Geometry and Topology Seminar

Thursday October 18, 15:00, room MS.04

Louis Kauffman (UIC and INI)

Jones Polynomial, Potts Model and Khovanov Homology

Abstract: The Jones polynomial invariant in knot theory and the Potts model in statistical mechanics are closely related through the bracket state sum model -- a partition function defined on knot diagrams that specializes to the Jones polynomial and can, by different specialization, represent the dichromatic and Tutte polynomials for plane graphs. Via this connection, one can use knot and link diagrams to represent the partition function for the Potts model. The loops in the bracket expansion then correspond to boundaries of regions of constant spin in the Potts model. These states (loop collections) in the bracket model are elevated to a category whose homology is Khovanov homology, an invariant more powerful than the Jones polynomial. From the point of view of the physics of the Potts model it is natural to ask for a physical interpretation of this homology theory based on states delineating regions of constant spin. We will raise these questions and discuss how Khovanov homology and its graded Euler characteristic look from the point of view of the Potts model. We will also point out how this way of thinking leads to a quantum-information theoretic reformulation of Khovanov homology and the Jones polynomial.

Thursday November 1, 15:00, room MS.04

Rob Kirby (Berkeley)

Morse 2-functions and trisections of 4-manifolds

Abstract: In joint work with David Gay, we show existence and uniqueness of Morse 2-functions (also known as broken Lefschetz fibrations), and then existence and uniqueness (up to stabilization) of trisections of 4-manifolds (analogous to Heegaard splittings of 3-manifolds).

Thursday November 1, 16:00, room MS.04

Robert Tang (Warwick)

Singular Euclidean surfaces and the curve complex

Abstract: In my talk, I will describe a family of metrics on a surface S obtained by gluing Euclidean rectangles along their edges in a certain way. One can ask how the set of short curves on S behaves as we vary the side lengths of the rectangles in a sensible way. I will approach this question from the point of view of the curve complex - a simplicial complex which encodes intersection information about simple closed curves on S. I will then use these results to approximate some coarse geometric notions in the curve complex such as "quasiconvex hulls" and nearest point projections using intersection numbers conditions.

Thursday November 8, 15:00, room MS.04

TBA (TBA)

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Abstract: TBA

Thursday November 15, 15:00, room MS.04

TBA (TBA)

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Abstract: TBA

Thursday November 22, 15:00, room MS.04

Lars Louder (Warwick)

Hierarchies for finitely presented groups

Abstract: A hierarchy of a finitely generated group is a tree of groups obtained by repeatedly passing to one-ended factors of vertex groups of nontrivial (minimal) graphs of groups decompositions over slender edge groups. We show that hierarchies of finitely presented groups relative to slender subgroups having infinite dihedral quotients are finite. This is joint work with Nicholas Touikan.

Thursday November 29, 15:00 - 17:00, room MS.04

Mehmet Haluk Şengün (Warwick)

On arithmetic hyperbolic 3-manifolds

Abstract: In this talk, I will focus on hyperbolic 3-manifolds which are "arithmetic". These manifolds provide a rich source of (counter-)examples for many topological phenomena. Moreover, their arithmetic nature allows one to use results from number theory and the theory of automorphic representations to deduce interesting results of topological nature, for example, related to the Virtual Betti Number conjecture and the Virtual Fibration conjecture.

In the first half of the talk, I will present the basic notions. As the talk is intended for an audience of topologists, only a minimal background in the number theory will be assumed. The second part will be devoted to exploiting the arithmetic nature of these manifolds to deduce results of interest for 3-manifold theory.

Thursday December 6, 15:00, room MS.04

Richard Evan Schwartz (Brown and Oxford)

Polytope exchange transformations, lattices, and renormalization

Abstract: Polytope exchange transformations arise when one has a partition of a big polytope by smaller polytopes in two different ways. The transformation is a self-map of the big polytope, which translates each piece in the one partition to its location in the other. I'll give some examples of these things, based on collections of lattices and their fundamental domains, and I'll explain what happens for the simplest case of the construction - a 1-parameter family of polygon exchange transformations which turns out have hidden hyperbolic-geometry symmetry in it.