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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.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

In the talk, I plan to give a definition of loose Legendrian knots inside contact manifolds of dimension 5 or greater. The definition is significantly different from the 3 dimensional case, in particular loose knots exist in local charts. I'll discuss an h-principle for such knots. This implies their classification, a bijective correspondence with their formal (algebraic topology) invariants. I'll also discuss applications of this result, comparisons with 3D contact toplogy, and some open questions.

Series: Geometry Topology Seminar

For a genus g surface with s > 0 punctures and 2g+s > 2, decorated Teichmuller space (DTeich) is a trivial R_+^s-bundle over the usual Teichmuller space, where the fiber corresponds to families of horocycles peripheral to each puncture. As proved by R. Penner, DTeich admits a mapping class group-invariant cell decomposition, which then descends to a cell decomposition of Riemann's moduli space. In this talk we introduce a new cellular bordification of DTeich which is also MCG-invariant, namely the space of filtered screens. After an appropriate quotient, we obtain a cell decomposition for a new compactification of moduli space, which is shown to be homotopy equivalent to the Deligne-Mumford compactification. This work is joint with R. Penner.

Series: Geometry Topology Seminar

We will use a new concordance invariant, epsilon, associated to the knot Floer complex, to define a smooth concordance homomorphism. Applications include a new infinite family of smoothly independent topologically slice knots, bounds on the concordance genus, and information about tau of satellites. We will also discuss various algebraic properties of this construction.