Minimal Numbers of Linear Constituents in Sylow restriction for S_n and A_n

Bim Gustavsson (University of Birmingham)
Thursday, 09th October 2025 15:10-16:00

Abstract:
For a natural number n, let P_n denote a Sylow p-subgroup of the symmetric group S_n. In 2017 E. Giannelli and G. Navarro proved that if χ is an irreducible character of S_n with degree divisible by p, then the restriction of χ to P_n has at least p different linear constituents. This talk will focus on the identifying the set of characters of S_n whose restriction to a Sylow p-subgroup have at most p distinct linear constituents. We will also talk about A_n if time allows.

Stacking quantum spin systems and the triviality of invertible phases

Naomi Wray (Cardiff University)
Thursday, 16th October 2025 15:10-16:00

Abstract:
The superselection structure of 2-dimensional topologically ordered systems is well studied; this can be thought of as the classification of anyon (quasi-particle) types arising in a quantum spin system. We ask about the relation between sectors (read: anyon types) in a stack of two such systems and sectors of each 2-dimensional layer. Constructing sectors in the stacked theory from individual sectors is relatively simple and leads to important conclusions about the triviality of invertible states – those which, when stacked with a suitable state, are in the trivial phase. The converse argument of each stacked sector decomposing to individual layers is more subtle. In this talk, I will introduce the setting and provide some intuition for discussing these objects before giving the complete superselection structure of stacked quantum lattice systems.
This talk is based on joint work with Sven Bachmann, Alan Getz, and Pieter Naaijkens.

From Stability of Geometric Inequalities to Optimal Partial Transport and Beyond

Prachi Sahjwani (Cardiff University)
Thursday, 23rd October 2025 15:10-16:00

Abstract:
This talk will serve as both an overview of my past and current research interests. I will begin with results from my PhD about the stability of geometric inequalities in various curved spaces, including quermassintegral inequalities in hyperbolic space and Minkowski-type inequalities in warped product spaces. After introducing the notion of stability through the classical isoperimetric inequality, I will outline the main techniques used in these results.
In the second part, I will discuss some of my ongoing projects, including work on optimal partial transport problems in the infinite Wasserstein setting, capillary-type flows in Minkowski space, and dimension estimation problems.

On the Baum–Connes conjecture (with coefficients) for lattices in simple Lie groups of real rank one

Shintaro Nishikawa (University of Southampton)
Thursday, 30th October 2025 15:10-16:00

Abstract:
The Baum–Connes conjecture (BC) is a central problem in noncommutative geometry, proposing a topological description of the K-theory of group C*-algebras, which are inherently analytic in nature. The Baum–Connes conjecture with coefficients (BCC) is a stronger form of this statement. The BCC remains completely open for higher-rank simple Lie groups, and even in the real rank one case, its status has been subtle.
In this talk, I will give a broad overview of the conjecture and its historical developments. I will then explain why the Baum–Connes conjecture with coefficients holds for all lattices in simple Lie groups of real rank one, based on recent joint work with Nansen Petrosyan.

Graded Lie algebras and families of algebraic curves

Beth Romano (King’s College London)
Thursday, 6th November 2025 15:10-16:00

Abstract:
The theory of graded Lie algebras is a beautiful part of algebra that has applications to a wide range of mathematics, including various parts of number theory. Motivated by questions in arithmetic statistics, Jef Laga and I have adapted a construction of Slodowy to the setting of graded Lie algebras to produce families of algebraic curves. In this talk, I will give an introduction to some of these ideas via examples.

Leverhulme Lecture: Homotopy groups of Cuntz classes in C*-algebras

Andrew Toms (Purdue University)
Thursday, 13th November 2025 15:10-16:00

Abstract:
The Cuntz semigroup of a C*-algebra A consists of equivalence classes of positive elements, where equivalence means roughly that two positive elements have the same rank relative to A.  It can be thought of as a generalization of the Murray von Neumann semigroup to positive elements and is an incredibly sensitive invariant.  We give a brief introduction to this object and its relevance to the classification theory of separable nuclear C*-algebras.  We then present a calculation of the homotopy groups of these Cuntz classes as topological subspaces of A when A is classifiable in the sense of Elliott.

Stable subgroups in graph products

Alice Kerr (MPI for Mathematics in the Sciences, Leipzig)
Thursday, 20th November 2025 15:10-16:00

Abstract:
The study of hyperbolic groups and their subgroups are central to the area of geometric group theory. A particularly nice class of subgroups are the quasiconvex ones: they are the subgroups that are themselves hyperbolic, and in some sense inherit that hyperbolicity from the group itself. In the more general setting of finitely generated groups, this notion is captured by the stable subgroups. It turns out that in many cases stable subgroups coincide with other natural classes of subgroups, but there are also many cases for which such characterisations are not known. We will discuss this problem for graph products, which are a generalisation of direct products, free products, and certain Coxeter and Artin groups.

No knowledge of geometric group theory will be assumed. This is based on joint work with Sahana Balasubramanya, Marissa Chesser, Johanna Mangahas, and Marie Trin.

4D/3D QFT and representation theory

Tomoyuki Arakawa (Kyoto University)
Thursday, 27th November 2025 15:10-16:00

Abstract:
4D/3D quantum field theory in theoretical physics is conceptually rich and gives rise to many interesting mathematical structures, even though a fully rigorous mathematical formulation of the theories themselves is still lacking. A relatively recent discovery by Beem et al. shows that to every 4D N=2 superconformal field theory one can associate a representation-theoretic object called a vertex algebra, which serves as an invariant (or observable) of the theory. Although vertex algebras are inherently algebraic, those arising as invariants of 4D QFT display striking connections with certain geometric objects that also appear as invariants of the same physical theories.
Similarly, to each 3D N=4 gauge theory one can associate two vertex algebras ”the A-twisted and B-twisted boundary VOAs” which may be viewed as refinements of the Higgs and Coulomb branches.
In this talk, I will discuss some representation-theoretic aspects of these phenomena.

Linear logic, representations of vertex algebras and surface diagrams

Christoph Schweigert (Universität Hamburg)
Thursday, 4th December 2025 15:10-16:00

Abstract:
Surprisingly similar structures arise in categorical logic and in the representation categories of vertex algebras. We explain these connections and demonstrate how to perform computations within these structures using a three-dimensional graphical calculus, illustrated through physical three-dimensional models produced by a 3D printer.

Nielsen Realization for Rational 4-Manifolds

Tudur Lewis (University of Bristol)
Thursday, 5th February 2026 15:10-16:00

Abstract:
The cyclic Nielsen realization problem for a closed, oriented manifold asks whether any mapping class of finite order can be represented by a homeomorphism of the same order. This talk focuses on the Nielsen problem for smooth 4-manifolds underlying del Pezzo surfaces. I will present joint work with Seraphina Lee and Sidhanth Raman on the Nielsen realization problem for irreducible mapping classes.

Topological Invariants for G-kernels

Ulrich Pennig (Cardiff University)
Thursday, 12th February 2026 15:10-16:00

Abstract:
A G-kernel is a group homomorphism from a (discrete) group G to Out(A), the outer automorphism group of a C*-algebra A. There are cohomological obstructions to lifting such a G-kernel to a group action. In the setting of von Neumann algebras, G-kernels on the hyperfinite $\mathrm{II}_1$-factor have been completely understood via deep results of Connes, Jones and Ocneanu. In the talk I will explain how G-kernels on C*-algebras and the lifting obstructions can be interpreted in terms of cohomology with coefficients in crossed modules. G-kernels, group actions and cocycle actions then give rise to induced maps on classifying spaces. This project is joint work with S. Giron Pacheco and M. Izumi. A follow-up project that I will mention is joint work with my current PhD student Georgios Tridimas.

Boundary actions of outer automorphism groups of Thompson-like groups

Chris Bruce (University of Newcastle)
Thursday, 26th February 2026 15:10-16:00

Abstract:
By constructing an explicit topologically free boundary action, we prove that the outer automorphism groups of the topological full groups of the Cuntz–Krieger groupoids are C*-simple, meaning that their reduced C*-algebras are simple (this is an extreme case of non-amenability). These groups generalise the outer automorphism groups of the Higman–Thomson groups, so our result applies, in particular to, Out(V), where V is Thompson’s group. This is joint work with Xin Li and Takuya Takeishi.

Representations of $GL_n(\mathbb{Z}_p)$, tensor categories, and finite rings

Ehud Meir (University of Aberdeen)
Thursday, 5th March 2026 15:10-16:00

Abstract:
Zelevinsky showed that the representation theory of the groups $GL_n(\mathbb{F}_q)$, where $\mathbb{F}_q$ is a finite field, gives rise to a very special type of Hopf algebra, called positive self adjoint Hopf algebra (or PSH algebra). This Hopf algebra enables us to split the representation theory of these groups into arithmetic part (the cuspidal representations) and combinatorial part.
If we replace $\mathbb{F}_q$ with $\mathbb{Z}/p^r$ where r>1, the representation theory becomes much more complicated.
However, a PSH-algebra is still present, and it enables us to split the representation theory into a combinatorial part and a much more complicatred arithmetic / mysterious part.
In this talk I will tell you how and why we get a PSH algebra. The proof requires going beyond traditional representation theory techniques, and studying general finite rings and tensor categories.
This talk is based on a joint work with Tyrone Crisp and Uri Onn.

Derived Categories and Noncommutative Crepant Resolutions

Charlotte Llewellyn (University of Glasgow)
Thursday, 12th March 2026 15:10-16:00

Abstract:
The existence of non-singular models for threefold algebraic varieties was established by Zariski in 1944. Since then, considerable effort has been devoted to extending and refining this result. One such development, due to Van den Bergh, is the notion of a noncommutative crepant resolution (NCCR): a noncommutative algebra defined to mimic crepant resolutions at the level of derived categories.

In this talk we survey the notion of an NCCR, tracing its development from its geometric origins, through the categorical framework of perverse sheaves, to the module-theoretic definition used today. We conclude by explaining how this technology lends itself naturally to the construction of perverse schobers on the A_{2} hyperplane arrangement.

Operator Algebras in the South Meeting

Thursday, 19th March 2026 – Friday, 20th March 2026

Homotopy classification of 4-manifolds

Daniel Kasprowski (University of Southampton)
Thursday, 16th April 2026 15:10-16:00

Abstract:
I will explain how to classify 4-manifolds with 3-manifold fundamental group up to homotopy equivalence and give applications to the stable and unstable homeomorphism classification. This talk is based on joint work with Jonathan Hillman, Mark Powell and Aru Ray.

Synthetic Lorentzian geometry and curvature bounds

Tobias Beran (Cardiff University)
Thursday, 23rd April 2026 15:10-16:00

Abstract:
I introduce Lorentzian length spaces and synthetic curvature comparison using triangle comparison. For such spaces with curvature bounded above, curves can be majorised with a curve in a model space. If time permits, I will briefly discuss current research about Lorentzian length spaces with curvature bounded below being covered by coordinates almost everywhere.

A transport-equation approach to translating soliton of the mean curvature flow.

Marie-Amelie Lawn (Imperial College London)
Thursday, 7th May 2026 15:10-16:00

Abstract:
Mean curvature flow is a central geometric evolution equation: it evolves a submanifold in the direction of its mean curvature vector. Translating solitons, which are solutions moving by ambient translation without changing shape, are among its basic special solutions. In this talk I will discuss a construction of such solitons using submersions. Starting from a hypersurface in a lower-dimensional warped product, we ask when it can be lifted through a pseudo-Riemannian submersion to a genuine translating soliton upstairs. The point is that the translator equation upstairs differs from the corresponding equation downstairs by a fibre term. Thus, the defect of the downstairs hypersurface can, in favourable cases, be absorbed by choosing the geometry of the fibres appropriately. For graphs, this condition becomes a first-order transport equation on the base. This gives a practical way of producing new examples, and also brings in questions about characteristic curves, critical points, global solvability and completeness. I will explain the construction and illustrate it with examples from Riemannian and Lorentzian geometry, including translators arising from cohomogeneity-one actions, and related examples in Generalised Robertson–Walker spacetimes.

The Boolean growth polytopes and their triangulations

Matjaž Konvalinka (University of Ljubljana)
Thursday, 28th May 2026 15:10-16:00

Abstract:
I will present a new object, the Boolean growth polytope, a special class of alcoved polytopes which generalizes both Stanley’s order polytope and the polytope of alternating sign matrices (and many of the classes of alternating sign matrices which are symmetric in some way). I will present some of its properties, like its vertices, facets, volume and Ehrhart polynomial. The proof of the last two is via an explicit triangulation. This is joint work with Roger Behrend.

$\mathbb{Z}_p$ Torus actions on Positively Curved Manifolds

Catherine Searle (Wichita State University)
Thursday, 4th June 2026 11:10-12:00

Abstract:
We consider $\mathbb{Z}_p$ torus actions with a fixed point on closed, positively curved $n$-manifolds. We show that we can lower the approximately $3n/8$ bound obtained by Fang and Rong and by Ghazawneh and still obtain the same homotopy equivalence classification. In particular, and of independent interest, we generalize a geometric embedding result of Wilking’s to establish a generalized p-ary Elias-Bassalygo error correcting code bound that allows us to improve the lower bound for $3\leq p\leq 19$. This is joint work with Muhammad Abdullah.

Coxeter Polar foliations and their invariants

Gustavo May (University of Parma)
Thursday, 2nd July 2026 15:10-16:00

Abstract: 
Singular Riemannian foliations (SRFs) generalize isometric group actions and encode rich transverse geometry. In the polar case, the leaf space carries a natural orbifold structure. Our first observation is that every orbifold Riemannian metric on the leaf space lifts to a Riemannian metric on the ambient space that is adapted to the foliation.
As an application, we study the subclass of polar SRFs for which, in particular, the leaf space is a Coxeter orbifold. We show that the stratification of the leaf space and an appropriate choice of invariants completely determine the foliation up to foliated diffeomorphisms.
This is part of an ongoing collaboration with Diego Corro.