Christopher Drupieski: The Lie superalgebra of transpositions

28th May, 2024

In this talk I will report on work, joint with Jonathan Kujawa, to answer a series of questions originally posed by MathOverflow user WunderNatur in August 2022: Considering the group algebra ℂSn of the symmetric group as a superalgebra (by considering the even permutations in Sn to be of even superdegree and the odd permutations in Sn to be of odd superdegree), and then in turn considering ℂSn as a Lie superalgebra via the super commutator, what is the structure of ℂSn as a Lie superalgebra, and what is the structure of the Lie sub-superalgebra of ℂSn generated by the transpositions? The non-super versions of these questions were previously answered by Ivan Marin, with very different results. Time permitting, some thoughts on analogues of these questions for Weyl groups of types B/C and D may also be discussed.

Eric Friedlander: Infinite-dimensional modules for linear algebraic groups

28th May, 2024

We investigate infinite-dimensional modules for a linear algebraic group 𝔾 over a field of positive characteristic p. For any subcoalgebra C ⊂ 𝒪(𝔾) of the coordinate algebra of 𝔾, we consider the abelian subcategory CoMod(C) ⊂ Mod(𝔾) and the left exact functor (−)C : Mod(𝔾) → CoMod(C) that is right adjoint to the inclusion functor. The class of cofinite 𝔾-modules is introduced using finite-dimensional subcoalgebras of 𝒪(𝔾). We categorify a construction of Hardesty-Nakano-Sobaje, thereby supplementing cofinite type in providing invariants for proper mock injective 𝔾-modules.

Jie Du: The quantum queer supergroup via their q-Schur superalgebras

28th May, 2024

Using a geometric setting of q-Schur algebras, Beilinson-Lusztig-MacPherson discovered a new basis for quantum 𝔤𝔩n (i.e., the quantum enveloping algebra Uq(𝔤𝔩n) of the Lie algebra 𝔤𝔩n) and its associated matrix representation of the regular module of Uq(𝔤𝔩n). This beautiful work has been generalized (either geometrically or algebraically) to quantum affine 𝔤𝔩n, quantum super 𝔤𝔩m|n, and recently, to some i-quantum groups of type AIII.

In this talk, I will report on a completion of the work for a new construction of the quantum queer supergroup using their q-Schur superalgebras. This work was initiated 10 years ago, and almost failed immediately after a few months’ effort, due to the complication in computing the multiplication formulas by odd generators. Then, we moved on testing special cases or other methods for some years and regained confidence to continue. Thus, it resulted in a preliminary version which was posted on arXiv in August 2022.

The main unsatisfaction in the preliminary version was the order relation used in a triangular relation and the absence of a normalized standard basis. It took almost two more years for us to tune the preliminary version up to a satisfactory version, where the so-called SDP condition, involving further combinatorics related to symmetric groups and Clifford generators, and an extra exponent involving the odd part of a labelling matrix play decisive roles to fix the problems.

Vyacheslav Futorny: Harish-Chandra representations of map Lie superalgebras

28th May, 2024

We will discuss the classification of simple modules with finite weight multiplicities over basic classical map superalgebras. Any such module is parabolically induced from a simple cuspidal bounded module over a cuspidal map subsuperalgebra. Moreover, any simple cuspidal bounded module is isomorphic to an evaluation module.

Yvain Bruned: Novikov algebras and multi-indices in regularity structures

27th May, 2024

In this talk, we will present multi-Novikov algebras, a generalization of Novikov algebras with several binary operations indexed by a given set, and show that the multi-indices recently introduced in the context of singular stochastic partial differential equations can be interpreted as free multi-Novikov algebras. This is parallel to the fact that decorated rooted trees arising in the context of regularity structures are related to free multi-pre-Lie algebras.

Paul Balmer: The Tate Intermediate Value Theorem

27th May, 2024

In this joint work with Beren Sanders, we analyze the question of (topologically) gluing an open piece and a closed piece of the spectrum of a tt-category, that is, of deciding which points in the open specialize to which points of the closed piece. We show that a critical role is played by the support of the Tate ring associated with this decomposition. I will take this opportunity to remind the audience of the very general notion of support for big objects in tt-categories that follows from the theory of homological residue fields.

Kent Vashaw: A Chinese remainder theorem and Carlson’s theorem for monoidal triangulated categories

27th May, 2024

Carlson’s Connectedness Theorem for cohomological support varieties is a fundamental result which states that the support variety for an indecomposable module of a finite group is connected. For monoidal triangulated categories, the Balmer spectrum is an intrinsic geometric space associated to the category which generalizes the notion of cohomological support for finite groups. In this talk, we will discuss a generalization of the Carlson Connectedness Theorem: that the Balmer support of any indecomposable object in a monoidal triangulated category with a thick generator is a connected subset of the Balmer spectrum. This is shown by proving a version of the Chinese remainder theorem in this context, that is, giving a decomposition for a Verdier quotient of a monoidal triangulated category by an intersection of coprime thick tensor ideals.

Milen Yakimov: Non-commutative tensor triangular geometry and finite tensor categories

27th May, 2024

Describing the thick ideals of a monoidal triangulated category is a key component of the analysis of the category. We will show how this can be done by non-commutative tensor triangular geometry (NTTG), thus extending the celebrated Balmer’s theorem from the symmetric case. We will then use NTTG to analyse the stable categories of finite tensor categories, which play an important role in representation theory, mathematical physics and quantum computing. We will present general results linking this approach to the traditional one through cohomological support, based on a notion of categorical centers of cohomology rings of monoidal triangulated categories.

Charlotte Chan: Generic character sheaves on parahoric subgroups

27th May, 2024

Lusztig’s theory of character sheaves for connected reductive groups is one of the most important developments in representation theory in the last few decades. I will give an overview of this theory and explain the need, from the perspective of the representation theory of p-adic groups, of a theory of character sheaves on jet schemes. Recently, R. Bezrukavnikov and I have developed the 'generic' part of this desired theory. In the simplest non-trivial case, this resolves a conjecture of Lusztig and produces perverse sheaves on jet schemes compatible with parahoric Deligne-Lusztig induction. This talk is intended to describe in broad strokes what we know about these generic character sheaves, especially within the context of the Langlands programme.

Arun Ram: Lusztig varieties and Macdonald polynomials

27th May, 2024

In type A, the Macdonald polynomials and the integral from Macdonald polynomials are related by a plethystic transformation. We interpret this plethystic transformation geometrically as a relationship between nilpotent parabolic Springer fibres and nilpotent Lusztig varieties. This points the way to a generalization of modified Macdonald polynomials and integral form Macdonald polynomials to all Lie types. But these generalizations are not polynomials, they are elements of the Iwahori-Hecke algebra of the finite Weyl group. This work concerns the generalization of, and connection between, a 1997 paper of Halverson-Ram (which counts points of nilpotent Lusztig varieties over a finite field) and a 2017 paper of Mellit (which counts points of nilpotent parabolic affine Springer fibres over a finite field).

Jon Carlson: Endotrivial modules for finite groups of Lie type

27th May, 2024

Suppose that G is a finite group and that k is a field of characteristic p > 0. A kG-module M is an endotrivial module if Homk(M,M) ≅ M∗ ⊗ M ≅ k ⊕ (proj). The endotrivial modules form the Picard group of self-equivalences of the stable category and have been classified for many families of groups. In this lecture I will describe some progress in the classification of endotrivial kG-modules in the case that G is a group of Lie type. We concentrate on the torsion subgroup of the group endotrivial modules, as the torsion free part was determined in earlier work. The torsion part consists mainly of modules whose restrictions to the Sylow subgroup of G are stably trivial. In most cases such modules have dimension 1, but the exceptions are notable.

Lenhard Ng: New Algebraic Invariants of Legendrian Links

24th May, 2024

For the past 25 years, Legendrian contact homology has played a key role in contact topology. I'll discuss a package of new invariants for Legendrian knots and links that builds on Legendrian contact homology and is derived from rational symplectic field theory. This includes a Poisson bracket on Legendrian contact homology and a symplectic structure on augmentation varieties. Time permitting, I'll also describe an unexpected connection to cluster theory for a family of Legendrian links associated to positive braids.

Timothy Logvinenko: The Heisenberg category of a category

23rd May, 2024

In the 90s, Nakajima and Grojnowski identified the total cohomology of the Hilbert schemes of points on a smooth projective surface with the Fock space representation of the Heisenberg algebra associated to its cohomology lattice. Later, Krug lifted this to derived categories and generalized it to the symmetric quotient stacks of any smooth projective variety.

On the other hand, Khovanov introduced a categorification of the free boson Heisenberg algebra, i.e., the one associated to the rank 1 lattice. It is a monoidal category whose morphisms are described by a certain planar diagram calculus which categorifies the Heisenberg relations. A similar categorification was constructed by Cautis and Licata for the Heisenberg algebras of ADE type root lattices.

We show how to associate the Heisenberg 2-category to any smooth and proper DG category and then define its Fock space 2-representation. This construction unifies all the results above and extends them to what can be viewed as the generality of arbitrary non-commutative smooth and proper schemes.

Anastasia Doikou: Parametric set-theoretic Yang-Baxter equation: p-racks, solutions and quantum algebras

20th May, 2024

The theory of the parametric set-theoretic Yang-Baxter equation is established from a purely algebraic point of view. We introduce generalizations of the familiar shelves and racks named parametric (p)-shelves and racks. These objects satisfy a 'parametric self-distributivity' condition and lead to solutions of the Yang-Baxter equation. Novel, non-reversible solutions are obtained from p-shelf/rack solutions by a suitable parametric twist, whereas all reversible set-theoretic solutions are reduced to the identity map via a parametric twist. The universal algebras associated to both p-rack and generic parametric set-theoretic solutions are next presented and the corresponding universal R-matrices are derived. By introducing the concept of a parametric coproduct we prove the existence of a parametric co-associativity. We show that the parametric coproduct is an algebra homomorphism and the universal R-matrices intertwine with the algebra coproducts.

Tomasz Brzezinski: Lie brackets on affine spaces

13th May, 2024

We first explore the definition of an affine space which makes no reference to the underlying vector space and then formulate the notion of a Lie bracket and hence a Lie algebra on an affine space in this framework. Since an affine space has neither distinguished elements nor additive structure, the concepts of antisymmetry and Jacobi identity need to be modified. We provide suitable modifications and illustrate them by a number of examples.

Viktor Ginzburg: Invariant Sets and Hyperbolic Periodic Orbits

10th May, 2024

The presence of hyperbolic periodic orbits or invariant sets often has an affect on the global behaviour of a dynamical system. In this talk we discuss two theorems along the lines of this phenomenon, extending some properties of Hamiltonian diffeomorphisms to dynamically convex Reeb flows on the sphere in all dimensions. The first one, complementing other multiplicity results for Reeb flows, is that the existence of a hyperbolic periodic orbit forces the flow to have infinitely many periodic orbits. This result can be thought of as a step towards Franks’s theorem for Reeb flows. The second result is a contact analogue of the higher-dimensional Le Calvez-Yoccoz theorem proved by the speaker and Gurel and asserting that no periodic orbit of a Hamiltonian pseudo-rotation is locally maximal.

Gyujin Oh: Derived Hecke action for weight 1 modular forms via lassicality

9th May, 2024

It is known that a p-adic family of modular forms does not necessarily specialize into a classical modular form at weight 1, unlike the modular forms of weight 2 or higher. We will explain how this obstruction to classicality leads to a 'derived' action on modular forms of weight 1, which can be understood as the so-called derived Hecke operator at p. We will also investigate the role of the derived action in the study of p-adic periods of the adjoint of the weight 1 modular forms.