Noncommutative Geometry

Speaker: Ali Moatadelro (Western)

"Representation theory of compact quantum groups with examples, lecture 6, Irreducible representations of semisimple Lie agbras. "

Time: 09:30

Room: MC 107

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.

Geometry and Topology

Speaker: Clark Barwick (MIT)

"Algebraic K-theory of $\infty$-categories"

Time: 15:30

Room: MC 107

In joint work with John Rognes, we show how to transfer the technologies and results of Quillen and Waldhausen in higher algebraic $K$-theory to the context of $\infty$-categories. Analogues of the $S_{\bullet}$ and $Q$-constructions — as well as versions of the additivity, localization, and d evissage theorems — are among the results we find in this new context. As a motivation for this work, we discuss a conjecture of Hopkins, Waldhausen, and Rognes on the algebraic $K$-theory of $BP\langle n\rangle$.

Analysis Seminar

Speaker: Ilya Kossovskiy (Western)

"Envelopes of holomorphy for real submanifolds in a complex space II"

Time: 15:30

Room: MC 107

One of the most impressive phenomena in several complex variables is the phenomenon of forced analytic continuation for holomorphic functions. The biggest domain, to which the family of all holomorphic functions extends, is called the envelope of holomorphy of a domain or of a real submanifold in a complex space. Envelopes of holomorphy have some nice geometric description, making them in a sense similar to convex hulls of domains and submanifolds in a Euclidian space. In the present talk we discuss some classical theorems for domains of holomorphy as well as some new results for real submanifolds in a complex space.

Noncommutative Geometry

Speaker: Ali Fathi (Western)

"Cyclic cohomology 9, K-theory for C^*-algebras (II): Basic K-groups, continued."

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

Colloquium

Speaker: Heydar Radjavi (Waterloo)

"When small parts imply small wholes"

Time: 15:30

Room: MC 107

Roughly speaking, this talk is about questions of the following general type on collections of matrices or linear operators: if we know something about the collection to be "small" in some sense, when can we say the collection is itself small? A classical example of the kind of results we are interested in is the old theorem that if a group G of complex matrices is irreducible (i.e., the members of G have no nontrivial invariant subspace in common), and if the traces of members of G form a finite set, then G is itself finite. We'll consider (a) measures of smallness other than finiteness, e.g., boundedness; and (b) linear functionals other than just trace.

Algebra Seminar

Speaker: Ekaterina Shemyakova (Western)

"Darboux transformation methods for integrable equations: some classical and own results "

Time: 14:30

Room: MC 107

Darboux transformation methods for integrable equations can be classified as differential equations, algebra, analysis, geometry of surfaces, mathematical physics or theoretic computer algebra. In my talk I shall give some introdution into the area intertwining some classical results with some of my own.