# Lecture Series

Venue: HIM lecture hall, Poppelsdorfer Allee 45 (if not stated otherwise)

## Batu Güneysu: Hida Distributions

- Wednesday, May 7, 14:00
- Friday, May 9, 14:00

## Mike Hopkins / Dan Freed: TQFT and state sums

- Tuesday, May 20, 16:30 (Mike Hopkins)
- Wednesday, May 21, 16:30 (Dan Freed)
- Monday, May 26, 16:30 (Mike Hopkins)
- Tuesday, May 27, 16:30: Orientifolds and Topology (Dan Freed)

## Peter Teichner / Stephan Stolz: Euclidean Field Theories and Generalized Cohomology

- Monday, May 26, 11:00
- Tuesday, May 27, 11:00
- Thursday, May 29, 11:00
- Friday, May 30, 11:00

## Ingo Runkel: Two-dimensional CFT from three-dimensional TFT

- Tuesday, June 3, 11:00
- Thursday, June 5, 11:00

## Alexander Rosenberg: Geometry of noncommutative ’spaces’ and schemes

- Friday, June 6, 14:00
- Friday, June 13, 14:00, HIM seminar room (Pop. Allee 45, basement)

Abstract: The first talk, will be dedicated to basic notions of noncommutative algebraic geometry - ’spaces’ represented by categories and morphisms of ’spaces’ represented by (isomorphism classes of) functors. We introduce and discuss noncommutative schemes and more general locally affine ’spaces’. The notion of a noncommutative scheme will be illustrated with important examples related to quantized enveloping algebras: the quantum flag varieties and the associated "quantum D-schemes" represented by the categories of (twisted) quantum D-modules. Examples of locally affine noncommutative ’spaces’ which are not schemes are noncommutative Grassmannians and flag varieties and their generalizations.

In the second talk, we recover geometry behind the pseudo-geometric picture sketched in the first lecture. For simplicity, we start with introducing underlying topological spaces (spectra) of ’spaces’ represented by abelian categories and describing their main properties. One of the consequences of these properties is the reconstruction theorem for commutative schemes, which can be regarded as a test for the noncommutative theory. It says, in particular, that any quasi-separated commutative scheme can be reconstructed uniquely up to isomorphism from its category of quasi-coherent sheaves. The noncommutative fact behind the reconstruction theorem is the "geometric realization" of a noncommutative scheme as a locally affine stack of local categories on its underlying topological spaces. The latter is a noncommutative analog of a locally affine locally ringed topological space, i.e. a geometric scheme. We conclude this short introduction to the geometry of noncommutative ’spaces’ and schemes with a sketch of the first notions and facts of pseudo-geometry and spectral theory of ’spaces’ represented by triangulated categories.

Noncommutative algebraic geometry gives new insights and immediate applications to representation theory, in particular to representation theory of quantized enveloping algebras and other algebras of mathematical physics. If time permits, I will try to give a flavor of this part of the story.

## Eli Hawkins: Geometric Quantization and Groupoids

- Friday, June 20, 14:00
- Friday, June 27, 14:00

Abstract: The mathematical idea of (strict deformation) quantization is to deform from a commutative algebra of functions on a manifold to a noncommutative C*-algebra. This is an abstraction of the transition from classical to quantum physics. In part 1 of this talk, I will describe different geometric examples of quantization, constructed using geometric quantization and groupoids. To describe these examples, I will explain prequantization, polarization, Lie groupoids, Lie algebroids, and convolution algebras.

In part 2, I will show how these examples can be unified through a general construction using symplectic groupoids. This involves my new concept of groupoid polarization. This general construction is still incomplete, but it holds the possibility of quantizing most Poisson manifolds.

## Masoud Khalkhali (University of Western Ontario): Introduction to Noncommutative Geometry

- Wednesday, June 25, 14:00
- Thursday, June 26, 14:00
- Tuesday, July 1, 16:00
- Thursday, July 3, 14:00

Abstract: This series of lectures is intended as an introduction to some of the basic ideas of noncommuttive geometry. No previous familiarity with the subject is assumed. Some of the topics to be discussed include: Noncommutative spaces and noncommutative quotients, Cyclic cohomology, Connes-Chern character and Connes’ index formula, applications.

References include:

[1] A. Connes, Noncommutative Differential Geometry, Chapter I: The Chern character in K homology; Chapter II: de Rham homology and noncommutative algebra, Publ. Math. IHES no. 62 (1985), 41-144; also avaialable at http://www.alainconnes.org/en/

[2] A. Connes, Noncommutative Geometry. Academic Press, Inc., 1994, also available at http://www.alainconnes.org/en/

[3] A. Connes and M. Marcolli, A walk in the Noncommutative Garden. An Invitation to Noncommutative Geometry, World Scientific 2008 (edited by M. Khalkhali and M. Marcolli); also available at http://arxiv.org/abs/math/0601054

[4] M. Khalkhali, Lectures on Noncommutative Geometry. An Invitation to Noncommutative Geometry, World Scientific 2008, also available at http://arxiv.org/PS_cache/math/pdf/0702/0702140v2.pdf

[5] M.Khalkhali, From Cyclic Cohomology to Hopf Cyclic Cohomology in 5 Lectures. available at http://www.math.vanderbilt.edu/~bisch/ncgoa07/talks/khalkhali_NCGOA07.pdf

## Jinhyun Park (Purdue University): Motivization for additive K-theory

- Friday, July 4, 14:00
- Friday, July 18, 14:00

Abstract: The first talk is an introduction to the subject. The second talk is about some recent ideas. I will begin with some basics of the algebraic K-theory of Quillen and additive K-theories. How Hochschild/cyclic homologies can be seen as additive K-theories is explained by recalling Leibniz/Lie algebra homologies. I will explain what "motivization" for these theories means. Then, I will sketch motivic analogues of the Connes B-operation and the shuffle product in cyclic homology theory in terms of "additive higher Chow groups". As a corollary, we obtain a motivic CDGA of absolute Kähler differentials.

## Alan Carey (Canberra): Semifinite NCG and some applications

- Monday, July 14, 14:00
- Tuesday, July 15, 11:00
- Thursday, July 17, 14:00

Abstract: I will explain the motivation for developing the idea of semifinite noncommutative geometry, give an outline of the local index formula for semifinite spectral triples and then discuss some recent generalizations and applications.

## Matthias Lesch: Pseudodifferential Operators and Regularized Traces

- Monday, July 14, 16:30
- Thursday, July 17, 16:30
- Monday, July 21, 16:30

Abstract: In these informal lectures I will give a leisurely introduction to pseudodifferential operators. I will discuss in particular the famous regularized traces (a la Wodzicki and Kontsevich-Vishik). Then I will report on my recent joint work with Henri Moscovici and Markus Pflaum on regularized traces in the parameter dependent pseudodifferential calculus. While "ordinary" traces give rise to K-theory invariants I will advertise that the mentioned traces give rise to *relative* K-theory invariants. There is an intriguing formula which expresses the Atiyah-Patodi-Singer eta-invariant in terms of a Chern-character like formula involving a regularized trace.

## Henri Moscovici (Ohio State University): Index theory and characteristic classes for non-commutative spaces

- Monday, July 21, 14:00
- Tuesday, July 22, 14:00
- Thursday, July 24, 14:00
- Friday, July 25, 14:00

Abstract: This series of four expository lectures will be devoted to the methods that allow the explicit calculation of the characteristic classes of noncommutative spaces. The first will cover the developments that led to the noncommutative version of the local index formula [1]; the second and the third will highlight its applications to the transverse geometry of foliations, which led in turn to the detection of a novel form of transverse symmetry and of cyclic cohomology [2], as well as to the resurgence of the classical pseudogroups of Lie and Cartan morphed into Hopf algebras [4]; the fourth will discuss the extension of the theory of characteristic classes to twisted spectral triples [3].

[1] A. Connes and H. Moscovici, *The local index formula in noncommutative geometry*, Geom. Funct. Anal. 5 (1995), no. 2, www.alainconnes.org/en/downloads.php

[2] A. Connes and H. Moscovici, *Hopf algebras, cyclic cohomology and the transverse index theorem*, Comm. Math. Phys. 198 (1998), no. 1, arxiv.org/abs/math/9806109

[3] A. Connes and H. Moscovici, *Type III and spectral triples*, Traces in Geometry, Number Theory and Quantum Fields, Aspects of Mathematics E 38, Vieweg Verlag 2008, pp. 57--71; arxiv.org/abs/math/0609703

[4] H. Moscovici and B. Rangipour, *Hopf algebras of primitive Lie pseudogroups and Hopf cyclic cohomology*, arxiv.org/abs/0803.1320

## Katia Consani (J. Hopkins University): An introduction to Endomotives

**Tuesday, July 22, 11:15****Wednesday, July 23, 14:00****Thursday, July 24, 11:15**