Algebra Seminar with Vladimir Dotsenko, Sarah Whitehouse, Peder Thompson, Florian Eisele and Henning Krause

Programme

9:45-10:45
Vladimir Dotsenko (Strasbourg): Yamaguti algebras, noncrossing partitions, and sl_2

11:00-12:00
Sarah Whitehouse (Sheffield): Homotopy theory and geometry related to multicomplexes

Lunch at Rullan

13:15-14:15
Peder Thompson (Vaesteraas): Cosyzygy modules

14:30-15:30
Florian Eisele (Manchester): Conjectures in the modular representation theory of finite groups and tame blocks

Coffee break

16:00-17:00
Henning Krause (Bielefeld): Linear representations of posets - from concrete to abstract

Abstracts:

Yamaguti algebras, noncrossing partitions, and sl_2

Abstract: Among various types of algebras originating in differential geometry, those known as Lie-Yamaguti algebras still remain somewhat mysterious algebraically. Last year, A.Das introduced their 'associative' version, the Yamaguti algebras, which are supposed to serve as envelopes of Lie-Yamaguti algebras. I'll report on joint work with Frédéric Chapoton, which exhibits relationships between this algebraic structure, combinatorics of noncrossing partitions, and the Lie algebra sl_2.

Homotopy theory and geometry related to multicomplexes

Abstract: A multicomplex is a variant of a bicomplex and these structures arise naturally in many geometric, topological and algebraic contexts, for example, from filtered simplicial sets. After a quick overview of work on homotopy theory related to spectral sequences, I will explain some recent joint work with Joana Cirici and Muriel Livernet. This explores homotopy theories related to the two spectral sequences of a truncated multicomplex. There are potential applications to the study of homotopy types of almost and generalized complex manifolds.

Cosyzygy modules

Abstract: In a projective resolution of a module, n-th syzygy modules are unique up to stable isomorphism. On the other hand, since not every module embeds into a projective, the dual "cosyzygy" modules are less well-behaved. We characterize all possible cosyzygy modules of a given finitely generated module by relating them to a "canonical" cosyzygy module. This construction yields a characterization of Gorenstein rings and highlights some interesting open questions. The talk is based on joint work with P. A. Bergh and D. A. Jorgensen.

Conjectures in the modular representation theory of finite groups and tame blocks

Abstract: Even though group algebras are a type of finite-dimensional algebra, their representation theory is usually studied using entirely different methods than those used in the representation theory of finite-dimensional algebras. Block algebras of finite and tame representation type are the exception to this. A (near) classification of the latter was achieved by Erdmann using Auslander-Reiten theory, and was later refined using group-theoretic methods. Surprisingly, this did not settle all big open conjectures in modular representation theory for these blocks. In this talk, I will report on recent joint work with Eaton, Kessar, Linckelmann and Schaeffer Fry that finally settles Donovan's conjecture for these blocks. I will also report on some past work of myself on the silting theory of these blocks, which has some connection to the Donald-Flanigan conjecture.

Linear representations of posets - from concrete to abstract

Abstract: For any finite totally ordered set and any field, the finite dimensional linear representations form an abelian category. Various classes of subcategories admit natural combinatorial descriptions, and counting them yields familiar integer sequences. Surprisingly, in some cases new integer sequences arise. These examples will be discussed in the first part of my talk. The formulation of such counting problems leads to the construction of a universal category of poset representations which does not depend on the choice of any ring of coefficients. The second part of the talk is devoted to a more abstract perspective, providing a characterisation of those length categories that arise as categories of linear poset representations.

These seminars is held on Zoom: https://uu-se.zoom.us/j/64555726999, (please contact the organiser for passcode)

This is a seminar in our algebra seminar series.