2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023
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.
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.
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.
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.
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.
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.
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".
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.
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.
-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.
Genevieve Walsh, Tufts University
John Etnyre, Georgia Tech
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.
Andrew Hicks, Drexel University
Nancy Hingston, The College of New Jersey
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.
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.
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.
Mehdi Yazdi, Princeton University
Abstract: In 1976, Thurston proved that taut foliations on closed hyperbolic 3â€“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.
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.
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.
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.
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.
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 \(S\), \(\ell(X)=\ell(Y)\)).
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.
Nick Miller, Purdue University
Matthew Stover, Temple University
This is part II, where I will talk about character varieties in characteristic p.
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.
Mark Pengitore, Purdue University
In this talk, we give polynomial upper and lower bounds for conjugacy separability of cocompact lattices in nilpotent Lie groups.
Martin Deraux, UniversitÃ© 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.
2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023