Geometry/Topology Seminar
Winter 2018
Thursdays (and sometimes Tuesdays) 34pm, in
Ryerson 358

 Thursday January 25 at 34pm in Ry 358
 Akhil Matthew, University of Chicago
 Rigidity and continuity in algebraic Ktheory

Abstract: Let A be a commutative ring complete with
respect to an ideal I. A basic continuity question in
algebraic Ktheory asks how the Ktheory of A (usually with
finite coefficients) compares to the Ktheory of the tower
A/I^{n}. For instance, with mod p coefficients for p
invertible on A, it is a consequence of the
GabberGilletThomasonSuslin 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 (Ffiniteness). Our methods are
based on a generalization of the rigidity theorem using the
theory of topological cyclic homology and general structural
properties of algebraic Ktheory. This is joint work with
Dustin Clausen and Matthew Morrow.

 Thursday February 8 at 34pm in Ry 358
 Phil Tosteson, University of Michigan
 Representation stability in the homology of DeligneMumford 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 34pm (pretalk 1:302:30) in Eck 203
 Craig Westerland, University of Minnesota
 Structure theorems for braided Hopf algebras

Abstract: The PoincaréBirkhoffWitt and
MilnorMoore 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
PoincaréBirkhoffWitt and MilnorMoore 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 34pm in Ry 358
 Eric Ramos, University of Michigan
 Families of nested graphs with compatible symmetricgroup actions

Abstract: For fixed positive integers n and k, the
Kneser graph KGn,k has vertices labeled by kelement 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 FImodule 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 Homcomplexes and configuration spaces
of the graphs, and algebra, concerning the changing
behaviors in the graph spectra.

 Thursday March 1 at 34pm 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 \tauHölder 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 Sanghyun Kim.

 Thursday March 8 at 34pm in Ry 358
 Federico RodriguezHertz, Penn State
 Some problems on Anosov flows in higher dimensions.

Abstract: In the 70's Franks and Williams built an
example of a notransitive 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 34pm 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
MikhalkinZharkov and further studied by
AnBakerKuperbergShokrieh. From a combinatorial point of
view, break divisors can be used to give a
“volumetheoretic proof” of Kirchhoff’s MatrixTree
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 nonArchimedean 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
MikhalkinZharkov) 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
ShokriehWu).

 Tuesday March 20 at 34pm 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 krational 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 34pm 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, AllcockCarlsonToledo (2003)
realized the moduli of smooth cubic threefolds as an
arrangement complement in a 10dimensional arithmetic ball
quotient and studied its compactifications (both GIT and
SatakeBailyBorel) and recently, LazaPearlsteinZhang
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
AllcockCarlsonToledo and PearlsteinLazaZhang mentioned
above.
Due to the high number of requests, we are no longer accepting speakers via selfinvitations.
For questions, contact