Torus actions on the Cuntz algebras, bundles,  and their reciprocal algebras.

Taro Sogabe (Kyoto University)
Thursday, 30th January 2025 15:10-16:00

Abstract:
In the operator algebra theory, it is often an interesting problem to find the torus  actions on an operator algebra and investigate their fixed point algebras.
The Cuntz algebra has a natural torus action whose fixed point algebra is the UHF algebra (i.e., infinite tensor product of the matrix algebra).
I would like to explain the Cuntz algebras and these torus actions, and I will introduce some problems related to the action including, the existence of some torus actions on the bundles of Cuntz algebras,  the position of the torus action in the fundamental group of the automorphism group of the Cuntz algebra.
If time permits, I will talk about a gauge action on the reciprocal Cuntz(–Krieger) algebras which was investigated in our ongoing joint work with Kengo Matsumoto.

Vinberg degeneration of Harish-Chandra transform

Roman Gonin (Cardiff University)
Thursday, 6th February 2025 15:10-16:00

Abstract:
Perverse sheaves play an important role in topology and algebraic geometry, with rich applications in Geometric Representation Theory (GRT). A key approach within GRT involves realising algebras and actions on K-theory. Omitting the K-theory functor leads naturally to a categorification. The Hecke algebra is fundamental in representation theory, particularly in the study of quantum groups. The monodromic Hecke category is a natural extension of its categorification.

In this talk, I will provide a gentle introduction to perverse sheaves, assuming no prior knowledge. We discuss the monodromic Hecke category, the Harish-Chandra transform as its categorical centre, and the Vinberg degeneration of the transform. The talk is based on a joint work with Kostiantyn Tolmachov and Andrei Ionov.

C*-algebraic factorization homology

Lucas Hataishi (University of Oxford)
Thursday, 13th February 2025 15:10-16:00

Abstract:
In this talk I will discuss an approach to analyze a class of framed 2-dimensional TQFTs arising from the representation theory of quasitriangular locally compact quantum groups.
The TQFTs are constructed via factorization homology with values in C*-categories, but its is expected that their values can be computed by means of actions of quantum groups on C*-algebras.
I will focus the exposition on the case of quasitriangular compact quantum groups, where the strategy can be fully implemented without being obscured by issues of analytical nature.

Categorification in representation theory

Vanessa Miemietz (University of East Anglia)
Thursday, 20th February 2025 15:10-16:00

Abstract:
I will explain some of the motivation and the ideas behind the theory of (finitary) additive 2-representations of (finitary) additive 2-categories, which attempts to categorify the representation theory of (finite dimensional) algebras. I will illustrate these on the example of the categorification of Hecke algebras via Soergel bimodules.

Mean curvature flow with generic initial data

Felix Schulze (University of Warwick)
Thursday, 27th February 2025 15:10-16:00

Abstract:
Mean curvature flow is the gradient flow of the area functional where an embedded hypersurface evolves in direction of its mean curvature vector. This constitutes a natural geometric heat equation for hypersurfaces, which ideally will evolve the embedding into a nicer shape. But due to the nonlinear nature of the equation singularities are guaranteed to form. Nevertheless, a key observation in geometry and physics is that generic solutions, obtained by small perturbations, can exhibit simpler singularities. In this direction, a conjecture of Huisken posits that a generic mean curvature flow encounters only the simplest singularities. We will discuss work together with Chodosh, Choi and Mantoulidis which together with recent work of Bamler-Kleiner establishes this conjecture for embedded hypersurfaces in R^3.

New methods in diffusion geometry

Iolo Jones (Durham University)
Thursday, 6th March 2025 15:10-16:00

Abstract:
Diffusion geometry is a new framework for geometric and topological data analysis that defines Riemannian geometry for probability spaces. This lets us apply the wealth of theory and methods from classical differential geometry as tools for data analysis. In this talk, I will outline the basic theory of diffusion geometry, like the construction of vector fields and differential forms. I will also survey a range of new data analysis tools, including vector calculus, solving spatial PDEs on data, finding integral curves and geodesics, and finding circular coordinates for de Rham cohomology classes. In the very special case of data from manifolds, we can compute the curvature tensors and dimension. These methods are highly robust to noise and fast to compute when compared with comparable methods like persistent homology.

Enumerating defect zero blocks

Emily Norton (University of Kent)
Thursday, 13th March 2025 15:10-16:00

Abstract:
A staircase partition cannot be tiled in such a way that upon removing a domino-shaped tile from the staircase, you still have a partition. We say that the staircase partition is a 2-core partition. The notion of an e-core partition is similar, but with e-ribbons in place of dominoes. The e-core partitions describe blocks in the representation theory of symmetric groups in positive characteristic, but also rational Cherednik algebras and Hecke algebras at roots of unity. In the modular representation theory of the finite general linear group, the e-core partitions describe the unipotent blocks.  In 1996, Granville and Ono proved that there exists an e-core partition of every size n if e is at least 4 (when e is 2 or 3, there are infinitely many values of n without an e-core partition of size n). We may restate Granville and Ono’s result as saying that in quantum characteristic at least 4, there exists a defect 0 unipotent block of GL(n,q) for every natural number n. We may then ask if there is an analogue of this theorem for other finite classical groups, for cyclotomic Hecke algebras at appropriate parameters, etc. This a project with Thomas Gerber.

Pursuing Coxeter Theory for Kac-Moody Affine Hecke Algebras

Dinakar Muthiah (University of Glasgow)
Thursday, 20th March 2025 15:10-16:00

Abstract:
I will discuss Kac-Moody Affine Hecke Algebras, which were first constructed as Iwahori-Hecke algebras for p-adic Kac-Moody groups. In the case where the Kac-Moody group is itself affine type, the Kac-Moody Affine Hecke Algebra is a slight variation of Cherednik’s DAHA. However, unlike the DAHA, the Kac-Moody Affine Hecke algebra is realized as a convolution algebra.
In particular, it has a “T”-basis corresponding to double cosets. For usual Affine Hecke algebras, this T-basis reflects the Coxeter group structure of the affine Weyl group. However, the Kac-Moody Affine Hecke algebra is not a Coxeter Hecke algebra. Despite this, many Coxeter-like phenomena abound in the Kac-Moody Affine setting. I will present recent results and conjectures with Anna Puskás about pursuing an analogue of Coxeter theory for Kac-Moody Affine Hecke algebras. I will also discuss earlier work with Dan Orr and work in progress with Auguste Hébert.

Cohen-Macaulay simplicial complexes and duality groups

Thomas Wasserman (University of Oxford)
Thursday, 27th March 2025 15:10-16:00

Abstract:
Duality groups are groups that admit a Poincaré-duality-like relationship between their cohomology and their homology twisted by a module known as the dualising module. Many interesting groups are (virtually) duality groups, like free groups and their (outer) automorphisms, mapping class groups, and the general linear groups over the integers. Knowing duality is useful for homology computations, and the dualising module often has a nice interpretation. In this talk I will discuss work with Ric Wade where we show that the (co)homology of simplicial complexes that are locally Cohen-Macaulay (look like manifolds with codimension one singularities) carry a duality similar to that for duality groups. I will discuss applications of this to outer automorphisms of free groups.

Generating functions of permutation classes

Robert Brignall (Open University)
Thursday, 3rd April 2025 15:10-16:00

Abstract:
The combinatorial study of permutations, fondly called `permutation patterns’, is largely dominated by questions of enumeration. A classic question is the following: given some set of permutations $B$, how many permutations of length $n$ avoid all the permutations in $B$? Asymptotically, the celebrated Marcus–Tardos Theorem tells us that for any such instance of the question where $B$ is non-empty, there exists a constant $c$ so that there are at most $c^n$ such permutations of each (large) length $n$. More precisely, we may construct the generating function to be the formal power series in which the coefficient of $z^n$ is the number of permutations that avoid everything in $B$.

This talk will explore the nature of these generating functions. We will look through history, from the 1990s when the ill-fated Noonan-Zeilberger Conjecture was formulated, through positive results that guarantee certain types of well-behaved generating functions in certain cases (for example, rational, algebraic  or $D$-finite), and cautionary tales along the way. We’ll finish by looking at what’s known about the form of generating functions when the constant $c$ guaranteed by Marcus–Tardos is `small’, and progress towards a conjecture that might just hold if $c<4$.

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Mike Hill (University of Minnesota)

Thursday, 8th May 2025 15:10-16:00