Noncommutative Geometry

Speaker: Ali Moatadelro (Western)

"Representation theory of compact quantum groups with examples, lecture 3. Irreducible representations of SU(3), continued."

Time: 09:30

Room: MC 106

In this series of lectures, we will discuss basic examples of compact quantum groups and their (finite dimensional) representations. We will start with reviewing the classical theory. We shall classify all finite dimensional irreducible representations of compact Lie groups SU(2) and SU(3). Then we will proceed to the general theory of representation of compact Lie groups and will discuss several important results including the highest weight theory, the Peter-Weyl decomposition theorem, and also the Borel-Weil-Bott construction of representations. Finally, we will see how much of the theory holds in the quantum case.

Analysis Seminar

Speaker: Rahim Moosa (Waterloo)

"Real-analytic versus complex-analytic families of complex-analytic sets"

Time: 15:30

Room: MC 107

Suppose M is a compact complex manifold. Model theory (a branch of mathematical logic) provides at least two approaches to the study of the complex-analytic subsets of Cartesian powers of M, roughly corresponding to whether one focuses on the real or complex structure on M. We can view M as definable in the structure R_an; that is, as a real globally subanalytic manifold. On the other hand, we can work in the Zariski-type structure CCM where M is the universe and there are predicates for all complex-analytic subvarieties of Cartesian powers of M. The two approaches lead to different notions of a "definable family" of complex-analytic subsets. I will give a geometric characterization, obtained in joint work with Sergei Starchenko in 2008, of when these two notions coincide, in terms of the Barlet or Douady spaces. As a consequence one has that for M Kaehler the two notions coincide.

Graduate Seminar

Speaker: Richard Gonzales (Western)

"The equivariant Chern character"

Time: 16:30

Room: MC 107

A classical result of Atiyah and Hirzebruch establishes a deep connection between K-theory and cohomology, via the Chern character. The purpose of my talk is to describe this relation in precise terms, and give an overview of its generalizations to the equivariant setting. Along the way we introduce a new class of objects, coming from algebraic geometry, on which many of these classical topological techniques could be successfully applied.

Noncommutative Geometry

Speaker: Arash Pourkia (Western)

"Cyclic Cohomology 4"

Time: 14:30

Room: MC 107

Cyclic (co)homology is the noncommutative analogue of de Rham (co)homology and as such plays an important role in noncommutative geometry and its applications (in operator algebras, index theory, ...) A variant of it, topological Hochschild and cyclic homology, plays an important role in algebraic K-theory as well.

We will give a series of lectures on the subject (2 hours per week), starting from basic material and gradually building towards more advanced stuff. Outline: 1. Basic homological algebra in abelian categories 2. Hochschild (co)homology; computations (Hochschild-Kostant-Rosenberg, group algebras) 3. Cyclic (co)ohomology, Connes' spectral sequence; computations (relation with de Rham, group algebras); cyclic category. 4. K-theory and K-homology, 5. Connes-Chern character 6. An index formula 7. Applications to idempotent conjectures.

The basic texts to follow are: 1. Cyclic Homology, J. L. Loday 2. Noncommutative Geometry, A. Connes 3. Noncommutative Differential Geometry, Publication math. IHES, 1985, A. Connes. 4. Basic noncommutative geometry, Masoud Khalkhali

Algebra Seminar

Speaker: Marc Moreno Maza (Western)

"Triangular decomposition of semi-algebraic systems"

Time: 14:30

Room: MC 107

Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. We propose adaptations of these tools focusing on solutions of the real analogue: semi-algebraic systems.

We show that any such system can be decomposed into finitely many so-called "regular semi-algebraic systems". We propose two specifications of such a decomposition and present corresponding algorithms. Under some assumptions, one type of decomposition can be computed in singly exponential time with respect to the number of variables. We have implemented our algorithms and the experimental results illustrate their effectiveness. A software demonstration will conclude this talk.

This is a joint work with Changbo Chen (UWO), James H. Davenport (Bath U.), John P. May (Maplesoft), Bican Xia (Peking U.) and Xiao Rong (UWO).

The corresponding article is published in the Proceedings of the 2010 International Symposium of Symbolic and Algebraic Computation (ISSAC'10) and available at ww.csd.uwo.ca/~moreno/Publications/118_paper.pdf