Geometry/Topology Seminar
Winter 2018
Thursdays (and sometimes Tuesdays) 34pm, in
Eckhart 308

 Thursday January 25 at 34pm in Eck 308
 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 Eck 308
 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 Eck 308
 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 Eck 308
 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 15 at 34pm in Eck 308
 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 Eck 308
 Victoria Hoskins, Freie Universität Berlin
 TBA

Abstract: TBA

 Tuesday April 03 at 34pm in Eck 308
 Manuel Krannich, Copenhagen
 Topological moduli spaces, E_{2}algebras, and homological stability

Abstract: Since the seventies, many families of
topological moduli spaces have been proven to stabilize
homologically, including moduli spaces of Riemann surfaces
(Harer), unordered configuration spaces (McDuff, Segal), and
moduli spaces of higherdimensional manifolds (Galatius,
RandalWilliams). From the perspective of homotopy theory, a
common structure these examples share is that of an
E_{2}algebra, or at least of a module over such an
algebra. In this talk, I will introduce a framework which
provides a uniform treatment of classical and new (twisted)
homological stability results from this perspective. If time
permits, I will also discuss how these results imply
representation stability for related moduli spaces.

 Thursday April 26 at 34pm in Eck 308
 Ramanujan Santharoubane, University of Virginia
 TBA

Abstract: TBA

 Thursday May 03 at 34pm in Eck 308
 Richard Canary, Michigan
 TBA

Abstract: TBA

 Thursday May 24 at 34pm in Eck 308
 Dmitri Gekhtman, Caltech
 TBA

Abstract: TBA
Due to the high number of requests, we are no longer accepting speakers via selfinvitations.
For questions, contact