Geometry/Topology Seminar
Fall 2019
Thursdays (and sometimes Tuesdays) 3:404:30pm, in
Ryerson 358

 Thursday September 26 at 4:005:00pm in Ry 358
 Daniel Woodhouse, University of Oxford
 Action rigidity of free products of hyperbolic manifold groups.

Abstract: Gromov's program for understanding
finitely generated groups up to their large scale geometry
considers three possible relations: quasiisometry, abstract
commensurability, and acting geometrically on the same
proper geodesic metric space. A *common model geometry* for
groups G and G' is a proper geodesic metric space on which G
and G' act geometrically. A group G is *action rigid* if any
group G' that has a common model geometry with G is
abstractly commensurable to G. We show that free products of
closed hyperbolic manifold groups are action rigid. As a
corollary, we obtain torsionfree, Gromov hyperbolic groups
that are quasiisometric, but do not even virtually act on
the same proper geodesic metric space. This is joint work
with Emily Stark.

 Thursday October 3 at 3:404:30pm in Ry 358
 Lvzhou Chen, University of Chicago
 Big mapping class groups and rigidity of the simple circle.

Abstract: Surfaces of infinite type, such as the
plane minus a Cantor set, occur naturally in dynamics.
However, their mapping class groups are much less studied
and understood compared to the mapping class groups of
surfaces of finite type. Many fundamental questions remain
open. We will discuss the mapping class group G of the plane
minus a Cantor set, and show that any nontrivial Gaction on
the circle is semiconjugate to its action on the socalled
simple circle. Along the way, we will discuss some
structural results of G to address the following questions:
What are some interesting subgroups of G? Is G generated by
torsion elements? This is joint work with Danny Calegari.

 Thursday October 10 at 3:404:30pm in Ry 358
 Antoni Rangachev, UChicago
 Lipschitz geometry and Zariski equisingularity

Abstract: After briefly reviewing the history and
importance of various equisingularity theories, I will
describe an approach to showing that Zariski equisingular
families of hypersurfaces are Lipschitz equisingular in all
dimensions. It is based on the wellknown observation that
Zariski equisingular families are birational to families of
quasiordinary singularities having the same characteristic
exponents. A singularity is called quasiordinary if there
exists a finite map from it to an affine space whose
discriminant is contained in a normal crossing divisor. I
will show that two germs of quasiordinary singularities are
biLipschitz equivalent if and only if they have the same
normalized characteristic exponents. This is joint work with
H. Mourtada and B. Teissier.

 Tuesday October 15 at 3:404:30pm in Ry 358
 Gavril Farkas, Humbolt Universitat zu Berlin
 Topological invariants of groups via Koszul modules.

Abstract: I will discuss the deep connection between
the structure of the equations of certain algebraic
varieties and Alexander invariants of groups. On the
topological side this produced a universal bound on the
nilpotency index of Torelli groups or fundamental group of
nonfibred Kähler groups, whereas on the algebrogeometric
side it has led to a simple proof of Green's Conjecture on
syzygies of canonical curves. Joint work with Aprodu,
Papadima, Raicu and Weyman.

 Monday October 21 at 1:302:30pm in Eckhart 203
 Margaret Bilu, Courant Institute of Mathematical Sciences (NYU)
 Motivic Euler products in motivic statistics.

Abstract:  NOTE SPECIAL TIME AND PLACE The
Grothendieck group of varieties over a field k is the
quotient of the free abelian group of isomorphism classes of
varieties over k by the socalled cutandpaste relations.
It moreover has a ring structure coming from the product of
varieties. Many problems in number theory have a natural,
more geometric counterpart involving elements of this ring.
Thus, Poonen's Bertini theorem over finite fields has a
motivic analog due to Vakil and Wood, which expresses the
motivic density of smooth hypersurface sections as the
degree goes to infinity in terms of a special value of
Kapranov's zeta function. I will present a broad
generalization of Vakil and Wood's result, which implies in
particular a motivic analog of Poonen's Bertini theorem with
Taylor conditions, as well as motivic analogs of many
generalizations and variants of Poonen's theorem. A key
ingredient for this is a notion of motivic Euler product
which allows us to write down candidate motivic
probabilities.

 Monday October 21 at 2:303:30pm in Eckhart 203
 Sean Howe, University of Utah
 A(nother) conjecture about zeta functions, or, "it's zeta functions all the way down."

Abstract:  NOTE SPECIAL TIME AND PLACE We
conjecture a unification of arithmetic and motivic
statistics over finite fields through a natural analytic
topology on the ring of zeta functions. A key step will be
to explain exactly what it means to evaluate the zeta
function of a zeta function at a zeta function.

 Thursday October 24 at 3:404:30pm in Ry 358
 Khashayar Filom, Northwestern
 On the topology of dynamical moduli spaces of rational maps

Abstract: The dynamical moduli space of rational
maps of degree d, defined as the space of Möbius conjugacy
classes of degree d holomorphic selfmaps of the Riemann
sphere, is a ubiquitous object in complex and arithmetic
dynamics. Using the techniques of Geometric Invariant
Theory, Silverman constructs this orbit space as an affine
variety of dimension 2d2 which admits a model over the
rationals. In the case of degree two, Milnor identifies this
space with the affine plane. I will present the results of a
joint work with Maxime Bergeron and Sam Nariman regarding
the topology of these moduli spaces. We compute the
fundamental group of the dynamical moduli space and show
that the space is rationally acyclic while its cohomology
groups with finite coefficients could be nontrivial. As an
application, the ranks of certain rational homotopy groups
of the parameter space of rational maps (within the unstable
range) will be computed.

 Thursday November 21 at 3:404:30pm in Ry 358
 Michelle Chu, University of Illinois at Chicago
 Arithmetic hyperbolic 3manifolds

Abstract: The study of virtual properties of
3manifolds groups has played a key role in major recent
developments in 3manifold topology. In this talk I will
motivate and introduce the study of arithmetic hyperbolic
manifolds and discuss some recent results on quantifying
their virtual properties.

 Thursday December 05 at 3:404:30pm in Ry 358
 Tuomas Sahlsten, Manchester
 Quantum chaos on random surfaces of large genus

Abstract: Quantum chaos and problems in the field
(quantum unique ergodicity, random wave conjecture) provide
a challenging frontier that have numerous connections to
various fields around mathematics. Some of the central works
in the field are in the context of arithmetic surfaces and
also recently in large random graphs. Inspired by these
works, we study quantum chaos on random surfaces when
modelled under WeilPetersson probability in the moduli
space of genus g Riemann surfaces. We employ the recent
advancements of Maryam Mirzakhani on computing expectations
of geometrictopological observables on the moduli space to
establish quantum ergodicity and new bounds for
L^{p} norms of eigenfunctions of the Laplacian for
large genus random surfaces. Based on joint works with Cliff
Gilmore (Cork), Etienne Le Masson (Cergy), and Joe Thomas
(Manchester).
Due to the high number of requests, we are no longer accepting speakers via selfinvitations.
For questions, contact