- You are here:
- GT Home
- Home
- News & Events

Series: Geometry Topology Seminar

The curve complex C(S) of a closed orientable surface S of genusg is an infinite graph with vertices isotopy classes of essential simpleclosed curves on S with two vertices adjacent by an edge if the curves canbe isotoped to be disjoint. By a celebrated theorem of Masur-Minsky, thecurve complex is Gromov hyperbolic. Moreover, a pseudo-Anosov map f of Sacts on C(S) as a hyperbolic isometry with "north-south" dynamics and aninvariant quasi-axis. One can define an asymptotic translation length for fon C(S). In joint work with Chia-yen Tsai, we prove bounds on the minimalpseudo-Anosov asymptotic translation lengths on C(S) . We shall alsooutline related interesting results and questions.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

I will talk about the asymptotic geometry of Teichmuller space equipped with the Weil-Petersson metric. In particular, I will give a criterion for determining when two points in the asymptotic cone of Teichmuller space can be separated by a point; motivated by a similar characterization in mapping class groups by Behrstock-Kleiner-Minsky-Mosher and in right angled Artin groups by Behrstock-Charney. As a corollary, I will explain a new way to uniquely characterize the Teichmuller space of the genus two once punctured surface amongst all Teichmuller space in that it has a divergence function which is superquadratic yet subexponential.

Series: Geometry Topology Seminar

I will discuss the following geometric problem. If you are
given an abstract 2-dimensional simplicial complex that is homeomorphic to a
disk, and you want to (piecewise linearly) embed the complex in the plane so
that the boundary is a geometric square, then what are the possibilities
for the areas of the triangles?
It turns out that for any such simplicial complex there is a
polynomial relation that must be satisfied by the areas. I will report on joint work with
Jamie Pommersheim in which we attempt to understand various features of this
polynomial, such as the degree. One thing we do not know, for
instance, if this degree is expressible in terms of other known integer invariants
of the simplicial complex (or of the underlying planar graph).

Series: Geometry Topology Seminar

We will discuss SL(N,C) representations of 3-manifolds, and their complex volumes, theoretically and computationally.

Series: Geometry Topology Seminar

The Neumann-Zagier equations are well-understood objects of classical hyperbolic geometry. Our discovery is that they have a nontrivial quantum content, (that for instance captures the perturbation theory of the Kashaev invariant to all orders) expressed via universal combinatorial formulas. Joint work with Tudor Dimofte.

Series: Geometry Topology Seminar

The outer automorphism group Out(F) of a non-abelian free group
F of finite rank shares many properties with linear groups and the mapping
class group Mod(S) of a surface, although the techniques for studying
Out(F) are often quite different from the latter two. Motivated by
analogy, I will present some results about Out(F) previously well-known
for the mapping class group, and highlight some of the features in the
proofs which distinguish it from Mod(S).

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

In a joint work with D.Tamarkin we study analytic continuability of solutions of theLaplace-transformed Schroedinger equation by methods of Kashiwara-Schapira style microlocal theoryof sheaves.

Series: Geometry Topology Seminar

In the past two years, Church, Farb and others have developed the concept of 'representation stability', an analogue of homological stability for a sequence of groups or spaces admitting group actions. In this talk, I will give an overview of this new theory, using the pure string motion group P\Sigma_n as a motivating example. The pure string motion group, which is closely related to the pure braid group, is not cohomologically stable in the classical sense -- for each k>0, the dimension of the H^k(P\Sigma_n, \Q) tends to infinity as n grows. The groups H^k(P\Sigma_n, \Q) are, however, representation stable with respect to a natural action of the hyperoctahedral group W_n, that is, in some precise sense, the description of the decomposition of the cohomology group into irreducible W_n-representations stabilizes for n>>k. I will outline a proof of this result, verifying a conjecture by Church and Farb.