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

 Tuesday April 03 at 34pm in Ry 358
 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. (This
talk is joint with Topology Seminar.)

 Tuesday April 10 at 34pm in Eck 203
 Ben Knudsen, Harvard
 Edge stabilization in the homology of graph braid groups.

Abstract: We discuss a novel type of stabilization
map on the configuration spaces of a graph, which increases
the number of particles occupying an edge. Through these
maps, the homology of the configuration spaces forms a
module over the polynomial ring generated by the edges of
the graph, and we show that this module is finitely
generated, implying eventual polynomial growth of Betti
numbers over any field. Moreover, the action lifts to an
action at the level of singular chains, which contains
strictly more information; indeed, we show that this
differential graded module is almost never formal over the
ring of edges. These results, along with numerous
calculations, arise from consideration of an explicit chain
complex, which is a structured enhancement of a cellular
model first considered by Swiatkowski. We arrive at this
model through a localtoglobal approach combining ideas
from factorization homology and discrete Morse theory. (This
talk is joint with Topology Seminar. THERE WILL BE A PRETALK
AT 1:30 IN ECKHART 203)

 Thursday April 26 at 34pm in Ry 358
 Ramanujan Santharoubane, University of Virginia
 Quantum representations of surface groups.

Abstract: I will show how we can produce exotic
representations of surface groups from the
WittenReshetikhinTuraev TQFT. These representations have
infinite images and give points on character varieties that
are fixed by the action of the mapping. Moreover we can
approximate these representations by representations into
finite groups in order to build exotic regular finite covers
of surfaces. These covers have the following property: the
integral homology is not generated by pullbacks of simple
closed curves on the base. This is joint work with Thomas
Koberda.

 Thursday May 03 at 34pm in Ry 358
 Richard Canary, Michigan
 Dynamics and the Hitchin component

Abstract: Hitchin discovered a component of the
space of representations of a surface group into PSL(n,R),
which bears many resemblances to the Teichmuller space of
Fuchsian representations of the surface group into PSL(2,R).
Labourie introduced dynamical techniques to show that these
Hitchin representations are discrete, faithful
quasiisometric embeddings. Sambarino associated Anosov
flows to Hitchin representations whose periods record the
spectral data of the representation. In this talk, we will
see how to use these flows to attach and study dynamical
quantities to Hitchin representations, e.g. entropies,
Liouville currents and associated Liouville volumes. We will
also discuss rigidity results and natural Riemannian metrics
on the Hitchin component. (These results are joint work with
Martin Bridgeman, Francois Labourite and Andres Sambarino.)

 Tuesday May 08 at 34pm in Ry 358
 Miklós Abért, MTA Alfréd Rényi Institute of Mathematics
 Limits of eigenfunctions in negative curvature.

Abstract: Abstract: Let M be a compact manifold with
negative curvature. The Quantum Unique Ergodicity conjecture
of Rudnick and Sarnak says that eigenfunctions of the
Laplacian on M get equidistributed as the eigenvalue tends
to infinity. The Quantum Ergodicity theorem by Shnirelman,
Colin de Verdière, and Zelditch says that this is true for
a density 1 subsequence of eigenvalues. On the other hand, a
conjecture of Berry, that has not been formulated in a
mathematically precise way, says that high energy
eigenfunctions behave like Gaussian random Euclidean waves.
Using the framework of BenjaminiSchramm convergence of
Riemannian manifolds, we give a mathematically exact
formulation of Berry’s conjecture. This allows us to
establish a relation to Quantum Unique Ergodicity and to
prove Berry’s conjecture in a specific setting. We also
investigate the case when the limit lives on a negatively
curved symmetric space, like covering towers of hyperbolic
surfaces. Our approach is related to the recent result of
Backhausz and Szegedy on almost eigenfunctions of random
regular graphs. A major difference is that instead of
expander graphs, here one deals with a very nonexpander
(hyperfinite) sequence of manifolds. In particular, opposed
to the BackhauszSzegedy theorem, Berry's conjecture will
not hold for almost eigenfunctions.

 Wednesday May 09 at 34pm in Ryerson 358
 Dave Witte Morris, Lethbridge
 Efficient bounded generation

Abstract: A subset X "boundedly generates" a group G
if every element g of G is the product of a bounded number
of elements of X. This is a very powerful notion in abstract
group theory, but geometric group theorists may need the
bounded generation to be "efficient", which means that each
g can be written as a bounded number of elements of X whose
sizes are bounded by a constant times the word length of g.
Twentyfive years ago, Lubotzky, Mozes, and Raghunathan
observed that SL(n,Z) is efficiently boundedly generated by
the elements of its natural SL(2,Z) subgroups. We will
explain the proof of this result, and discuss a recent
generalization to other arithmetic groups.

 Thursday May 10 at 34pm in Ry 358
 Mikolaj Fraczyk, MTA Alfréd Rényi Institute of Mathematics
 Growth of mod2 homology of higher rank locally symmetric spaces.

Abstract: We study the geometry of lengthminimizing
representatives of mod2 homology classes for a locally
symmetric space M modeled after a fixed higher rank
symmetric space X . Using the presence of large
2dimensional local flats in M one can show that every first
mod2 homology class in M is represented by a cycle of total
length at most o(volume(M)). In this talk I will sketch a
short proof of this bound and explain how it implies that
the first mod2 Betti number of M is o(volume(M)). Because
of our geometric approach those results are valid even for
sequences of noncommensurable locally symmetric spaces  a
case that wasn't treated before in the context of modp
homology groups. If time permits I will discuss the
difficulties encountered in adapting our argument to the
modp setting for p odd and describe the link between this
problem and certain invariant random patterns on higher rank
symmetric spaces.

 Tuesday May 15 at 34pm in Ry 358
 Eduard Looijenga, Tsinghua University
 Representations of mapping class groups associated with
surfaces endowed with a fixed automorphism group.
(joint work with Marco Boggi)

Abstract: Riemann surfaces endowed with a faithful
action of a finite group of a given topological type have
their own moduli space. These have monodromy representations
which commute with the action of the finite group and give
rise to virtual representations of the mapping class group
of the orbit surface. With the help of Hodge theory we
obtain some properties of these (topologically defined)
representations. NOTE: These two talks will be in an
informal style, where the audience is particularly
encouraged to ask questions.

 Thursday May 17 at 34pm in Ry 358
 Eduard Looijenga, Tsinghua University
 Representations of mapping class groups associated with
surfaces endowed with a fixed automorphism group.
(joint work with Marco Boggi)

Abstract: Riemann surfaces endowed with a faithful
action of a finite group of a given topological type have
their own moduli space. These have monodromy representations
which commute with the action of the finite group and give
rise to virtual representations of the mapping class group
of the orbit surface. With the help of Hodge theory we
obtain some properties of these (topologically defined)
representations. NOTE: These two talks will be in an
informal style, where the audience is particularly
encouraged to ask questions.

 Thursday May 24 at 34pm in Ry 358
 Dmitri Gekhtman, Caltech
 Holomorphic retracts of Teichmuller space

Abstract: The Teichmuller space of a closed surface
carries a natural complex structure, whose analytic
properties reflect the topology and geometry of the surface.
In this talk, we discuss the problem of classifying the
holomorphic retracts of Teichmuller space. Our approach
hinges on the analysis of two dynamical flows  one in the
moduli space of halftranslation surfaces, and the other in
the space of bounded holomorphic functions on the polydisk.
We will also discuss connections to the mapping class group.

 Tuesday May 29 at 34pm (pretalk 1:302:30) in Eck 203
 Sander Kupers, Harvard
 Cellular E_{2}algebras and the unstable homology of mapping class groups

Abstract: We discuss joint work with Soren Galatius
and Oscar RandalWilliams on the application of
higheralgebraic techniques to classical questions about the
homology of mapping class groups. This uses a new
"multiplicative" approach to homological stability  in
contrast to the "additive" one due to Quillen  which has
the advantage of providing information outside of the stable
range.
Due to the high number of requests, we are no longer accepting speakers via selfinvitations.
For questions, contact