On May 31st and June 1st, we will hold a special meeting of the ARTIN network, in Edinburgh, to welcome permanent appointments who have arrived since 2020. Thanks to generous funding from the Glasgow Mathematical Journal Trust, the Edinburgh Mathematical Society, and the ERC Starting Grant of Alexander Shapiro, we expect to be able to fund accommodation and travel for many participants from the ARTIN network institutions, and you are all warmly encouraged to attend.
We will have talks by: Nora Szakacs (Manchester), Raquel Coelho Simões (Lancaster), Alexander Shapiro (Edinburgh), Tudor Dimofte (Edinburgh), Dinakar Muthiah (Glasgow), Florian Eisele (Manchester), Anna Puskas (Glasgow), Pavel Safronov (Edinburgh),
The talks will take place in the ICMS conference room Bayes 5.10, which is on the downtown campus and is not near to JCMB where the School of Mathematics is headquartered.
The schedule of the workshop is:
Tuesday May 31st
9-11 Coffee/tea
11-12 Talk Dimofte
12-1 Talk Puskas
1-2 Lunch
2-3 Talk Muthiah
3-4 Coffee/tea
4-5 Talk Szakacs
Wednesday June 1st
10-11 Talk Eisele
11-12 Coffee/tea
12-1 Talk Safronov
1-2 Lunch
2-3 Talk Coelho Simões
3-4 Coffee/tea
4-5 Talk Shapiro
You can register for the workshop using the form below; this is important even for locals so I can organize the catering. Please email me at djordan@ed.ac.uk with any questions!
There will not be an organized dinner on Tuesday evening.
The titles and abstracts I have received so far are:
Raquel Coelho Simões (Lancaster)
Title: Simple-mindedness in negative Calabi-Yau categories
Abstract: Cluster-tilting theory provides effective techniques for studying positive Calabi-Yau (CY) triangulated categories. The main objects of study here are cluster-tilting objects, which can be viewed as a generalisation of projective modules.
Simple-minded systems (SMSs) were introduced by Koenig-Liu as an abstraction of non-projective simple modules in stable module categories. The idea was to use SMSs as a way to get around the lack of projective generators to help develop a Morita theory for stable module categories.
In this talk, we will see how SMSs in negative CY triangulated categories play the role of cluster-tilting objects in the positive CY setup. This is based on joint work with David Pauksztello and David Ploog.
Florian Eisele (Manchester)
Title: What we know about Donovan’s conjecture
Abstract: One of the major open questions in the representation theory of finite groups over fields of characteristic p>0 is Donovan’s conjecture, which constrains the representation theory of a group G and the structure of its group algebra in terms of the isomorphism type of its Sylow p-subgroups. I will talk about some recent results by myself and others on this conjecture and related questions.
Dinakar Muthiah (Glasgow)
Title: Two approaches to double loop groups
Abstract: Many of my mathematical interests are in one way or another about double loop groups. I will discuss a bit about two approaches that have been fruitful: one via p-adic loop groups, and the other via Coulomb branches.
Alexander Shapiro (Edinburgh)
Title: Positive representation theory
Abstract: Positive representations of quantum groups are defined via cluster structure on character varieties. I will discuss these representations, as well as 2d topological modular functor, and a``continuous'' analogue of a braided monoidal category they give rise to.
Nora Szakacs (Manchester)
Title: The large scale geometry of inverse semigroups
Abstract: Since Gromov's work in the 90s, studying groups as metric spaces has become a main trend in group theory. In turns out that there is a strong link between some of the algebraic properties of the group sand some of the large scale geometric properties of the metric associated to it, and in some cases, this links to properties of a C*-algebra associated to the large scale geometry and the group. In the past few years, some of these results have been extended to so-called inverse semigroups by Chung, Gray, Martínez, Lledo, Silva, and the author. Inverse semigroups are a generalization of groups which encapsulate partial symmetries similarly as to how groups encapsulate symmetries. During the talk, we will give an introductory exposition of these results.
Tudor Dimofte (Edinburgh)
Title: 3d topological twists
Abstract: Topological twisting, introduced by Witten in 1988, is a technique to isolate sectors of supersymmetric quantum field theories that behave topological and can, in principle, give rise to extended TQFT's in the (now) mathematical sense. Famous examples include topological twists of 2d sigma-models (leading to TQFT's fundamental for mirror symmetry/homological mirror symmetry), and topological twists of 4d gauge theories (capturing Donaldson/Seiberg-Witten invariants, as well as geometric Langlands duality). I'd like to discuss topological twists of three-dimensional gauge theories, which sit halfway between these famous examples, and have turned out to be intimately connected with geometric representation theory. I'll review recent progress and open problems in extracting/defining fully extended 3d TQFT's of cohomological type from 3d gauge theories; and will discuss some recent work of mine relating twisted 3d theories with non-semisimple 3d TQFT's of Costantino, Geer, Patureau-Mirand and others.
Anna Puskas (Glasgow, coming soon)
Title: Hecke algebras and Whittaker functions
Abstract: Certain objects from representation theory, for example Whittaker functions, can be constructed via an action of a Hecke algebra. This provides crucial insight into the behaviour of these objects beyond the classical setting: for metaplectic and/or infinite-dimensional groups. In the infinite-dimensional setting the behaviour of the Hecke algebras themselves is somewhat mysterious. We will see how Hecke algebras are used in the construction of metaplectic Whittaker functions, and discuss what can be learned about them in the single and double affine setting.
Pavel Safronov (Edinburgh)
Title: Skein modules and perverse sheaves
Abstract: Skein modules of 3-manifolds were introduced to generalize the Jones polynomial for links in the 3-sphere to other manifolds. I will review a connection of skein modules to character varieties of 3-manifolds and quantum groups. I will also explain a connection between the skein modules for generic quantum parameters and the cohomology of a certain perverse sheaf on the character variety.