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).
 

 

Algebraic Gluing of Holomorphic Discs in K3 Surfaces and Tropical Geometry

Speaker: 

Yu-Shen Lin

Institution: 

Harvard CMSA

Time: 

Monday, October 30, 2017 - 4:00pm

Location: 

RH 340P

We will start from the motivation of the tropical geometry. Then
we will explain how to use Lagrangian Floer theory to establish the
correspondence between the weighted counting of tropical curves to the
counting of holomorphic discs in K3 surfaces. In particular, the result
provides the existence of new holomorphic discs which do not come easily
from direct gluing argument.

Vector bundles and A^1-homotopy theory

Speaker: 

Marc Hoyois

Institution: 

USC

Time: 

Monday, March 12, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

The study of vector bundles on algebraic varieties is a classical topic at the intersection of geometry and commutative algebra. In its algebraic form it is the study of finitely generated projective modules over commutative rings. There are many long-standing conjectures and open questions about algebraic vector bundles, such as: is every topological vector bundle over complex projective space algebraic? In recent years, there have been a number of significant developments in this area made possible using the A^1-homotopy theory of algebraic varieties introduced by Morel and Voevodsky in the late 90s. The talk will provide some background on such questions and discuss some recent joint work with Aravind Asok and Matthias Wendt.

Subgroups of the mapping class group via algebraic geometry

Speaker: 

Nick Salter

Institution: 

Harvard University

Time: 

Monday, November 27, 2017 - 4:00pm

Location: 

RH 340P

This talk will be a discussion of some interesting and novel subgroups of the mapping class group that arise via algebro-geometric constructions. Our talk will focus on the special case of how the theory of plane algebraic curves (essentially just polynomials in two variables!) interacts with the mapping class group in subtle ways. The motivating question can be formulated simply as, ``which mapping classes (of a surface of genus g) arise as one-parameter families of polynomials in two variables?’’ Perhaps surprisingly, the answer turns out to be ``either none at all, or else virtually all of them”. No familiarity with algebraic geometry will be assumed. 

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