GAPT Seminar
Autumn 2024/Spring 2025
Seminar Talks – Autumn 2024
Nonstable K-theory for Z-stable C*-algebras
Shanshan Hua (University of Oxford)
Thursday, 3rd October 2024 15:10 – 16:00
In Jiang’s unpublished paper (1997), it is shown that any $\mathcal{Z}$-stable C*-algebra $A$ is $K_1$-injective and $K_1$-surjective, which means that $K_1(A)$ can be calculated by looking at homotopy equivalence classes of $U(A)$, without matrix amplifications. Moreover, for such $A$, higher homotopy groups of $U(A)$ are isomorphic to $K_0(A)$ or $K_1(A)$, depending on the dimension of the higher homotopy group. In this talk, I will present Jiang’s result for $\mathcal{Z}$-stable C*-algebras. Moreover, I will explain briefly our new strategies to reprove his theorems using an alternative picture of the Jiang-Su algebra as an inductive limit of generalized dimension drop algebras.
Stability of geometric inequalities in various spaces
Prachi Sahjwani (Cardiff University)
Thursday, 10th October 2024 15:10 – 16:00
In this talk, I will discuss the stability of two inequalities: the “Alexandrov-Fenchel inequalities in hyperbolic space” and “Minkowski’s inequality in a general warped product space.” I will give a brief overview of both inequalities and their respective stability problems. To explain what I mean by stability, I will first discuss it in the context of the isoperimetric inequality, which is a special case of the Alexandrov-Fenchel inequalities. I will also briefly discuss the proofs in both cases. This is joint work with Prof. Dr. Julian Scheuer and is based on the work of Wang/Xia on Alexandrov-Fenchel inequalities and the work of Brendle/Hung/Wang and Scheuer on Minkowski’s inequality.
Topological Data Analysis of Abelian Magnetic Monopoles in Gauge Theory
Xavier Crean (Swansea University)
Thursday, 17th October 2024 15:10 – 16:00
It has been widely argued that non-trivial topological features of the Yang-Mills vacuum are responsible for colour confinement. However, both analytical and numerical progress have been limited by the lack of understanding of the nature of relevant topological excitations in the full quantum description of the model. In this talk, we shall explain how Topological Data Analysis (TDA) may be used to quantitatively analyse Abelian magnetic monopoles across the deconfinement phase transition in lattice gauge theory. The talk will give a background to the physics and introduce the necessary methods from TDA.
Lusztig-Spaltenstein induction of nilpotent orbits
Lewis Topley (University of Bath)
Thursday, 24th October 2024 15:10 – 16:00
Parabolic induction is the most ubiquitous construction in Lie theoretic representation theory, allowing us to construct representations from a nice class subobjects, whenever we work with Lie algebras, algebraic groups, quantum groups, Weyl groups or finite groups of Lie type. In fact, this may be the only technique for generating interesting representations (discuss). At the same time representations are often associated to nilpotent orbits in Lie algebras. Examples of the latter relationship include Springer theory, Whittaker modules and finite W-algebras. The analogue of parabolic induction for nilpotent orbits is called Lusztig-Spaltenstein induction, and I like it very much. In this talk I will introduce this construction, and explain a theorem of mine (joint with Neil Saunders) which describes the combinatorial behaviour with respect to Springer fibres in type A.
New quantum projective spaces from deformations of q-polynomial algebras
Mykola Matviichuk (Imperial)
Thursday, 31st October 2024 15:10 – 16:00
I will discuss how to construct a large collection of “quantum projective spaces”, in the form of Koszul, Artin-Schelter regular quadratic algebras with the Hilbert series of a polynomial ring. I will do so by starting with the toric ones (the q-polynomial algebras), and then deforming their relations using a diagrammatic calculus, proving unobstructedness of such deformations under suitable nondegeneracy conditions. Time permitting, I will show that these algebras coincide with the canonical quantizations of corresponding families of quadratic Poisson structures. This provides new evidence to Kontsevich’s conjecture about convergence of his deformation quantization formula. This is joint work with Brent Pym and Travis Schedler.
Quantum Markov Semigroups and Modified Logarithmic Sobolev Inequalities
Angela Capel Cuevas (Cambridge)
Thursday, 7th November 2024 15:10 – 16:00
A dissipative evolution of an open quantum many-body system weakly coupled to an environment can be modelled by a quantum Markov semigroup, and its mixing time can be bounded using optimal constants of certain quantum functional inequalities, such as the modified logarithmic Sobolev constant. In this talk, we will review the mathematical formalism of dissipative evolutions governed by Lindbladians, and we will summarize the current state of the art on mixing times when the system has an associated commuting Hamiltonian.
Deformations of Lipschitz Homeomorphisms
Mohammad Al Attar (Durham University)
Thursday, 14th November 2024 15:10 – 16:00
The theory of deforming homeomorphisms dates back to the 60’s with Černavskii showing the local contractibility of homeomorphism groups of certain classes of manifolds. In this talk I will discuss the history of the theory of deforming homeomorphisms and then discuss my research on deforming Lipschitz homeomorphisms, which extends the work of Siebenmann to the Lipschitz category.
Applications of topology in tumours, glaciers, and Chicago
Gillian Grindstaff (Oxford)
Thursday, 21st November 2024 15:10 – 16:00
Topological data analysis (TDA) has been widely applied to perform generalised hole detection and shape profiling across a range of disciplines. The flagship technique in TDA is persistent homology of a Vietoris-Rips complex, where the shape of data is completely unknown, but filtrations over a known domain or incorporating additional structure can also be incredibly powerful in isolating topological signal from noise. In this talk we will tour three spatial applications: agent-based tumour-immune modeling, arctic melt-pond evolution, and resource deserts in urban areas. For each, we craft filtrations tailored to the available information, assumptions, and hypotheses, and interpret the results. We highlight computational limits and scientific questions such as the interpretation of persistence diagrams, comparison with benchmark techniques, model fitting, and feature extraction.
Parabolic Induction in Lie Theory
Matthew Westaway (Bath)
Thursday, 28th November 2024 15:10 – 16:00
Parabolic induction is an important tool in Lie theory arising in different contexts including nilpotent orbits, ideals in universal enveloping algebras, representations of reduced enveloping algebras, and representations of finite W-algebras. This talk will provide a general introduction to parabolic induction, requiring no prior background. Then in some of these contexts, we will look at more recent results regarding when this procedure is injective or surjective. This talk will be based on joint work with S. Goodwin and L. Topley.
From Classical to Free Entropy
Jennifer Pi (University of Oxford)
Thursday, 5th December 2024 15:10 – 16:00
There are two main notions of entropy from probability theory, which coincide in the classical setting. These two constructions led Voiculescu to define different notions of entropy in the setting of free probability, called microstates and non-microstates free entropy. In this talk, I’ll discuss the classical setting as motivation, then move to the free setting. In joint work with David Jekel, we relate each notion of free entropy back to an appropriate limit of classical entropic quantities, which proves an inequality between the two notions of free entropy.
Seminar Talks – Spring 2025
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 $c<4$.
Quantization of integrable system via deformation quantization of Poisson vertex algebras
Simone Castellan (University of Glasgow)
Thursday, 8th May 2025 15:10-16:00
Abstract:
Following the work of De Sole, Kac, Valeri, classical Hamiltonian PDEs can be formalized using Poisson vertex algebras. This framework is important for two reasons. For start, it gives us new tools to construct and study classical integrable systems. Moreover, Poisson vertex algebras have a well-known canonical quantization, vertex algebras. This suggests a way to quantize a large class of classical integrable systems. In this talk I will explain the classical framework and discuss the ideas for the quantization. No prior knowledge on (Poisson) vertex algebras is assumed.