Spring 2016

** Organizers: Gennadi Kasparov, Ioana
Suvaina, ****Rares Rasdeaconu**

** Fridays,
3:10-4:00pm in SC 1310 (unless otherwise noted) **

**Friday, February 5th**

__Speaker:__** Kun Wang,
Vanderbilt University**

__Title____:__** Topological
rigidity for closed aspherical manifolds fibering over the
unit circle** ** **

__Abstract__**: **The Borel conjecture
in manifold topology predicts that every closed aspherical
manifold is

topologically rigid, i.e. every homotopy equivalence between
any two closed aspherical manifolds

is homotopic to a homeomorphism. There are variants of the
Borel conjecture, such as the simple

Borel conjecture and the bordism Borel conjecture,
corresponding to other types of topological rigidity.

In this talk, I consider topological rigidity for closed
aspherical manifolds that fiber over the unit circle.

We show that, in dimensions greater than or equal to 5, both
the simple Borel conjecture and the

bordism Borel conjecture hold for such an aspherical manifold,
provided the fundamental group

of the fiber belongs to a large class of groups, including
Gromov hyperbolic groups, CAT(0) groups,

and lattices in virtually connected lie groups. The main
ingredients in proving this rigidity result

are some general results that we obtain in algebraic
L-theory. These results also have some applications

to the Novikov conjecture.

**
Friday, February 12th**

__Speaker:__** Matthieu
Jacquemet, Vanderbilt University**

__Title____:__** Around
hyperbolic Coxeter polyhedra I** **
**

__Abstract__**: **Unlike their
spherical and Euclidean cousins, hyperbolic Coxeter polyhedra
do not exist

any more in higher dimensions, and are far from being
classified. In this first talk, we intend to give

a survey on their existence and classification. No particular
background will be assumed.

**
Friday, February 19th**

__Speaker:__** Matthieu
Jacquemet, Vanderbilt University**

__Title____:__** Around hyperbolic
Coxeter polyhedra II** **
**

__Abstract__**: **In this second talk,
we shall discuss recent results related to two natural classes
of

hyperbolic polyhedra : simplices, and Coxeter cubes. It time
permits, we shall outline a couple

of open problems which could be attacked by using these new
results.

**
February 25th, 2016 (Thursday), 4:10 pm** (

__Speaker__: **JEFF CHEEGER****,
NYU **

__Title__: **REGULARITY OF MANIFOLDS WITH
BOUNDED RICCI CURVATURE AND THE CODIMENSION 4 CONJECTURE**

__Abstract__**: **Let X^n denote the Gromov-Hausdorff
limit of a noncollapsing sequence of Riemannian manifolds with
uniformly bounded Ricci

curvature. Around 1990, early workers, in particular, Mike
Anderson, conjectured that apart from a (possibly empty)
closed subset S of

(Hausdorff) codimension greater or equal to 4, X^n is a smooth
riemannian manifold. The example of limits of scaled down
4-dimensional

complete noncompact Ricci flat spaces showed that such a
result would be sharp. We will try to explain the statement of
the conjecture and

some of the ideas in the proof. This is joint work with Aaron
Naber.

Tea at 3:30 pm in SC 1425. (Contact Person:
Marcelo Disconzi)

SPECIAL EVENT: Shanks
Workshop on Geometric Analysis, **March 11-12, 2016,
Vanderbilt University**

**
Friday, March 18th**

__Speaker:__** Yoshiyasu
Fukumoto, Kyoto University, Japan**

__Title:__ **On the Strong Novikov
Conjecture for Locally Compact Groups in Low Degree
Cohomology Classes**

G-manifold X to the K-theory of the C*-algebra of the group G. The action of G on X is assumed to be proper and cocompact. Under

the assumption that the Kronecker pairing of a K-homology class with a low-dimensional cohomology class is non-zero, we prove that

the image of this class under the index map is non-zero. Neither discreteness of the locally compact group G nor freeness of the action

of G on X are required. The case of free actions of discrete groups was considered earlier by B. Hanke and T. Schick.

** Friday, April 1st**

__Speaker:__** Chris
Leininger, UIUC**

__Title:__ **Pseudo-Anosov homeomorphisms
and homology**

__Abstract__**: **The mapping
torus of a pseudo-Anosov homeomorphism admits a hyperbolic
metric by work of Thurston. Pseudo-Anosov

homeomorphisms themselves have interesting geometric and
dynamical properties which can be distilled into a single
invariant called

the stretch factor. I will explain what pseudo-Anosov
homeomorphisms are through examples, and describe the various
interpretations

of the stretch factor. After recalling some motivating
connections between the stretch factor and the action on
homology, I will describe,

with the help of hyperbolic geometry, a new link between these
two. This is joint work with Ian Agol and Dan Margalit.

**
Wednesday, April 6th (at the "Group Theory and Topology"
Seminar), 4:10pm in SC 1310 **

__Speaker:__** Viatcheslav
Kharlamov, IRMA, Strasbourg, France**

__Title:__ **Real cubic projective
hypersurfaces**

__Abstract__**: **Cubic hypersurfaces
is one of the classical objects of study in real algebraic
geometry. While the case of cubic

surfaces can be considered as rather well understood, the case
of cubics of dimensions five and higher remain still largely

open (even over the complex field). The topological and
deformation classifications of real cubic hypersurfaces in
dimensions

3 and 4 was achieved only recently. In a joint work with S.
Finashin (work in progress) we suggest a solution of one
further,

related, problem, that of the topological and deformation
classifications of pairs consisting of a real cubic 3-fold and
a real

straight line contained in it. Our approach is based on a
certain, spectral, correspondance between such pairs and plane

quintics equipped with a real theta-characteristic. This
correspondance allows to disclose some new phenomena both on
the

cubic and quintic sides.

**
Friday, April 8th**

__Speaker:__** Viatcheslav
Kharlamov, IRMA, Strasbourg, France**

__Title:__ **Real rational symplectic
4-manifolds**

__Abstract__**: **The foundational
results of Gromov-Taubes and Seiberg-Witten allowed to
understand rather well the structure of

rational and ruled symplectic 4-manifolds and, in particular,
to prove that every such symplectic manifold is Kaehler. The

aim of our joint work with V. Shevchishin (work in progress)
is to show at what extent the latter result can be extended

to rational symplectic manifolds equipped with an
anti-symplectic involution. Our approach is based on
appropriate real

versions of Lalonde-McDuff inflation and rational blow-ups. It
shows, in particular, that the classification of
anti-symplectic

involutions on real rational symplectic 4-folds is very
similar to that of the classification of real rational
surfaces.

**
Friday, April 15th**

__Speaker:__** Jonathan
Campbell, University of Texas at Austin**

__Title:__ **The K-Theory of Varieties**

__Abstract__**: **The Grothendieck ring
of varieties is a fundamental object of study for algebraic
geometers - it is a universal home

for the Euler characteristic and is related to birational
invariants of varieties. I'll introduce this object, and show
how it

arises from higher algebraic K-theory (which I will also
introduce). I'll also present applications of this result:
lifting so-called

motivic measures (including the zeta-function!) to the
infinite loop space level.

**Tuesday, April 19th,
4:10-5:00pm, SC 1310 (special day and time!)**

__Speaker:__** Hang Wang,
University of Adelaide, Australia**

__Title:__ **A fixed-point theorem on
noncompact manifolds**

__Abstract__**: **The Lefschetz number
of an isometry of a compact manifold measures of the "size" of
the fixed-point set. This is

incorporated in the Atiyah-Segal-Singer fixed point theorem,
by computing the equivariant index of an elliptic operator

on a compact manifold, equipped with a compact Lie group
action. In this talk the Atiyah-Segal-Singer fixed point
formula

is generalized to noncompact manifolds. We use tools from
operator algebra to deal with elliptic operators having
infinitely

dimensional kernels and explore applications in representation
theory of some noncompact Lie groups. This is joint work

with Peter Hochs.

**Friday, April 22nd**

__Speaker:__** Rafael
Guglielmetti, University of Fribourg, Switzerland**

__Title:__ **Computing invariants of
hyperbolic Coxeter groups and polyhedra**

__Abstract__**: **Given a hyperbolic
Coxeter group G and its associated polyhedron P, we are
interested in computing different invariants

of these two objects. Concerning P, we want to answer to the
following questions: is it compact? has it finite volume? what
is its

f-vector? Regarding G, we would like to compute its Euler
characteristic (which is related to the volume of P when the
dimension

is even), its growth series and growth rate and we want to
know if it is arithmetic or not. We will explain explain how
these

questions can be answered using the Coxeter graph of P and see
that these computations can be handled by a computer.

Old Seminar Web-Pages: Fall 2009, Fall
2010