April 27, 2016 at 14:30, Wachman 617
The simple loop conjecture for 3-manifolds modeled on Sol

Drew Zemke, Cornell University

The simple loop conjecture for 3-manifolds states that every 2-sided immersion of a closed surface into a 3-manifold is either injective on fundamental groups or admits a compression. This can be viewed as a generalization of the Loop Theorem to immersed surfaces. We will give a brief history of this problem and outline a solution when the target 3-manifold admits a geometric structure modeled on Sol.

• Thursday August 18, 2016 at 14:30, Wachman 617
Rips complex for relatively hyperbolic groups

Piotr Przytycki, McGill University

We will describe a Rips complex, a thickening of the Cayley graph of a relatively hyperbolic group G, with a graph-theoretic property called dismantlability. This guarantees fixed-point properties and implies that the Rips complex is a classifying space for G (with respect to appropriate family). This is joint work with Eduardo Martinez-Pedroza.

• Wednesday September 14, 2016 at 14:30, Wachman 617
On Thurston's Euler class one conjecture

Mehdi Yazdi, Princeton University

Abstract: In 1976, Thurston proved that taut foliations on closed hyperbolic 3–manifolds have Euler class of norm at most one, and conjectured that, conversely, any Euler class with norm equal to one is Euler class of a taut foliation. I construct counterexamples to this conjecture and suggest an alternative conjecture.

• Friday September 23, 2016 at 14:30, Bryn Mawr College, Park Science Building 328
Legendrian graph surfaces (PATCH)

Roger Casals, MIT

In this talk we discuss Legendrian surfaces in the standard contact 5-sphere. The goal is to present ideas relating cubic planar graphs and Legendrian surfaces, elaborating on earlier work of E. Zaslow and D. Treumann. In particular, we will talk about Legendrian singularities, count trees and introduce a combinatorial invariant in graph theory. This is work in progress with E. Murphy.

There will also be a background talk at 9:30 AM.

• Friday September 23, 2016 at 16:00, Bryn Mawr College, Park Science Building 328
Packings of hyperbolic surfaces (PATCH)

Jason DeBlois, University of Pittsburgh

In the background talk (11:00 AM), I'll introduce packing problems in general and some famous packing problems in particular. I'll discuss the related meshing problem, some of its standard solutions the Delaunay and Voronoi triangulations, and some of their advantages and shortcomings.

In the research talk (4:00 PM), I'll specialize to the problem of packing disks on complete hyperbolic surfaces of finite area. I'll exhibit the best density bounds that I know, and I'll show that they are sharp in some cases and not sharp in others.

• Wednesday October 5, 2016 at 14:30, Wachman 527
Surface group actions on products of trees

Matthew Stover, Temple University

Let G be the fundamental group of a closed Riemann surface of genus g > 1. Does G admit a properly discontinuous action on a (finite) product of (finite-valence) trees? This remains open. I will discuss a number of results, joint with David Fisher, Michael Larsen, and Ralf Spatzier, related to this question.

• Wednesday October 19, 2016 at 14:30, Wachman 617
Finding geodesics in the curve graph

Mark Bell, University of Illinois

The curve graph associated to a surface records the pairs of essential closed curves that are disjoint. The graph is connected but, unfortunately, locally infinite. Thus standard pathfinding algorithms struggle to compute paths through this graph. We will discuss some of the techniques of Leasure, Shackleton, Watanabe and Webb for overcoming this local infiniteness, enabling geodesics to be constructed.

We will finish with a new refinement that allows such geodesics to be found in polynomial time (in terms of their length). An important corollary of which, is a new (polynomial-time) algorithm to determine the Nielsen--Thurston type of a mapping class via its action on the curve graph. This is joint work with Richard Webb.

• Friday October 21, 2016 at 14:00, Wachman 617
Computer driven questions, pre-theorems and theorems in geometry (PATCH)

Moira Chas, Stony Brook University

In the background talk (9:30-10:30am), I will introduce several numbers can be associated to free homotopy class $X$ of closed curves on a surface $S$, with boundary and negative Euler characteristic. Among these are:

- the self-intersection number of $X$ (this is the smallest number of times a representative of the X crosses itself),

- the word length of $X$ (given a minimal set of generators of the fundamental group, this is the smallest number of generators in a word representing the deformation or conjugacy class) and

- the length of the geodesic corresponding to $X$ (given a hyperbolic metric on $S$ with geodesic boundary)

- the number of free homotopy classes of a given word length the mapping class group orbit of $X$.

The interrelations of these numbers exhibit many patterns when explicitly determined or approximated by running a variety of algorithms in a computer.

In the research talk (2:00-3:00pm), we will discuss how these computations lead to counterexamples to existing conjectures and to the discovery of new patterns . Some of these new patterns, so intricate and unlikely that they are certainly true (even if not proven yet), are "pre-theorems". Many of these pre-theorems later became theorems. An example of such a theorem states that the distribution of the self-intersection of free homotopy classes of closed curves on a surface, appropriately normalized, sampling among given word length, approaches a Gaussian when the word length goes to infinity. An example of a counterexample (no pun untended!) is that there exists pairs of length equivalent free homotopy classes of curves on a surface S that have different self-intersection number. (Two free homotopy classes $X$ and $Y$ are length equivalent if for every hyperbolic metric on $S$, $\ell(X)=\ell(Y)$).

• Friday October 21, 2016 at 15:30, Wachman Hall 617
Cellular Sheaves in Applications (PATCH)

Robert Ghrist, University of Pennsylvania

Background talk (11am-12pm): Homological Inference

In this background talk, we'll recall what makes homological methods work so well for problems of inference (in Science as well as in Mathematics): the fundamentals of functoriality, exactness, and naturality, are the engines of inference. We'll show what basic commutative diagrams can do by demonstrating a new proof of the classic Hex Theorem from game theory using only exactness and diagram chasing.

Research talk (3:30-4:30pm): Cellular Sheaves in Applications

In this talk, I'll argue that the recent advances in applied algebraic topology (persistent homology especially) point to cellular co/sheaves as good structures for modelling data tethered to spaces; and co/homology as an especially useful compression of such data. I'll survey a few simple applications, then dig into one less-simple application from game theory.

• Thursday October 27, 2016 at 11:30, Wachman 527
Arithmetic progressions in the primitive length spectrum

Nick Miller, Purdue University

• Wednesday November 2, 2016 at 14:30, Wachman 617
Surface group actions on products of trees II

Matthew Stover, Temple University

This is part II, where I will talk about character varieties in characteristic p.

• Wednesday November 9, 2016 at 14:30, Wachman 617
Essential surfaces from intersections in the character variety

Michelle Chu, University of Texas

I will describe the SL2(C) character variety for a family of hyperbolic two-bridge knots. These character varieties have multiple components which intersect at points corresponding to non-integral irreducible representations. As such, these points carry lots of interesting topological information. In particular, they are associated to splittings along Seifert surfaces.

• Wednesday November 30, 2016 at 14:30, Wachman 617
Effective Conjugacy Separability of Lattices in Nilpotent Lie groups

Mark Pengitore, Purdue University

In this talk, we give polynomial upper and lower bounds for conjugacy separability of cocompact lattices in nilpotent Lie groups.

• Wednesday December 14, 2016 at 14:30, Waschman 617
Non-arithmetic lattices

Martin Deraux, Université Grenoble Alpes

I will present joint work with Parker and Paupert, that allowed us to exhibit new commensurability classes of non-arithmetic lattices in the isometry group of the complex hyperbolic plane. If time permits, I will also explain close ties between our work and the theory of discrete reflection groups acting on other 2-dimensional complex space forms.