Geometry Seminar

                                                                                                                      Vanderbilt University
                                                                                                                             Spring  2017

   Organizers:  Anna Marie Bohmann, Ioana Suvaina, Rares Rasdeaconu

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

                Friday, February 3rd

                Speaker: Rudy Rodsphon, Vanderbilt University

                Title:  Dirac operators and index theory

In order to set the ground for next week's talk, We shall give a gentle introduction to Dirac operators,
                Clifford algebras and index theory. The talk should be accessible to everyone.  (Contact Person: Ioana Suvaina)


                Friday, February 10th

                Speaker: Rudy Rodsphon, Vanderbilt University

                Title:  Quantizations and index theory

A way to describe succinctly (local) index theory on closed spin manifolds is the following slogan of Quillen:
                Dirac operators are a "quantization" of connections, and index theory is a "quantization" of the Chern character. For non
                necessarily spin manifolds, pseudodifferential operators and their symbolic calculus play a crucial role in the original proofs
                of the index theorem. However, symbols may also be viewed as a deformation quantization of functions on the cotangent bundle,
                which has led to other fruitful approaches to index theory through another "quantization" process. Even if both viewpoints originate
                from physics (more precisely from quantum mechanics), the methods used involve a priori quite different technologies. The upshot
                of the talk will be to see that these different theories might have more to tell to each other, and that far reaching index problems may
                be solved very directly from such an interaction.   (Contact Person: Ioana Suvaina)

                Friday, February 17th

                Speaker: Hang Wang, University of Adelaide, Australia

                Title:  Index Theory and Character Formula

                Abstract: This talk will focus on the link between geometry and representation theory of Lie groups in the context of operator algebras.
                Weyl character formula describes characters of irreducible representations of compact Lie groups. This formula can be obtained using
                geometric method, for example, from the Atiyah-Bott fixed-point theorem. Harish-Chandra character formula, the noncompact analogue
                of the Weyl character formula, can also be studied from the point of view of index theory. We apply orbital integrals on K-theory of
                Harish-Chandra Schwartz algebra of a semisimple Lie group G, and then use geometric method to deduce Harish-Chandra character
                formulas for discrete series representations of G. This joint work with Peter Hochs (arXiv:1701.08479).  (Contact Person: Gennadi Kasparov)

                Friday, March 3rd

                Speaker: Angelica Osorno, Reed College

                Title:  Algebraic models of homotopy types

                Abstract: One of the goals of algebraic topology is to classify topological spaces up to homotopy. This task becomes more manageable
                when we restrict to spaces that only have finitely many  non-vanishing homotopy groups. In this talk I will give a historical account of the
                different algebraic models that have been developed to classify finite homotopy types, with a special emphasis on recent joint work with
                N. Gurski, N. Johnson and Marc Stephan on modeling stable 2-types. (Contact Person: Anna Marie Bohmann)


                March 10-11, 2017 Shanks Workshop on Real Algebraic Geometry  
Location: Stevenson Center 1432 (Contact Person: Rares Rasdeaconu)



                Thursday, March 16th, Colloquium Talk, 4:10-5pm in SC5211 (tea in SC1425 starting 3:30pm)

                Speaker: Viatcheslav Kharlamov, Strasbourg University, France

                Title:  The 16th Hilbert Problem: Disclosed and Hidden

                Abstract: The talk will be focused on the first part of this problem: the part devoted by Hilbert to topological properties of real algebraic
                curves and surfaces. It is this part, together with 9 other problems of the famous list that was chosen by Hilbert for the oral presentation.
                In this talk we will present certain milestones achieved in the directions influenced by this problem. In particular, we will mention those
                which allowed to respond to at least those of Hilbert questions he posed precisely. We will try to explain at least some of the multitude of
                new ideas, methods and theories disclosed (giving preference to topological and geometrical settings), but also to list selected, still open,
                questions. (Contact Person: Rares Rasdeaconu)


                Friday, March 17th

                Speaker: Radu Ionas, C.N. Yang Institute for Theoretical Physics, Stony Brook University

                Title:  Hidden hyperkaehler symmetries and gravitational instantons

                Abstract: I will present a generalization of the Gibbons-Hawking Ansatz to a class of hyperkaehler metrics with hidden symmetries,
                which I will then use to construct explicit generic metrics on gravitational instantons of type D_k. (Contact Person: Ioana Suvaina)


                March 25-26, 2017 Shanks Workshop on Homotopy Theory
Location: TBA (Contact Person: Anna Marie Bohmann)

                Friday, April 7th (3:10pm-4pm) - Doubleheader

                Speaker: Olguta Buse,  IUPUI

                Title:  On contact embeddings in low dimensions

                Abstract: Transversal knots in contact thee dimensional manifolds have standard solid tori neighboorhoods. We introduce the concepts
                of  capacity  and shape for  a three dimensional contact manifold (M, \xi) relative to a transversal knot K to study the sizes of these tori. 
                We will explain the connection with the existing literature   and compute  the shape  in the case of lens spaces L(p,q) with a toric contact
                structure.  The main tool used here are  rational surgeries which will be explained through their toric interpretations based on the continuous
                fraction expansions of p/q. This is joint work with D. Gay. (Contact Person: Ioana Suvaina)

Friday, April 7th (4:10pm-5pm) - Doubleheader    

Speaker: Rodrigo Perez,  IUPUI
Title:  Dynamics of bi-reversible automata

Abstract: We will review the proof that Grigorchuk's group has intermediate growth, in order to motivate the wreath product notation for
self-similar groups. This class of groups is easily defined:

Let T be an infinite binary tree. Each state of an automaton with 2 inputs defines an automorphism of T (as a graph). The set of all such
automorphisms generates a group associated with the automaton, and such groups are called "self-similar".

Many other interesting groups have a self-similar structure. A classical example is the lamplighter group, and Nekrashevych introduced
the family of iterated monodromy groups, associated to post-critically finite rational maps on the Riemann sphere. It is a curious fact
that many of the most interesting self-similar groups are associated to bi-reversible automata. The question arises of constructing random
bi-reversible automata in order to generate examples of groups with potentially interesting properties. We will tackle this question. This
is joint work with joint with D. Savchuk (Contact Person: Ioana Suvaina)

Friday, April 14th

Speaker: Mehdi Lejmi,  CUNY Bronx Community College

Title:  The Chern-Yamabe problem

Abstract: On an almost Hermitian manifold the Chern connection is the unique Hermitian connection with J-anti-invariant torsion.
In this talk, we compare the Chern scalar curvature with the Riemannian one. Moreover, we study an analog of Yamabe problem by
looking for an almost Hermitian metric with constant Chern scalar curvature in a conformal class extending results of Angella,
Calamai and Spotti to the non-integrable case. This is joint work with Markus Upmeier. (Contact Person: Ioana Suvaina)

Thursday, May 4th

Speaker: Alexander Engel, Texas A&M University

Title:  Wrong way maps in homology of groups

Abstract: Given a certain geometric situation (a manifold embedded in a specific way into another manifold), we will construct
wrong way maps in the homology of the corresponding fundamental groups and discuss some applications (which are mainly
index theoretic in spirit). (Contact Person: Rudy Rodsphon)


 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, Spring 2016, Fall 2016