Geometry/Topology Seminar

Winter 2018

Thursdays (and sometimes Tuesdays) 3-4pm, in Ryerson 358

- Thursday January 25 at 3-4pm in Ry 358
- Akhil Matthew, University of Chicago
- Rigidity and continuity in algebraic K-theory
- Abstract: Let A be a commutative ring complete with respect to an ideal I. A basic continuity question in algebraic K-theory asks how the K-theory of A (usually with finite coefficients) compares to the K-theory of the tower A/Iⁿ. For instance, with mod p coefficients for p invertible on A, it is a consequence of the Gabber-Gillet-Thomason-Suslin rigidity theorem that the tower is constant and agrees with K(A). We show that continuity holds for an arbitrary noetherian complete ring satisfying a mild condition (F-finiteness). Our methods are based on a generalization of the rigidity theorem using the theory of topological cyclic homology and general structural properties of algebraic K-theory. This is joint work with Dustin Clausen and Matthew Morrow.
- Thursday February 8 at 3-4pm in Ry 358
- Phil Tosteson, University of Michigan
- Representation stability in the homology of Deligne-Mumford compactifications
- Abstract: The space \bar M_g,n is a compactification of the moduli space algebraic curves with marked points, obtained by allowing smooth curves to degenerate to nodal ones. We will talk about how the asymptotic behavior of its homology, H_i(\bar M_g,n), for n >> 0 can be studied using the category of finite sets and surjections.
- Tuesday February 13 at 3-4pm (pretalk 1:30-2:30) in Eck 203
- Craig Westerland, University of Minnesota
- Structure theorems for braided Hopf algebras
- Abstract: The Poincaré-Birkhoff-Witt and Milnor-Moore theorems are fundamental tools for understanding the structure of Hopf algebras. Part of the classification of pointed Hopf algebras involves a notion of “braided Hopf algebras.” I will present work in progress which will establish analogues of the Poincaré-Birkhoff-Witt and Milnor-Moore theorems in this setting. The main new tool is a notion of a braided Lie algebra defined in terms of braided operads. This can be used to establish forms of these results, and also presents an unexpected connection to profinite braid groups and related operads.
- Thursday February 22 at 3-4pm in Ry 358
- Eric Ramos, University of Michigan
- Families of nested graphs with compatible symmetric-group actions
- Abstract: For fixed positive integers n and k, the Kneser graph KGn,k has vertices labeled by k-element subsets of {1,2,...,n} and edges between disjoint sets. Keeping k fixed and allowing n to grow, one obtains a family of nested graphs, each of which is acted on by a symmetric group in a way which is compatible with all of the other actions. In this talk, we will provide a framework for studying families of this kind using the FI-module theory of Church, Ellenberg, and Farb, and show that this theory has a variety of asymptotic consequences for such families of graphs. These consequences span a range of topics including enumeration, concerning counting occurrences of subgraphs, topology, concerning Hom-complexes and configuration spaces of the graphs, and algebra, concerning the changing behaviors in the graph spectra.
- Thursday March 1 at 3-4pm in Ry 358
- Thomas Koberda, University of Virginia
- Diffeomorphism groups of intermediate regularity
- Abstract: Let M be the interval or the circle. For each real number \alpha \in [1,∞), write \alpha=k+\tau, where k is the floor function of \alpha. I will discuss a construction of a finitely generated group of diffeomorphisms of M which are C^k and whose k^th derivatives are \tau--Hölder continuous, but which are admit no algebraic smoothing to any higher Hölder continuity exponent. In particular, such a group cannot be realized as a group of C^k+1 diffeomorphisms of M. I will discuss the construction of countable simple groups with the same property, and give some applications to continuous groups of diffeomorphisms. This is joint work with Sang-hyun Kim.
- Thursday March 8 at 3-4pm in Ry 358
- Federico Rodriguez-Hertz, Penn State
- Some problems on Anosov flows in higher dimensions.
- Abstract: In the 70's Franks and Williams built an example of a no-transitive Anosov flow in dimension 3, in contrast to a theorem by Verjovsky stating that a codimension 1 Anosov flow is transitive if dimension is at least 4. The idea of the talk is to address the transitivity problem for Anosov flows in dimension larger than 4. I plan to explain why some natural constructions fail to produce the desired goal and to show how to build examples on higher dimensions. Finally the idea is to formulate several related problems. This is joint work with T. Barthelmé, C. Bonatti and A. Gogolev.
- Thursday March 15 at 3-4pm in Ry 358
- Matt Baker, Georgia Tech
- The geometry of break divisors
- Abstract: In the first part of the talk, I will introduce the concept of break divisors on a graph G or tropical curve C. Break divisors form a canonical set of effective representatives for Pic^g(G) and Pic^g(C), respectively, and were introduced by Mikhalkin-Zharkov and further studied by An-Baker-Kuperberg-Shokrieh. From a combinatorial point of view, break divisors can be used to give a “volume-theoretic proof” of Kirchhoff’s Matrix-Tree Theorem and a novel characterization of planarity for ribbon graphs. From a geometric point of view, I will discuss two applications of break divisors to the theory of algebraic curves. The first is a result of Amini describing the limiting distribution of higher Weierstrass points on an algebraic curve over a non-Archimedean field K. The limiting distribution, called the Zhang measure, has an interesting interpretation in terms of break divisors, and is analogous to the Bergman metric on a compact Riemann surface. A second application is the recent (independent) work of Tif Shen and Karl Christ relating break divisors to Simpson compactifications of Jacobians. If time permits, I’ll also discuss connections with tropical theta functions (work of Mikhalkin-Zharkov) and Kazhdan’s theorem realizing the hyperbolic metric on a compact Riemann surface as the limit of the Bergman measures of its finite covers (work of Shokrieh-Wu).
- Tuesday March 20 at 3-4pm in Ry 358
- Victoria Hoskins, Freie Universität Berlin
- Group actions on quiver varieties and applications
- Abstract: We study two types of actions on King's moduli spaces of quiver representations over a field k, and we decompose their fixed loci using group cohomology in order to give modular interpretations of the components. The first type of action arises by considering finite groups of quiver automorphisms. The second is the absolute Galois group of a perfect field k acting on the points of this quiver moduli space valued in an algebraic closure of k; the fixed locus is the set of k-rational points, which we decompose using the Brauer group of k, and we describe the rational points as quiver representations over central division algebras over k. Over the real numbers, we have two types of rational points arising from real and quaternionic quiver representations. Over the complex numbers, we describe the symplectic and holomorphic geometry of these fixed loci in hyperkaehler quiver varieties using the language of branes. This is joint work with Florent Schaffhauser.
- Thursday March 22 at 3-4pm in Ry 358
- Zhiwei Zheng, Tsinghua University
- Moduli spaces of cubic fourfolds with specified prime order automorphism groups, and their compactifications
- Abstract: The period map is a powerful tool to study moduli spaces of many kinds of objects related to K3 surfaces and cubic fourfolds, thanks to the global Torelli theorems. In this spirit, Allcock-Carlson-Toledo (2003) realized the moduli of smooth cubic threefolds as an arrangement complement in a 10-dimensional arithmetic ball quotient and studied its compactifications (both GIT and Satake-Baily-Borel) and recently, Laza-Pearlstein-Zhang studied the moduli of pairs consisting of a cubic threefold and a hyperplane section. I will talk about a joint work with Chenglong Yu about the moduli space of cubic fourfolds with a prime order automorphism group specified, and its compactification. We uniformly deal with a list of 14 examples, including two corresponding to the works by Allcock-Carlson-Toledo and Pearlstein-Laza-Zhang mentioned above.

Due to the high number of requests, we are no longer accepting speakers via self-invitations.

For questions, contact

Benson Farb, farb at math.uchicago.edu
Shmuel Weinberger, shmuel at math.uchicago.edu
Danny Calegari, dannyc at math.uchicago.edu
David Cohen, dbcohen at math.uchicago.edu
Maxime Bergeron, mbergeron at math.uchicago.edu