Talk_id  Date  Speaker  Title 
15383

Thursday 1/17 2:00 PM

Chris Kottke, New College Florida

Compactification of monopole moduli spaces
 Chris Kottke, New College Florida
 Compactification of monopole moduli spaces
 01/17/2019
 2:00 PM  3:00 PM
 C304 Wells Hall
I will discuss joint work with Michael Singer and Karsten Fritzsch on compactifications of the moduli spaces $M_k$ of $\mathrm{SU}(2)$ magnetic monopoles on $\mathbf{R}^3$ . Via a geometric gluing procedure, we construct manifolds with corners compactifying the $M_k$ , the boundaries of which represent monopoles of charge $k$ decomposing into widely separated ‘monopole clusters' of lower charge. The hyperkahler metric on $M_k$ has a complete asymptotic expansion, the leading terms of which generalize the asymptotic metric discovered by Bielawski, Gibbons and Manton in the case that the monopoles are all widely separated. From the structure of the compactification, we are able to make partial progress toward proving Sen's conjecture for $L^2$ cohomology of the moduli spaces.

16468

Thursday 1/24 2:00 PM

Rebecca Winarski, University of Michigan

Solving the Twisted Rabbit Problem using trees
 Rebecca Winarski, University of Michigan
 Solving the Twisted Rabbit Problem using trees
 01/24/2019
 2:00 PM  3:00 PM
 C304 Wells Hall
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 BartholdiNekyrashevych 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.

15394

Thursday 2/7 2:00 PM

Adam Sikora, SUNY at Buffalo

New Approach to Quantum Teichmuller Theory
 Adam Sikora, SUNY at Buffalo
 New Approach to Quantum Teichmuller Theory
 02/07/2019
 2:00 PM  3:00 PM
 C304 Wells Hall
The Jones polynomial invariant of links in R^3 extends to links in thickened surfaces, leading to the notion of the skein algebra of a surface, which is a version of ChekhovFock quantum Teichmuller space. The algebraic structure of skein algebras is quite rich and mysterious. We will approach it using the theory of measured foliations and pseudoAnosov diffeomorphisms of surfaces

17499

Thursday 2/14 2:00 PM

Melissa Zhang, Boston College

Localization in Khovanov homology
 Melissa Zhang, Boston College
 Localization in Khovanov homology
 02/14/2019
 2:00 PM  3:00 PM
 C304 Wells Hall
When a topological object admits a group action, we expect that our invariants reflect this symmetry in their structure. This talk will tour the expression of link symmetries in three generations of related invariants: the Jones polynomial; its categorification, Khovanov homology; and the youngest invariant in the family, the Khovanov stable homotopy type, introduced by Lipshitz and Sarkar. I will describe how to use LawsonLipshitzSarkar's Burnside functor construction of the LipshitzSarkar Khovanov homotopy type to produce localization theorems and Smithtype inequalities for the Khovanov homology of periodic links. This joint work with Matthew Stoffregen.

17505

Thursday 2/21 2:00 PM

Ilya Gekhtman , University of Toronto

Growth rates of invariant random subgroups of hyperbolic groups and rank 1 Lie groups.
 Ilya Gekhtman , University of Toronto
 Growth rates of invariant random subgroups of hyperbolic groups and rank 1 Lie groups.
 02/21/2019
 2:00 PM  3:00 PM
 C304 Wells Hall
Abstract: Invariant random subgroups (IRS) are conjugacy invariant probability measures on the space of subgroups of a given group G. They arise naturally as point stabilizers of probability measure preserving actions. The space of invariant random subgroups of SL_{2}R can be regarded as a natural compactification of the moduli space of Riemann surfaces, related to the DeligneMumford compactification. Invariant random subgroups can be regarded as a generalization both of normal subgroups and of lattices in topological groups. As such, it is interesting to extend results from the theories of normal subgroups and of lattices to the IRS setting.
Jointly with Arie Levit, we prove such a result: the critical exponent (exponential growth rate) of an infinite IRS in an isometry group of a Gromov hyperbolic space (such as a rank 1 Lie group, or a hyperbolic group) is almost surely greater than half the Hausdorff dimension of the boundary.
This generalizes an analogous result of MatsuzakiYabukiJaerisch for normal s
As a corollary, we obtain that if $\Gamma$ is a typical subgroup and $X$ a rank 1 symmetric space then $\lambda_{0}(X/\Gamma)<\lambda_{0}(X)$ where $\lambda_0$ is the bottom of the spectrum of the Laplacian. The proof uses ergodic theorems for actions of hyperbolic groups.
I will also talk about results about growth rates of normal subgroups of hyperbolic groups that inspired this work.

17509

Thursday 2/28 2:00 PM

Eva Belmont, Northwestern University

Localizing the E_2 page of the Adams spectral sequence
 Eva Belmont, Northwestern University
 Localizing the E_2 page of the Adams spectral sequence
 02/28/2019
 2:00 PM  3:00 PM
 C304 Wells Hall
The Adams spectral sequence is one of the central tools for calculating the stable homotopy groups of spheres, one of the motivating problems in stable homotopy theory. This talk focuses on the E_2 page, which can be calculated algorithmically in a finite range but whose largescale structure is too complicated to be understood in full. I will give an introduction to some features of the Adams E_2 page for the sphere at p = 3, and discuss an approach for calculating it in an infinite region. This approach relies on computing an analogue of the Adams spectral sequence in Palmieri's stable category of comodules, which can be regarded as an algebraic analogue of stable homotopy theory.

17537

Thursday 3/7 2:00 PM

SPRING BREAK

SPRING BREAK
 SPRING BREAK
 SPRING BREAK
 03/07/2019
 2:00 PM  3:00 PM
 C304 Wells Hall
No abstract available.

17538

Thursday 4/4 2:00 PM

Leonid Chekhov, MSU

$SL_k$ character varieties and quantum cluster algebras
 Leonid Chekhov, MSU
 $SL_k$ character varieties and quantum cluster algebras
 04/04/2019
 2:00 PM  3:00 PM
 C304 Wells Hall
briefly recall a combinatorial approach to the description and quantization of Teichmuller spaces of Riemann surfaces $\Sigma_{g,s}$ of genus $g$ with $s$ holes and algebras of geodesic functions on these surfaces. We describe sets of geodesic functions in W.Thurston shear coordinates based on an ideal triangle decomposition of Riemann surfaces with holes and demonstrate the polynomiality and positivity properties of the corresponding geodesic functions. In the algebraic setting, these sets are related to traces of monodromies of $SL_2$ connection on $\Sigma_{g,s}$, and Darbouxtype Poisson and quantum relations on shear coordinates were proven to generate Goldman brackets on geodesic functions. I will describe these structures and their recent generalizations to $SL_2$ and $SL_n$ (decorated) character varieties on Riemann surfaces $\Sigma_{g,s,n}$ with holes and $n$ marked points on hole boundaries and how it is interlaced with cluster algebras, reflection equations, and groupoids of upper triangular matrices. [Based on work in collaboration with M.Mazzocco, V.Roubtsov, and M.Shapiro.]

17510

Monday 4/8 3:00 PM

Mona Merling, University of Pennsylvania

The equivariant stable parametrized hcobordism theorem
 Mona Merling, University of Pennsylvania
 The equivariant stable parametrized hcobordism theorem
 04/08/2019
 3:00 PM  4:00 PM
 C304 Wells Hall
The stable parametrized hcobordism theorem provides a critical link in the chain of homotopy theoretic constructions that show up in the classification of manifolds and their diffeomorphisms. For a compact smooth manifold M it gives a decomposition of Waldhausen's A(M) into QM_+ and a delooping of the stable hcobordism space of M. I will talk about joint work with Malkiewich on this story when M is a smooth compact Gmanifold.

17498

Thursday 4/11 2:00 PM

Jennifer Hom, Georgia Tech

Heegaard Floer and homology cobordism
 Jennifer Hom, Georgia Tech
 Heegaard Floer and homology cobordism
 04/11/2019
 2:00 PM  3:00 PM
 C304 Wells Hall
We show that the threedimensional homology cobordism group admits an infiniterank summand. It was previously known that the homology cobordism group contains an infiniterank subgroup and a Zsummand. The proof relies on the involutive Heegaard Floer homology package of HendricksManolescu and HendricksManolescuZemke. This is joint work with I. Dai, M. Stoffregen, and L. Truong.

18566

Thursday 4/18 2:00 PM

Robert Bell, MSU

Quasipositivity in free groups and braid groups
 Robert Bell, MSU
 Quasipositivity in free groups and braid groups
 04/18/2019
 2:00 PM  3:00 PM
 C304 Wells Hall
I'll discuss joint work with Rita Gitik (UM) on the problem of recognizing quasipositive elements of a group G defined by
a finite presentation (X ; R). An element of G is quasipositive if it can be represented by a word that is a product of conjugates of positive powers of letters in X. The recognition problem is to determine whether or not a given word (using both positive and negative powers of letters in X) represents an element of G that is quasipositive. This problem was solved by Orevkov when G is free with basis X or when G is the 3strand braid group with its standard generating set. I'll present a new solution to the recognition problem for free groups and discuss some of the challenges posed by braid groups and related groups.

18560

Thursday 4/25 2:00 PM

Linh Truong, Columbia University

More concordance homomorphisms from knot Floer homology
 Linh Truong, Columbia University
 More concordance homomorphisms from knot Floer homology
 04/25/2019
 2:00 PM  3:00 PM
 C304 Wells Hall
I will describe an infinite family of integervalued concordance homomorphisms defined using knot Floer homology. These invariants have topological applications to concordance genus, concordance unknotting number, and bridge index. This is joint work with Irving Dai, Jen Hom, and Matt Stoffregen.
