Mathematics Colloquium Spring 2008

April 8, 2008

On Current Conjectures on Block Theory

Dr. Ahmad M. Alghamdi, Fulbright Scholar 

Group theory is that part of Mathematics which deals with symmetry. Representation of a group is to represent this group by matrices or linear transformations. Each representation can be decomposed into the so-called irreducible representations. The major contributions of this area of Mathematics is due to Frobinous, Schur, Brauer and Green. Nowadays, much of the research in block theory is devoted to prove conjectures which have been introduced in this field by J. Alperin, E. Dad, M. Broue and G. Robinson. We shall talk about Dade's Projective Conjecture and the Ordinary Weight Conjecture.

April 15, 2008

Endotrivial Modules

Prof. Jon Carlson, MSRI 

Endotrivial modules play an important role in the representation theory of finite groups over fields of finite characterictic. In this lecture, I will attempt to explain with some simple examples, what the modules are, how the modules arise and how they are constructed.

April 22, 2008

What does p-completion do to the classifying space of a finite group?

Dr. David Benson, MSRI 

After a gentle introduction to classifying spaces and p-completion, I shall discuss what happens when you p-complete the classifying space of a finite group. This leads to some interesting connections between algebra and topology. The punchline is an algebraic way to compute some topological invariants of a finite group. I'll end with some conjectures about the boundary between polynomial and exponential growth for these invariants.

April 29, 2008

Modular Representation Theory and Cohomology: An Elementary Approach.

Prof. Julia Pevtsova, MSRI 

I shall describe an approach to the study of modular representation theory via restrictions of representations to certain elementary subalgebras which are analogs of one-parameter subgroups. These subalgebras behave like a cyclic group Z/p and have ``easy" representation theory.

As an application, we can recover the algebraic variety associated to the cohomology ring of a finite group scheme G by purely representation-theoretic means, in particular generalizing Quillen's "stratification theorem" for group cohomology. As another application, I'll introduce a new class of modules, ``modules of constant Jordan type" and state some recent results and conjectures about them. 

Most of the results apply to any finite group scheme, but they are non-trivial even in the case of the finite group Z/p x Z/p, which is a baby example that will be used for illustrative purposes throughout the talk.

May 6, 2008

The Frobenius number of a block

Prof. Radha Kessar, University of Aberdeen 

Let p>0 be a prime number, k an algebraic closure of the field of p elements and B a finite dimensional k-algebra. The Frobenius number of B is the smallest positive integer, say n, such that B has a k-basis with respect to which all the multiplicative structure constants of B lie in the subfield of p^n elements. I will explain the role of Frobenius numbers in modular representation theory, state some open questions, and give a few answers.

May 13, 2008

NO COLLOQUIUM THIS WEEK

May 20, 2008

Submanifolds and G_2 Geometry

Dr. Jason Lotay, NSF Postdoctoral Fellow at MSRI 

In Berger's list of Riemannian holonomy groups, there is an exceptional case in dimension 7 given by the Lie group G_2. On a manifold with G_2 holonomy, one is naturally lead to consider two distinguished classes of submanifolds: associative 3-folds and coassociative 4-folds. These are calibrated, hence minimal, submanifolds which can be thought of as generalizations of interesting lower-dimensional geometries. I will give examples of these submanifolds and discuss some of their properties.

May 27, 2008

Representations of Galois algebras

Prof. Vyacheslav Futorny, Institute of Math and Statistics, Univ. of Sao Paulo 

The talk is based on joint results with S.Ovsienko. We will discuss representation theory of recently introduced class of Galois algebras which are certain invariant sub-algebras in skew group rings. Examples of Galois algebras include the universal enveloping algebra of gl(n), the corresponding W-algebras and the Generalized Weyl algebras. The structure of a Galois algebra is closely related with the Gelfand-Kirillov conjecture.