L2

I'll explain how the `Auslander philosophy' from finite dimensional algebras gives new methods to tackle problems in higher-dimensional birational geometry. The geometry tells us what we want to be true in the algebra and conversely the algebra gives us methods of establishing derived equivalences (and other phenomenon) in geometry. Algebraically two of the main consequences are a version of AR duality that covers non-isolated singularities and also a theory of mutation which applies to quivers that have both loops and two-cycles.

An overview of the early history of the soliton (1960-1970) and equipartition in nonlinear 1D lattices : From Fermi-Pasta-Ulam to Korteweg de Vries, to Nonlinear Schrodinger*…., and recent developments .

I shall give a gentle introduction to the cohomology of finite groups from the point of view of algebra, topology, group actions and number theory

The study of representation growth of infinite groups asks how the

numbers of (suitable equivalence classes of) irreducible n-dimensional

representations of a given group behave as n tends to infinity. Recent

works in this young subject area have exhibited interesting arithmetic

and analytical properties of these sequences, often in the context of

semi-simple arithmetic groups.

In my talk I will present results on the representation growth of some

classes of finitely generated nilpotent groups. They draw on methods

from the theory of zeta functions of groups, the (Kirillov-Howe)

coadjoint orbit formalism for nilpotent groups, and the combinatorics

of (finite) Coxeter groups.

This short course runs from Monday 29th June to Friday 3rd July. For details of the course and how to register, please visit http://www2.maths.ox.ac.uk/oxmos/meetings/moms/.