Representation Stability and Milnor Fibers

Speaker: 

Phil Tosteson

Institution: 

Michigan

Time: 

Monday, May 20, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

The Type  Milnor fiber is the subset of  defined by the equation .  It carries an action of the alternating group and the th roots of unity. We will discuss how tools from representation stability can be used to study the homology of the Milnor fiber for  and determine the stable limit.  This is joint work with Jeremy Miller. 

Solving the Twisted Rabbit Problem using trees

Speaker: 

Rebecca Winarski

Institution: 

University of Michigan

Time: 

Monday, January 28, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

The twisted rabbit problem is a celebrated problem in complex dynamics. Work of Thurston proves that up to equivalence, there are exactly three branched coverings of the sphere to itself satisfying certain conditions. When one of these branched coverings is modified by a mapping class, a map equivalent to one of the three coverings results. Which one?

After remaining open for 25 years, this problem was solved by Bartholdi-Nekyrashevych using iterated monodromy groups. In joint work with Belk, Lanier, and Margalit, we present an alternate solution using topology and geometric group theory that allows us to solve a more general problem.

Algebraic fibrations of Kahler groups

Speaker: 

Stefano Vidussi

Institution: 

UC Riverside

Time: 

Monday, October 22, 2018 - 4:00pm

Location: 

RH 340P

One of the major results in the study of 3-manifolds is the fact that most 3-manifolds have a finite cover that fibers over S1. One may ask what is the counterpart of this result for other classes of manifolds. In this talk we will discuss the case of smooth projective varieties (or more generally Kaehler manifolds) and present some geometric and group-theoretic aspects of  "virtual algebraic fibrations" of their fundamental groups.

The geometry of the cyclotomic trace

Speaker: 

Aaron Mazel-Gee

Institution: 

USC

Time: 

Monday, April 1, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

K-theory is a means of probing geometric objects by studying their vector bundles, i.e. parametrized families of vector spaces.  Algebraic K-theory, the version applying to varieties and schemes, is a particularly deep and far-reaching invariant, but it is notoriously difficult to compute.  The primary means of computing it is through its "cyclotomic trace" map K→TC to another theory called topological cyclic homology.  However, despite the enormous computational success of these so-called "trace methods" in algebraic K-theory computations, the algebro-geometric nature of the cyclotomic trace has remained mysterious.

In this talk, I will describe a new construction of TC that affords a precise interpretation of the cyclotomic trace at the level of derived algebraic geometry.  By the end of the talk, you will be able to take home with you a very nice and down-to-earth fact about traces of matrices.  No prior knowledge of algebraic K-theory or derived algebraic geometry will be assumed.

This represents joint work with David Ayala and Nick Rozenblyum.

Joint Los Angeles Topology Seminar at Caltech

Institution: 

Joint Seminar

Time: 

Monday, November 5, 2018 - 4:00pm to 6:00pm

Location: 

Linde 310

Chris Gerig (Harvard): SW = Gr
Whenever the Seiberg-Witten (SW) invariants of a 4-manifold X are defined, there exist certain 2-forms on X which are symplectic away from some circles. When there are no circles, i.e. X is symplectic, Taubes' ``SW=Gr'' theorem asserts that the SW invariants are equal to well-defined counts of J-holomorphic curves (Taubes' Gromov invariants). In this talk I will describe an extension of Taubes' theorem to non-symplectic X: there are well-defined counts of J-holomorphic curves in the complement of these circles, which recover the SW invariants. This ``Gromov invariant'' interpretation was originally conjectured by Taubes in 1995.

Biji Wong (CIRGET Montreal): A Floer homology invariant for 3-orbifolds via bordered Floer theory
Using bordered Floer theory, we construct an invariant for 3-orbifolds with singular set a knot that generalizes the hat flavor of Heegaard Floer homology. We show that for a large class of 3-orbifolds the orbifold invariant behaves like HF-hat in that the orbifold invariant, together with a relative Z_2-grading, categorifies the order of H_1^orb. When the 3-orbifold arises as Dehn surgery on an integer-framed knot in S^3, we use the {-1,0,1}-valued knot invariant epsilon to determine the relationship between the orbifold invariant and HF-hat of the 3-manifold underlying the 3-orbifold.

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