Subdivisional spaces and configuration spaces of graphs

Speaker: 

Gabriel Drummond-Cole

Institution: 

POSTECH IBS-CGP

Time: 

Monday, January 22, 2018 - 4:00pm to 5:00pm

Location: 

RH 340P

Configuration spaces of manifolds are often studied using the local model of configurations of Euclidean space. Configuration spaces of graphs have been studied as rigid combinatorial objects. I will describe a model for configuration spaces of cell complexes which combines the best features of both of these traditions, along with some applications in the homology of the configuration spaces of graphs. This is joint work with Byunghee An and Ben Knudsen.

Ordering actions on hyperbolic metric spaces

Speaker: 

Carolyn Abbott

Institution: 

UC Berkeley

Time: 

Monday, February 5, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

Every group admits at least one action by isometries on a hyperbolic metric space, and certain classes of groups admit many different actions on different hyperbolic metric spaces (in fact, often uncountably many).  One such class of groups is the class of so-called acylindrically hyperbolic groups, which contains many interesting groups, such as mapping class groups, Out(F_n), and right-angled Artin and Coxeter groups, among many others.  In this talk, I will describe how to put a partial order on the set of actions of a given group on hyperbolic spaces which, in some sense, measures how much information about the group the action provides.  This partial order defines a "poset of actions" of the given group.  I will then define the class of acylindrically hyperbolic groups and give some structural properties of the resulting poset of actions for such groups.  In particular, I will discuss for which (classes of) groups the poset contains a largest element.

Cohomology of arithmetic groups and characteristic classes of manifold bundles

Speaker: 

Bena Tshishiku

Institution: 

Harvard University

Time: 

Monday, January 8, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

A basic problem in the study of fiber bundles is to compute the ring H*(BDiff(M)) of characteristic classes of bundles with fiber a smooth manifold M. When M is a surface, this problem has ties to algebraic topology, geometric group theory, and algebraic geometry. Currently, we know only a very small percentage of the total cohomology. In this talk I will explain some of what is known and discuss some new characteristic classes (in the case dim M >>0) that come from the unstable cohomology of arithmetic groups. 

Joint Los Angeles Topology Seminar at Caltech

Institution: 

Joint Seminar

Time: 

Monday, December 4, 2017 - 4:00pm to 6:00pm

Location: 

201 E Bridge

Raphael Zentner (University of Regensburg): Irreducible SL(2,C)-representations of integer homology 3-spheres
We prove that the splicing of any two non-trivial knots in the 3-sphere admits an irreducible SU(2)-representation of its fundamental group. This uses instanton gauge theory, and in particular a non-vanishing result of Kronheimer-Mrowka and some new results that we establish for holonomy perturbations of the ASD equation. Using a result of Boileau, Rubinstein and Wang (which builds on the geometrization theorem of 3-manifolds), it follows that the fundamental group of any integer homology 3-sphere different from the 3-sphere admits irreducible representations of its fundamental group in SL(2,C).

Zhouli Xu (MIT): TBA
TBA

Joint Los Angeles Topology Seminar at UCLA

Institution: 

Joint Seminar

Time: 

Monday, November 6, 2017 - 4:00pm to 6:00pm

Location: 

MS 6627

Sheel Ganatra (USC): Liouville sectors and localizing Fukaya categories
We introduce a new class of Liouville manifolds-with-boundary, called Liouville sectors, and show they have well-behaved, covariantly functorial Fukaya/Floer theories. Stein manifolds frequently admit coverings by Liouville sectors, which can then be used to study the Fukaya category of the total space. Our first main result in this setup is a local criterion for generating (global) Fukaya categories. One of our goals, using this framework, is to obtain a combinatorial presentation of the Fukaya category of any Stein manifold. This is joint work with John Pardon and Vivek Shende.

Nathan Dunfield (UIUC): An SL(2, R) Casson-Lin invariant and applications
When M is the exterior of a knot K in the 3-sphere, Lin showed that the signature of K can be viewed as a Casson-style signed count of the SU(2) representations of pi_1(M) where the meridian has trace 0. This was later generalized to the fact that signature function of K on the unit circle counts SU(2) representations as a function of the trace of the meridan. I will define the SL(2, R) analog of these Casson-Lin invariants, and explain how it interacts with the original SU(2) version via a new kind of smooth resolution of the real points of certain SL(2, C) character varieties in which both kinds of representations live. I will use the new invariant to study left-orderability of Dehn fillings on M using the translation extension locus I introduced with Marc Culler, and also give a new proof of a recent theorem of Gordon's on parabolic SL(2, R) representations of two-bridge knot groups. This is joint work with Jake Rasmussen (Cambridge).
 

 

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