Mathematics Colloquium

As every algebra student knows, Maschke's theorem breaks down when the characteristic of the field divides the order of the group. This brings about modular representation theory, which is a beautiful subject in its own right with applications across mathematics and theoretical physics. The starting point is the block decomposition, and the subtlety is to understand the blocks and how they are related.

In the Langlands program one works with (infinite-dimensional) representations of infinite groups such as GL(n) over the p-adic numbers. Even for complex coefficients there is a fundamental block decomposition due to Bernstein, and a great deal is known about the individual blocks. In the talk I will try to convince the audience that mod p representations of p-adic Lie groups are important in number theory, and I will present new results (obtained in collaboration with Peter Schneider) on derived analogues of the Bernstein decomposition, and the Bernstein center, in the mod p setting.

Most of the talk will be accessible to first-year graduate students, and open-minded undergraduates, in all areas of mathematics.