| Talk_id | Date | Speaker | Title |
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15383
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Thursday 1/17
2:00 PM
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Chris Kottke, New College Florida
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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.
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16468
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Thursday 1/24
2:00 PM
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Rebecca Winarski, University of Michigan
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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 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.
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15394
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Thursday 2/7
2:00 PM
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Adam Sikora, SUNY at Buffalo
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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 Chekhov-Fock 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 pseudo-Anosov diffeomorphisms of surfaces
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17499
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Thursday 2/14
2:00 PM
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Melissa Zhang, Boston College
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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 Lawson-Lipshitz-Sarkar's Burnside functor construction of the Lipshitz-Sarkar Khovanov homotopy type to produce localization theorems and Smith-type inequalities for the Khovanov homology of periodic links. This joint work with Matthew Stoffregen.
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17505
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Thursday 2/21
2:00 PM
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Ilya Gekhtman , University of Toronto
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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 Deligne-Mumford 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 Matsuzaki-Yabuki-Jaerisch 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.
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17509
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Thursday 2/28
2:00 PM
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Eva Belmont, Northwestern University
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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 large-scale 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.
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17537
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Thursday 3/7
2:00 PM
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SPRING BREAK
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SPRING BREAK
- SPRING BREAK
- SPRING BREAK
- 03/07/2019
- 2:00 PM - 3:00 PM
- C304 Wells Hall
No abstract available.
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17538
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Thursday 4/4
2:00 PM
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Leonid Chekhov, MSU
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$SL_k$ 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 Darboux-type 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.]
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17510
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Monday 4/8
3:00 PM
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Mona Merling, University of Pennsylvania
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The equivariant stable parametrized h-cobordism theorem
- Mona Merling, University of Pennsylvania
- The equivariant stable parametrized h-cobordism theorem
- 04/08/2019
- 3:00 PM - 4:00 PM
- C304 Wells Hall
The stable parametrized h-cobordism 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 h-cobordism space of M. I will talk about joint work with Malkiewich on this story when M is a smooth compact G-manifold.
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17498
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Thursday 4/11
2:00 PM
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Jennifer Hom, Georgia Tech
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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 three-dimensional homology cobordism group admits an infinite-rank summand. It was previously known that the homology cobordism group contains an infinite-rank subgroup and a Z-summand. The proof relies on the involutive Heegaard Floer homology package of Hendricks-Manolescu and Hendricks-Manolescu-Zemke. This is joint work with I. Dai, M. Stoffregen, and L. Truong.
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18566
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Thursday 4/18
2:00 PM
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Robert Bell, MSU
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Quasi-positivity in free groups and braid groups
- Robert Bell, MSU
- Quasi-positivity 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 quasi-positive elements of a group G defined by
a finite presentation (X ; R). An element of G is quasi-positive 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 quasi-positive. This problem was solved by Orevkov when G is free with basis X or when G is the 3-strand 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.
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