“Average distances in the curve complex and in Teichmuller space”
Abstract:This talk will investigate the concept of average distance both in the curve complex and in Teichmuller space. More precisely, I will aim to address the question "What is the average distance between a pair of points in the sphere of radius R?" in both settings and to explain why the answer reflects a statistical sort of hyperbolicity. The bulk of the talk will be devoted to the curve complex setting, but time permitting I will also discuss related results for Teichmuller space. This is joint work with Moon Duchin and Howard Masur.
“Stable commutator length in cyclic amalgams”
Abstract: We show that stable commutator length is rational on free products of free abelian groups amalgamated over Z^k, a class of groups containing the fundamental groups of all torus knot complements. We consider a geometric model for these groups and parameterize all surfaces with specified boundary mapping to this space. Using this work we provide a topological algorithm to compute stable commutator length in these groups.
“Overtwisted discs in planer open books”
Abstract: For a planar open book, by using open book foliation technique, which is a generalization of Birman-Menasco's braid foliation machinery, we show that one can put an overtwisted disc in a (topologically) nice position. Such an overtwisted disc tells us how the monodromy of an open book twists curves. As a corollary, we give a topologicial argument for certain planar open books to support a tight contact structure.
“The solution to Siegel's Problem on small covolume lattices”
Abstract:We outline the history and the proof of the identification of the minimal covolume lattice of hyperbolic 3-space as the 3-5-3 Coxeter group extended by the involution preserving the symmetry of this diagram. This solves (in three dimensions) the problem posed by Siegel in 1945 (Siegel solved this problem in two dimensions by deriving the signature formula identifying the (2,3,7)-triangle group as having minimal co-area). There are strong connections with arithmetic hyperbolic geometry in the proof and the result has applications in the maximal symmetry groups of hyperbolic 3-manifolds (in much the same way that Hurwitz 84g-84 theorem and Siegel's result do).
November 8, 9:30am:
“Exploring parameterised complexity in computational topology”
Abstract: Topological decision problems in three dimensions are notoriously difficult: the mere existence of an algorithm is often a major result, and many important algorithms have yet to be studied from a complexity viewpoint. Even "simple" problems, such as unknot recognition (testing whether a loop of string is knotted), or 3-sphere recognition (testing whether a triangulated 3-manifold is topologically trivial), have best-known algorithms that are worst-case exponential time. In practice, however, some of these algorithms run surprisingly well in experimental settings, and we discuss some reasons why parameterised complexity now looks to be the "right" tool to explain this behaviour. We outline some initial forays into the parameterised complexity of topological problems, including both tractability and hardness results.
November 8, 1:15pm:
“Vanishing and nonvanishing in the high-dimensional cohomology of
Abstract: For O_k a ring of integers and p close to the vcd of SL_n(O_k), I will discuss a sequence of results and conjectures about H^p(SL_n(O_k);Q). This is joint work with Tom Church and Benson Farb.
November 15, 9:30am:
“Compressed words in Gromov hyperbolic groups”
Abstract: Markus Lohrey defined the compressed word problem for groups and gave a polynomial-time solution in the case of free groups. I'll explain how this leads to a polynomial-time algorithm to the usual word problem for automorphism groups of free groups. Time permitting, I will sketch a few of the ideas needed to go from the free group to Gromov hyperbolic groups in general. As a nice application, this gives a polynomial-time solution to the word problem for the mapping class group; this new solution is very different from the usual one!
November 15, 1:15pm:
“Congruence subgroups of braid groups”
Abstract: The integral Burau representation gives a symplectic representation of the braid group. In this talk we will discuss the resulting congruence subgroups of braid groups, that is, preimages of the principal congruence subgroups of the symplectic group. In particular, we will show that the level 4 congruence braid group is generated by squares of Dehn twists. One key tool is a "squared lantern relation" amongst Dehn twists. This is joint work with Dan Margalit.
“Constructing non-arithmetic lattices”
Abstract:I will survey the relationship between arithmetic groups and lattices in semi-simple Lie groups; the construction of non-arithmetic lattices in SU(n,1) for n=2, 3 by Mostow and Deligne in the 1980s. I will then describe a joint project with Martin Deraux and Julien Paupert to construct new examples in SU(2,1).
“A family of curve complexes and Chillingworth's winding numbers”
Abstract: Curve complexes have traditionally been used to study the mapping class group of a surface. For the Torelli group, i.e. the subgroup of the mapping class group that acts trivially on homology, standard techniques do not apply. In this talk it will be shown that a family of oriented curve complexes give a "linear approximation" to the Torelli group in the following way: the stable lengths of an element of the Torelli group acting on the family of curve complexes define a cohomology class on the surface, proportional to the Chillingworth class. The Chillingworth class is the dual of a tensor contraction of the Johnson homomorphism.
“Relative hyperbolicity of (random) right-angled Coxeter groups”
Abstract: Right-angled Coxeter groups (RACGs) form a rich class of groups inspired by groups generated by reflections across orthogonal axes, while relative hyperbolicity can be defined in terms of the existence of an "interesting" proper action on a Gromov-hyperbolic space. I will discuss the fact that relative hyperbolicity of a given RACG can be detected algorithmically reasonably quickly. Also, I will present a randomized model for RACGs that, depending on a certain parameter, generates either relatively hyperbolic or non-relatively hyperbolic RACGs with asymptotic probability 1. Joint with Jason Behrstock and Mark Hagen.
Other relevant information.Spring 2013, Fall 2012, Spring 2012, Fall 2011, 2010/11, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006.
- Columbia Symplectic Geometry/Gauge Theory Seminar
- All Columbia Math Dept Seminars
- CUNY Geometry and Topology Seminar
- CUNY Complex Analysis & Dynamics Seminar
- CUNY Magnus Seminar
- Princeton Topology Seminar.