Geometry Seminar

                                                                                                                      Vanderbilt University
                                                                                                                             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
(Colloquium talk) - SPECIAL EVENT



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

                  Abstract: The main result I will discuss is non-vanishing of the image of the index map from the G-equivariant K-homology of a
                  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

AbstractThe 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, Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Fall 2015