Event Details

Is convex geometry trying to teach us homological algebra?

An abelian category is one in which there are short exact sequences. In particular, each morphism has a kernel, a cokernel and the first isomorphism theorem holds. Examples include the category of abelian groups, the category of finitely generated modules over a finite-dimensional algebra, and coherent sheaves on a smooth projective variety.

For a given abelian category, closely related (pun intended) abelian categories can be constructed by a process called tilting. For a given abelian category H, we define a cone C(H) in a finite-dimensional real vector space. Running over the abelian categories K constructed from H by tilting, the cones C(K) fit together to form a fan. This fan has remarkable properties, for example, the convex geometry of C(K) can tell us whether the Jordan-Holder theorem holds in K, i.e. whether every object in K has a composition series.

In this talk I will outline the construction of the fan through some simple examples coming from representation theory and algebraic geometry and describe some of the interplay between the convex geometry and the homological algebra. The convex geometry is elementary so the only background I will need is second-year linear algebra. There will be lots of pictures and I will assume no background in homological algebra.

The talk will be based on some joint work with Nathan Broomhead, David Ploog and Jon Woolf.