Links between commutative algebra and representation theory

Objective: The aim of the meeting is to foster connections between the research areas of commutative algebra and the representation theory of algebras, with a programme designed to encourage interactions between participants working in these fields.

Speakers: 

• Clark Barwick (Edinburgh)

• Benjamin Briggs (Imperial College London)

• Raquel Coelho Simões (Lancaster University)

• Clover May (University of Bristol)

• Alice Rizzardo (University of Liverpool)

• Amit Shah (City St George's, University of London)

Abstracts:

Ben Briggs:

The fundamental group(s) of a finite dimensional algebra

I will explain how a presentation for a finite dimensional algebra A, by a quiver with relations, gives rise to a "fundamental group" for A, closely analogous to the fundamental group of a space. This is an old invariant studies by Farkas, Green, Marcos, de la Peña, Saorín, and Le Meur, among others.

Seemingly unrelated, the "first Hochschild cohomology" of A is the Lie algebra of derivations on A, modulo the inner ones. This is an important invariant, and there's still a lot we don't know about its Lie algebra structure. I'll explain how to prove that every maximal torus in this Lie algebra arises as the dual of the fundamental group of associated to some presentation of A. In other words, you can compute the rank of this Lie algebra "topologically". We can use this to build some useful derived invariants. For example, one consequence is that there are only finitely many monomial algebras in any derived equivalence class of finite dimensional algebras. This is all joint work with Lleonard Rubio y Degrassi.

Clover May:

Representations of Mackey functors and equivariant homotopy theory

An algebra’s representation type, tame or wild, determines whether it is possible to describe all its indecomposable representations in a meaningful way. Mackey functors, which are representations of the Mackey algebra, were introduced by Dress and Green to encode operations like restriction and induction in representation theory. They play a central role in equivariant homotopy theory, where homotopy groups are replaced by homotopy Mackey functors. In this talk, I will discuss the representation type and derived representation type of cohomological Mackey algebra using quivers, as well as some consequences in equivariant homotopy theory. This is joint work with Jacob Grevstad.

Alice Rizzardo: 

New examples of triangulated categories with non-unique enhancements

We all like it when a triangulated category has a unique enhancement, but how often does this happen? Examples of triangulated categories with multiple enhancements in the literature are rare and not very well behaved. In this talk we construct new examples of triangulated categories over a field that admit a t-structure, and that admit non-unique enhancements.

Amit Shah:

Higher homological algebra

Higher homological algebra is a relatively new research area that takes ideas from classical “1-dimensional” homological algebra and representation theory of finite-dimensional algebras, and generalises them to higher dimensions. In this talk, I'll try to give a little motivation and an introduction to higher homological algebra.

Raquel Coelho Simões: 

Mutation for simple-minded objects

Structural understanding of categories is often gained by studying their generators. There are usually two types of generators: projective-minded objects and simple-minded objects. Projective-minded objects play a central role in tilting theory and have led to the development of cluster-tilting theory. Mutation of projective-minded objects, which is a crucial aspect in the study of generators, is well understood. 

Mutation theory of simple-minded objects is much less well understood and more intricate despite simple-minded generators being more fundamental: they go back to Schur and Jordan-Holder. Indeed, there are contexts in which one has simple-minded generators but no projective-minded generators, e.g. length abelian categories without enough projectives or stable module categories of selfinjective algebras.

In this talk I will explain the relationship between mutation, tilting and reduction for simple-minded objects. This is based on joint work with Nathan Broomhead, David Pauksztello, David Ploog and Jon Woolf. 

Clark Barwick:

Factorization algebras in quite a lot of generality

Factorization algebras are algebraic gadgets that capture the structure one finds on the observables of a quantum field theory. They play a role in geometric representation theory — in particular via constructions like the affine Grassmannian.

In the last decade there has been a flurry of interest in arithmetic quantum field theories​. Since the 1960s, researchers have identified an analogy between various objects of arithmetic geometry and low-dimensional manifolds. For example, Spec of a number ring “looks like” an open 3-manifold, and primes therein “are” embedded knots. This story has become known as arithmetic topology​. The idea of arithmetic QFT is to enrich that analogy by importing tools from physics, just as with low-dimensional topology. One even dreams of using these tools to study number-theoretic questions (the behavior of L-functions, Langlands dualities, etc.).

But the objects of arithmetic geometry are not​ manifolds. The tools of topology and differential geometry do not work directly in arithmetic. So it’s unclear how to translate physical concepts to arithmetic settings.

To this end, we introduce a minimalist framework for factorization algebras, where the role of the spacetime manifold can be played by a geometric object of a very general sort.

Registration: The registration form is now closed. If you would like to attend please contact the organisers.