Geometry/Topology Seminar
Winter 2016
Thursdays (and sometimes Tuesdays) 34pm, in
Eckhart 308

 Thursday January 14 at 34pm in Eck 308
 Elizabeth Vidaurre, CUNY
 Cohomology of Polyhedral Product Spaces and Real Moment Angle Complexes

Abstract: Certain subspaces of a product of pairs of
spaces whose factors are labeled by the vertices of a
simplicial complex are referred to as "polyhedral product
spaces". Polyhedral products are given by taking the union
of subproducts associated to each simplex. Such polyhedral
products are realized by objects studied in combinatorics,
commutative algebra and algebraic geometry. In algebraic
geometry, the labeled pairs are 2disks and their
boundaries; the associated polyhedral product is called a
momentangle complex. The real versions of momentangle
complexes, where the pairs are intervals and their
boundaries, is also considered. We will study how the
cohomology of polyhedral products can be given in terms of
the underlying simplicial complex. We will illustrate this
by considering different classes of simplicial complexes in
different settings. *There will be a pretalk in Eckhart 203
from 1:30 pm  2:30 pm

 Thursday January 28 at 34pm in Eck 308
 James Davis, Bloomington
 Bordism of anharmonic manifolds

Abstract: A manifold is anharmonic if it is
L^{2}acyclic, i.e if its universal cover has no
L^{2} harmonic forms on it. One can ask when an
L^{2}acyclic manifold is a boundary of an
L^{2}acyclic manifold with the same fundamental
group. We develop the geometric topology necessary to answer
this question for manifolds of dimension greater than 4 and
virtually abelian fundamental group. The theme of the talk
will be the role of the signature of a manifold and the
development of new invariants for these manifolds. This is
joint work with Sylvain Cappell and Shmuel Weinberger.

 Tuesday February 2 at 34pm in Eck 308
 Egor Shelukhin, IAS
 Metrics on areapreserving diffeomorphism groups

Abstract: We discuss results on the largescale
geometry of various metrics defined on groups of
areapreserving diffeomorphisms of surfaces. This talk is
partially based on joint works (some of which are in
preparation) with Michael Brandenbursky, Jarek Kedra, and
Leonid Polterovich.

 Thursday February 4 at 34pm in Eck 308
 Craig Westerland, University of Minnesota
 FoxNeuwirth cells, quantum shuffle algebras, and the homology
of braid groups.

Abstract: The notion of a braided vector space V
comes from the Hopf algebra community, and examples abound,
from conjugacy classes in groups to braidings coming from
Cartan matrices. From this definition, the tensor powers of
V form a family of braid group representations. They also
may be assembled into a noncommutative, noncocommutative
braided Hopf algebra called a quantum shuffle algebra. Our
main result identifies the homology of the braid groups with
these coefficients as the cohomology of this algebra. Using
change of rings spectral sequences, we begin to get a handle
on these homology groups. This is joint work with TriThang
Tran and Jordan Ellenberg

 Tuesday February 16 at 34pm in Eck 308
 Tengren Zhang, Caltech
 Degeneration of Hitchin Representations

Abstract: I will describe an analog of the
FenchelNielsen coordinates on the Hitchin component, and
then use these coordinates to define a large family of
deformations in the Hitchin component called "internal
sequences". Then, I will explain some geometric properties
of these internal sequences, which allows us to conclude
some structural similarities and differences between the
higher Hitchin components and Teichmuller space.

 Thursday February 18 at 34pm in Eck 308
 Jae Choon Cha, POSTECH
 A topological approach to CheegerGromov universal bounds of
von Neumann rho invariants

Abstract: I will begin with a quick introduction to
the CheegerGromov rho invariants from a topological
viewpoint, and then explain recent results on how they are
related to triangulations, Heegaard splittings, and surgery
descriptions of 3manifolds. I will also discuss an
algebraic notion of controlled chain homotopy, which is one
of the key ingredients. Some applications will be given if
time permits.

 Tuesday February 23 at 34pm in Eck 308
 Isaac Mabillard, IST Austria

Whitney Trick and Counterexamples to the Topological Tverberg

Abstract: Let's assume that r balls intersect in
R^{d} in general positions and that their
intersection consists of two points of opposite intersection
signs. I'll describe a generalization of the classical
Whitney Trick to this situation: our goal is to eliminate
the pair of intersection points, by means of ambient
isotopies having "small" support. A neat application of this
"generalized Whitney" is the construction of counterexamples
to the topological Tverberg conjecture, which asserts that
for any continuous map from the Nsimplex to R^{d},
one can always find "a large number" of disjoint cells of
the Nsimplex that intersect in the image in R^{d}.
Due to the (co)dimension requirements of our current
techniques, we can only build counterexamples for d at least
12. So what happens in lower dimensions remains a mystery...
(Joint work with Uli Wagner)

 Thursday March 3 at 34pm in Eck 308
 Wouter van Limbeek, University of Michigan
 Symmetry gaps in Riemannian geometry and minimal orbifolds

Abstract: In 1893 Hurwitz showed that a hyperbolic
surface of genus at least 2 has isometry group of order at
most 84(g1). Do such bounds on the order of isometry groups
exist more generally? It was conjectured by FarbWeinberger
that this is the case for certain aspherical manifolds. In
this spirit we prove that the size of the isometry group of
an arbitrary closed manifold is bounded in terms of certain
geometric quantities (such as curvature and volume), unless
the manifold admits an action by a compact connected Lie
group. We give two applications of this result: First we
characterize locally symmetric spaces among all Riemannian
manifolds, and secondly, we generalize results of
KazhdanMargulis and Gromov on the existence of minimal
quotients of locally symmetric spaces and negatively curved
manifolds.

 Monday March 7 at 2:303:30 in Eck 202
 Harald Helfgott, University of Gottingen
 Growth in groups: a survey talk on a family of methods

Abstract: This will be an overview of methods
developed in the last few years to prove results on growth
in noncommutative groups. These techniques have their roots
in both additive combinatorics and group theory, as well as
other fields. We will discuss linear algebraic groups, with
SL_{2}(Z/pZ) as the basic example, as well as
permutation groups. (Joint Logic / Geometry and Topology
seminar)

 Tuesday March 15 at 34pm in THE BARN
 Yong Hou, Institute for Advanced Study
 Uniformization of Riemann surface and Hausdorff spectra, and a
question of Bers.

Abstract: In this talk I will discuss my joint work
with Jim Anderson on Uniformizations of closed Riemann
surfaces by Kleinian groups which are Schottky groups. In
particular, I will talk about the question of when can one
uniformized a given closed Riemann surface by a classical
Schottky groups, which is an old problem due to Bers that
has been studied by many people (ref: Maskit “remark on
msymmetric Riemann surface”, pp 433, Lipa's Legacy:
Proceedings of the Bers Colloquium, October 1920, 1995).
The idea that we take is through the study of Hausdorff
dimension spectra, (which will be defined in the talk) of
uniformization Kleinian groups. We will study
uniformizations of Riemann surface as infinite 3hyperbolic
handlebody markings. First we obtain geometric criterions on
the handle body which provides uniformization by classical
Schottky group. This will rely on our previous result on
Hausdorff dimension of Schottky groups. Secondly we relates
these geometrical condition via homological marking
condition on the Riemann surface. (IN THE BARN)

 Thursday March 31 at 34pm in Eck 308
 Kevin Kordek, Texas A&M
 Mapping class groups and the topology of the hyperelliptic locus

Abstract: The hyperelliptic mapping class group is
the subgroup of the mapping class group of a closed
orientable surface whose elements commute with a fixed
hyperelliptic involution. This group and its principal
congruence subgroups are important not only in geometric
topology and group theory, but also in algebraic geometry,
where they appear as fundamental groups of the components of
the hyperelliptic loci in various moduli spaces of Riemann
surfaces. In this talk, I will summarize what is known about
the grouptheoretic and topological structure of these
objects, describe a few open problems, and report on some
recent partial progress.

 Thursday April 14 at 34pm in Eck 308
 Haomin Wen, Notre Dame
 TBA

Abstract: TBA

 Thursday May 5 at 34pm in Eck 308
 Nate Harman, MIT
 TBA

Abstract: TBA
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