Spring 2017

** Organizers: Anna Marie Bohmann, Ioana
Suvaina, ****Rares Rasdeaconu**

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

Clifford algebras and index theory. The talk should be accessible to everyone.

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

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)

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)

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

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

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
**

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
**

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
**

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