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Series: Geometry Topology Seminar

I will talk about the long standing analogy between the mapping class group of a hyperbolic surface and the outer automorphism group of a free group. Particular emphasis will be on the dynamics of individual elements and applications of these results to structure theorems for subgroups of these groups.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

The general linear groups GL_n(A) can be defined for any ring A, and Quillen's definition of K-theory of A takes these groups as its starting point. If A is commutative, one may define symplectic K-theory in a very similar fashion, but starting with the symplectic groups Sp_{2n}(A), the subgroup of GL_{2n}(A) preserving a non-degenerate skew-symmetric bilinear form. The result is a sequence of groups denoted KSp_i(A) for i = 0, 1, .... For the ring of integers, there is an interesting action of the absolute Galois group of Q on the groups KSp_i(Z), arising from the moduli space of polarized abelian varieties. In joint work with T. Feng and A. Venkatesh we study this action, which turns out to be an interesting extension between a trivial representation and a cyclotomic representation.

Series: Geometry Topology Seminar

It is generally a difficult problem to compute the Betti numbers of a
given finite-index subgroup of an infinite group, even if the Betti
numbers of the ambient group are known. In this talk, I will describe a
procedure for obtaining new lower
bounds on the first Betti numbers of certain finite-index subgroups of
the braid group. The focus will be on the level 4 braid group, which is
the kernel of the mod 4 reduction of the integral Burau representation.
This is joint work with Dan Margalit.

Series: Geometry Topology Seminar

We discuss the growth of homonoly in finite coverings, and show that the growth of the torsion part of the first homology of finite coverings of 3-manifolds is bounded from above by the hyperbolic volume of the manifold. The proof is based on the theory of L^2 torsion.

Series: Geometry Topology Seminar

Peter Lambert-Cole: Mutant knots are notoriously hard to distinguish. Many, but not all, knot invariants take the same value on mutant pairs. Khovanov homology with coefficients in Z/2Z is known to be mutation-invariant, while the bigraded knot Floer homology groups can distinguish mutants such as the famous Kinoshita-Terasaka and Conway pair. However, Baldwin and Levine conjectured that delta-graded knot Floer homology, a singly-graded reduction of the full invariant, is preserved by mutation. In this talk, I will give a new proof that Khovanov homology mod 2 is mutation-invariant. The same strategy can be applied to delta-graded knot Floer homology and proves the Baldwin-Levine conjecture for mutations on a large class of tangles. -----------------------------------------------------------------------------------------------------------------------------------------------Alex Zupan: Generally speaking, given a type of manifold decomposition, a natural
problem is to determine the structure of all decompositions for a fixed
manifold. In particular, it is interesting to understand the space of
decompositions for the simplest objects. For example, Waldhausen's
Theorem asserts that up to isotopy, the 3-sphere has a unique Heegaard
splitting in every genus, and Otal proved an analogous result for
classical bridge splittings of the unknot. In both cases, we say that
these decompositions are "standard," since they can be viewed as generic
modifications of a minimal splitting. In this talk, we examine a
similar question in dimension four, proving that -- unlike the situation
in dimension three -- the unknotted 2-sphere in the 4-sphere admits a
non-standard bridge trisection. This is joint work with Jeffrey Meier.

Series: Geometry Topology Seminar

Heegaard Floer theory provides a powerful suite of tools for studying 3-manifolds and their subspaces. In 2006, Ozsvath, Szabo and Thurston defined an invariant of transverse knots which takes values in a combinatorial version of this theory for knots in the 3—sphere. In this talk, we discuss a refinement of their combinatorial invariant via branched covers and discuss some of its properties. This is joint work with Mike Wong.

Series: Geometry Topology Seminar

In a joint work with M. Yampolsky, we gave a classification of Thurston maps with parabolic orbifolds based on our previous results on characterization of canonical Thurston obstructions. The obtained results yield a solution to the problem of algorithmically checking combinatorial equivalence of two Thurston maps.

Series: Geometry Topology Seminar

The immersed Seifert genus of a knot $K$ in $S^3$ can be defined as the minimal genus of an orientable immersed surface $F$ with $\partial F = K$. By a result of Gabai, this value is always equal to the (embedded) Seifert genus of $K$. In this talk I will discuss the embedded and immersed cross-cap numbers of a knot, which are the non-orientable versions of these invariants. Unlike their orientable counterparts these values do not always coincide, and can in fact differ by an arbitrarily large amount. In further contrast to the orientable case, there are families of knots with arbitrarily high embedded 4-ball cross-cap numbers, but which are easily seen to have immersed cross-cap number 1. After describing these examples I will discuss a classification of knots with immersed cross-cap number 1. This is joint work with Seungwon Kim.