2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016
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.
Drew Zemke, Cornell University
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.
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.
Nancy Hingston, The College of New Jersey
Andrew Hicks, Drexel University
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.
John Etnyre, Georgia Tech
Genevieve Walsh, Tufts University
-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.
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.
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.
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".
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.
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.
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.
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.
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.
2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016