Titles and abstracts

• Chris Bowman: TBA

TBA

• Arthur Chong: TBA
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• Maud deVisscher: TBA

TBA

• Johannes Flake: TBA

TBA

• Jack Gidney: Normal Coverings of the Finite Classical Groups
  The normal covering number of a finite group is the smallest number of proper subgroups whose union of conjugates cover the group. By a theorem of Jordan, this number must be at least two for any finite group. Building on the work of Britnell and Maróti on the linear groups, I am investigating bounds on the normal covering number of the symplectic, orthogonal and unitary groups.
  
  

• Duncan Laurie: TBA
  TBA

• Ben Mills: TBA
  TBA
  
  

• Arshia Tajlili Moghanjoghi: TBA
  TBA
  
  

• Benjamin Morris: Interpolating between Temperley-Lieb and Brauer representation theory: generalised roots of unity
  In 2019, Kadar, Martin, and Yu, introduced a discrete family of diagram categories/algebras with the aim of describing inhomogeneity in lattice models via the transfer matrix algebra approach. Their construction introduces the notion of “height” which is a topological condition on the crossings of a Brauer diagram. The result is a family of nested algebras interpolating between the Temperley-Lieb (TL) and Brauer algebras, setting up a fascinating program in representation theory. In this talk, I will review the construction of these, “Kadar-Martin-Yu” (KMY) algebras, and present the first fruits of the latter program. A key result on semisimplicity establishes an analogy between the TL and KMY algebras. Namely, for each integer partition, there exists a Chebyshev series of polynomials which control (some) semisimplicity criteria for the KMY algebras; this can be seen as a direct generalisation of the classical “root of unity” criteria for TL algebras. I will also present some low rank calculations as evidence that the geometric understanding of the non-semisimple, TL and Brauer cases, can be extended to the intermediate, KMY cases.
  
  

• Bárbara Muniz: TBA
  TBA
  
  

• Lucia Noelle: Fock Spaces and Category O of rational Cherednik algebras
  A brief introduction to Category O for the cyclotomic rational Cherednik algebra and its relationship to Fock spaces.
  
  

• Theresa Ortscheidt: TBA
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• Willoughby Seago: TBA
  TBA
  
  

• Charles Senécal: TBA
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• Sasha Shapiro: TBA
  TBA
  
  

• Lei Shi: TBA
  TBA
  
  

• Jasper Stokman: Discrete quantum walks, Hecke algebras and Hall-Littlewood polynomials
  Quantum walks are quantum mechanical versions of random walks on graphs. A well-known example of a discrete-time quantum walk is the Grover walk, which generalizes Grover’s celebrated quantum algorithm for unstructured database searches.
  
  The transition operator of a discrete quantum walk is the composition of two involutions: the first involution represents the flipping of a quantum coin, the second represents the quantum step. The abstract group generated by two involutions is the infinite dihedral group, which is an example of an affine Weyl group. Group algebras of affine Weyl groups admit natural flat deformations called affine Hecke algebras, which play an important role in representation theory of reductive p-adic groups.
  
  In the first part of the talk I will give a short introduction to discrete quantum walks from the perspective of representation theory of the infinite dihedral group. In the second part of the talk I will introduce two-parameter generalisations of discrete quantum walks by replacing the role of the infinite dihedral group by its affine Hecke algebra. Finally, I will explain how perfect state transfer in these models can be described in terms of Hall-Littlewood polynomials.
  
  

• Joel Summerfield: The Hasse Diagram for the Closure Order on Decomposition Classes
  Decomposition classes provide a way of partitioning the Lie algebra of an algebraic group into finitely many pieces based on the Jordan decomposition. We can equip the set of decomposition classes with the closure order, and consider the corresponding Hasse diagram. In this talk, we will summarise general properties of these diagrams and provide a description of their unique minimal and maximal elements. Finally, in the case of good characteristic, we will classify the edges of the Hasse diagram into two distinct types: semisimple and nilpotent.
  
  

• James Timmins: TBA
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• Lewis Topley: Harish-Chandra bimodules in positive characteristics
  I will report on a joint work with Lewis Groves and Matt Westaway, soon to appear. Let g = Lie(G) be the Lie algebra of a simple algebraic group over an algebraically closed field k. A Harish-Chandra bimodule is a U(g)-U(g)-bimodule M with an action of G such that the adjoint g-action on M integrates to G. In 1980 Bernstein and Gelfand studied the category of such modules when char(k) = 0: they classified the simple objects and showed that the subcategory with a fixed generalised central character is equivalent to a block of category O for g. Their work later inspired Soergel to introduce his ubiquitous category of Soergel bimodules. More recently, HC bimodules have featured in works of Losev and Bezrukavnikov—Riche in the case char(k) = p > 0. In our work we classify the simple objects in the category, when p > 0 is very good for G.
  
  

• Angelina Vargulevich: TBA
  TBA
  
  

• Scott Warrander: Bar involutions and canonical bases in K-theory via stable envelopes
  Many modules over algebras with canonical basis theories, such as affine Hecke algebras and affine quantum groups, can be realised using equivariant K-theory. In the late 90s, Lusztig began studying these K-theory groups and trying to write down a geometric definition of their bases, characterising them by invariance under an appropriately defined bar involution. In this short talk I will discuss a new approach to defining bar involutions due to Hikita, using stable envelopes.
  
  

• Milen Yakimov: TBA
  TBA