Wed, 12 Nov 2025

16:00 - 17:00
L6

Cutting along hyperplanes

Ralfs Pundurs
(University of Birmingham)
Abstract

You can cut a cake in half, a pizza into slices, but can you cut an infinite group? I'll tell you about my sensei's secret cutting technique and demonstrate a couple of examples. There might be some spectral goodies at the end too. 

Wed, 05 Nov 2025

16:00 - 17:00
L6

Improving acylindrical actions on trees

Will Cohen
(Cambridge)
Abstract
Loosely speaking, an action of a group on a tree is acylindrical if long enough paths must have small stabilisers. Groups admitting such actions form a natural subclass of acylindrically hyperbolic groups, and interesting an feature of acylindrical actions on trees is that many interesting properties are inherited from their vertex stabilisers. In order to make use of this, it is important to have some degree of control over these stabilisers. For example, can we ask for these stabilisers to be finitely generated, or even malnormal (or finite-height)? Even stronger, if our group is hyperbolic, can we ask for the stabilisers to be quasiconvex?
 
In this talk, I will introduce acylindrical actions and some stronger and related concepts, and discuss a method known as the Dunwoody—Sageev resolution that we can use to move between these concepts and provide positive answers to the above questions in some cases.
Tue, 18 Nov 2025
16:00
L6

Matrix-product state skeletons in Onsager-integrable quantum chains

Imogen Camp
(Department of Physics)
Abstract

Matrix-product state (MPS) skeletons are connected networks of local one-dimensional quantum lattice models with ground states admitting an MPS representation with finite bond dimension. In this talk, I will discuss how such skeletons underlie certain families of models obeying the Onsager algebra, and how these simple ground states provide a route to explicitly computing correlation functions.

Wed, 29 Oct 2025
16:00
L6

Hard Unknot Diagrams and Arc Presentations

Sam Ketchell
(Mathematical Institute University of Oxford )
Abstract
Abstract: There exist diagrams of the unknot that are "hard" in the sense that any sequence of Reidemeister moves rearranging them into the standard unknot diagram must at some point increase the crossing number beyond what it was originally. I will discuss how such diagrams can be produced and what is known and unknown about them.
Then, I will define and discuss the arc index of a knot, an invariant that behaves more nicely than the crossing number from this perspective, and in some other ways.
Fri, 14 Nov 2025
13:00
L6

Towards Finite Element Tensor Calculus

Kaibo Hu
(Oxford University)
Abstract

Classical finite element methods discretize scalar functions using piecewise polynomials. Vector finite elements, such as those developed by Raviart-Thomas, Nédélec, and Brezzi-Douglas-Marini in the 1970s and 1980s, have since undergone significant theoretical advancements and found wide-ranging applications. Subsequently, Bossavit recognized that these finite element spaces are specific instances of Whitney’s discrete differential forms, which inspired the systematic development of Finite Element Exterior Calculus (FEEC). These discrete topological structures and patterns also emerge in fields like Topological Data Analysis.

In this talk, we present an overview of discrete and finite element differential forms motivated by applications from topological hydrodynamics, alongside recent advancements in tensorial finite elements. The Bernstein-Gelfand-Gelfand (BGG) sequences encode the algebraic and differential structures of tensorial problems, such as those encountered in solid mechanics, differential geometry, and general relativity. Discretization of the BGG sequences extends the periodic table of finite elements, originally developed for Whitney forms, to include Christiansen’s finite element interpretation of Regge calculus and various distributional finite elements for fluids and solids as special cases. This approach further illuminates connections between algebraic and geometric structures, generalized continuum models, finite elements, and discrete differential geometry.

Fri, 24 Oct 2025
13:00
L6

Generalized Persistent Laplacians and Their Spectral Properties

Arne Wolf
(Imperial College)
Abstract
Laplacian operators are classical objects that are fundamental in both pure and applied mathematics and are becoming increasingly prominent in modern computational and data science fields such as applied and computational topology and application areas such as machine learning and network science. In our recent paper, we introduce a unifying operator-theoretic framework of generalized Laplacians as invariants that encompasses and extends all existing constructions, from discrete combinatorial settings to de Rham complexes of smooth manifolds. Within this framework, we introduce and study a generalized notion of persistent Laplacians. While the classical persistent Laplacian fails to satisfy the desirable properties of monotonicity and stability - both crucial for robustness and interpretability - our framework allows to isolate and analyze these properties systematically.  We demonstrate that their component maps, the up- and down-persistent Laplacians, satisfy these properties individually. Moreover, we provide a condition for full monotonicity and show that the spectra of these separate components fully determine the spectra of the full Laplacians, making them not only preferable but sufficient for analysis. We study these questions comprehensively, in both the finite and infinite dimensional settings. Our work expands and strengthens the theoretical foundation of generalized Laplacian-based methods in pure, applied, and computational mathematics.


 

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