Past

We discuss the formal relationship between the absence of global symmetries and completeness, both in effective field theory and in quantum gravity. In effective field theory, we must broaden our notion of symmetry to include non-invertible topological operators. However, in gravity, the story is simplified as the result of charged gravitational solitons.

This session will take place live in L1 and online. A Teams link will be shared 30 minutes before the session begins.

How can we make maths more accessible, promote its many applications, and encourage more women to enter the field? These are the questions we aim to address with Mathematigals.

Caoimhe Rooney and Jessica Williams met in 2015 at the start of their PhDs in mathematics in Oxford, and in 2020, they co-founded Mathematigals. Mathematigals is an online platform producing content to demonstrate fun mathematical curiosities, showcase ways maths can be used in real life, and promote female mathematicians. Mathematigals primarily produces animated videos that present maths in a way that is engaging to the general public.

In this session, Jess and Caoimhe will talk about their initial motivation to begin Mathematigals, demonstrate the process behind their content creation, and describe their future visions for the platform. The session will end with an opportunity for the audience to provide feedback or ideas to help Mathematigals on their journey to encourage future mathematicians.

Locally analytic representations of $p$-adic Lie groups are of interest in several branches of number theory, for example in the theory of automorphic forms and in the $p$-adic local Langlands program. To better understand these representations, Ardakov-Wadsley introduced a sheaf of infinite order differential operators $\overparen{\mathcal{D}}$ on smooth rigid analytic spaces, which resulted in several Beilinson-Bernstein style localisation theorems. In this talk, we discuss the current research on analogues of a Riemann-Hilbert correspondence for $\overparen{\mathcal{D}}$-modules, and what this has to do with complete convex bornological vector spaces.

High-dimensional approximation of Hamilton-Jacobi-Bellman PDEs – architectures, algorithms and applications

Hamilton-Jacobi Partial Differential Equations (HJ PDEs) are a central object in optimal control and differential games, enabling the computation of robust controls in feedback form. High-dimensional HJ PDEs naturally arise in the feedback synthesis for high-dimensional control systems, and their numerical solution must be sought outside the framework provided by standard grid-based discretizations. In this talk, I will discuss the construction novel computational methods for approximating high-dimensional HJ PDEs, based on tensor decompositions, polynomial approximation, and deep neural networks.

Catheter ablation and antiarrhythmic drug therapy approaches for treatment of atrial fibrillation are sub-optimal. This is in part because it is challenging to predict long-term response to therapy from short-term measurements, which makes it difficult to select optimal patient-specific treatment approaches. Clinical trials identify patient demographics that provide prediction of long-term response to standard treatments across populations. Patient-specific biophysical models can be used to assess novel treatment approaches but are typically applied in small cohorts to investigate the acute response to therapies. Our overall aim is to use machine learning approaches together with patient-specific biophysical simulations to predict long-term atrial fibrillation recurrence after ablation or drug therapy in large populations.

In this talk I will present our methodology for constructing personalised atrial models from patient imaging and electrical data; present results from biophysical simulations of ablation treatment; and finally explain how we are combining these methodologies with machine learning techniques for predicting long-term treatment outcomes.

[REMOTE TALK]

In this talk, we will propose a new construction of the virtual fundamental classes of quasi-smooth derived schemes using the vanishing cycle complexes. This is based on the dimensional reduction theorem of cohomological Donaldson—Thomas invariants which can be regarded as a variant of the Thom isomorphism. We will also discuss a conjectural approach to construct DT4 virtual classes using the vanishing cycle complexes.

Zoom link: https://us02web.zoom.us/j/86267335498?pwd=R2hrZ1N3VGJYbWdLd0htZzA4Mm5pd…

Computing spectral properties of operators is fundamental in the sciences, with applications in quantum mechanics, signal processing, fluid mechanics, dynamical systems, etc. However, the infinite-dimensional problem is infamously difficult (common difficulties include spectral pollution and dealing with continuous spectra). This talk introduces classes of practical resolvent-based algorithms that rigorously compute a zoo of spectral properties of operators on Hilbert spaces. We also discuss how these methods form part of a broader programme on the foundations of computation. The focus will be computing spectra with error control and spectral measures, for general discrete and differential operators. Analogous to eigenvalues and eigenvectors, these objects “diagonalise” operators in infinite dimensions through the spectral theorem. The first is computed by an algorithm that approximates resolvent norms. The second is computed by building convolutions of appropriate rational functions with the measure via the resolvent operator (solving shifted linear systems). The final part of the talk provides purely data-driven algorithms that compute the spectral properties of Koopman operators, with convergence guarantees, from snapshot data. Koopman operators “linearise” nonlinear dynamical systems, the price being a reduction to an infinite-dimensional spectral problem (c.f. “Koopmania”, describing their surge in popularity). The talk will end with applications of these new methods in several thousand state-space dimensions.

Microscopic green algae show great diversity in structural complexity, and successfully evolved efficient swimming strategies at low Reynolds numbers. Gonium is one of the simplest multicellular algae, with only 16 cells arranged in a flat plate. If the swimming of unicellular organisms, like Chlamydomonas, is nowadays widely studied, it is less clear how a colony made of independent Chlamydomonas-like cells performs coordinated motion. This simple algae is therefore a key organism to model the evolution from single-celled to multicellular locomotion.

In the absence of central communication, how can each cell adapt its individual photoresponse to efficiently reorient the whole algae? How crucial is the distinctive Gonium squared structure?

In this talk, I will present experiments investigating the shape and the phototactic swimming of Gonium, using trajectory tracking and micro-pipette techniques. I will explain our model linking the individual flagella response to the colony trajectory. This eventually emphasises the importance of biological noise for efficient swimming.

The curve graph of a surface of finite type is a fundamental object in the study of its mapping class group both from the metric and the combinatorial point of view. I will discuss joint work with Valentina Disarlo and Thomas Koberda where we conduct a thorough study of curve graphs from the model theoretic point of view, with particular emphasis in the problem of interpretability between different curve graphs and other geometric complexes.   

Given a group G acting on a CAT(0) polygonal complex, X, it is natural to ask whether the structure of X allows us to deduce properties of G. We discuss some recent work on local properties that X may possess which allow us to answer these questions - in many cases we can in fact deduce that the group is a linear group over Z.

I'll explain 'symplectic duality', a surprising relationship between certain pairs of algebraic symplectic manifolds, under which Hamiltonian automorphisms of one are identified with Poisson deformations of the other, and which is ultimately characterized by a Koszul-type equivalence between categories of modules over their filtered quantizations. I'll outline why such relationships are expected from physics in terms of three dimensional mirror symmetry, and rediscover the Coulomb branch construction of Braverman-Finkelberg-Nakajima from this perspective. We'll see that this explicitly constructs the symplectic dual of any variety which is presented as the symplectic reduction of a vector space by a reductive group.
 

Many PDE models encode fundamental physical, geometric and topological structures. These structures may be lost in discretisations, and preserving them on the discrete level is crucial for the stability and efficiency of numerical methods. The finite element exterior calculus (FEEC) is a framework for constructing and analysing structure-preserving numerical methods for PDEs with ideas from topology, homological algebra and the Hodge theory. 

In this seminar, we present the theory and applications of FEEC. This includes analytic results (Hodge decomposition, regular potentials, compactness etc.), Hodge-Laplacian problems and their structure-preserving finite element discretisation, and applications in electromagnetism, fluid and solid mechanics. Knowledge on geometry and topology is not required as prerequisites.

References:

1. Arnold, D.N.: Finite Element Exterior Calculus. SIAM (2018) 

2. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15, 1 (2006) 

3. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bulletin of the American Mathematical Society 47(2), 281–354 (2010) 

4. Arnold, D.N., Hu, K.: Complexes from complexes. Foundations of Computational Mathematics (2021)

Here we present the findings of our summer research project: an 8-week investigation of C*-Algebras. Our aim was to explore when a group is uniquely determined by its reduced group C*-algebra; i.e. when the group is C*-rigid. In particular, we applied techniques from topology, algebra, and analysis to prove C*-rigidity for a certain class of Bieberbach groups.

Let $A$ be drawn uniformly at random from the set of all $n \times n$ symmetric matrices with entries in $\{-1,1\}$. We show that $A$ is singular with probability at most $e^{-cn}$ for some absolute constant $c>0$, thereby resolving a well-known conjecture. This is joint work with Marcelo Campos, Marcus Michelen and Julian Sahasrabudhe.
 

Let $G$ be a finitely presented residually finite group. We are interested in the growth of size of the torsion of $H^{ab}$ as a function of $|G:H|$ where $H$ ranges over normal subgroups of finite index in $G$. It is easy to see that this grows at most exponentially in terms of $|G:H|$. Of particular interest is the case when $G$ is an arithmetic hyperbolic 3-manifold group and $H$ ranges over its congruence subgroups. Proving exponential lower bounds on the torsion appears to be difficult and in this talk I will focus on the situation of finitely presented pro-$p$ groups.

In contrast with abstract groups I will show that in finitely presented pro-$p$ groups torsion in the abelianizations can grow arbitrarily fast. The examples are rather 'large' pro-$p$ groups, in particular they are virtually Golod-Shafarevich. When we restrict to $p$-adic analytic groups the torsion growth is at most polynomial.

TBA

Several problems of great importance in the study of glaciers and ice sheets involve processes of attachment and reattachment of the ice from the bedrock. Consider, for example, an ice sheet sliding from the continent into the ocean, where it goes afloat. Another example is that of subglacial cavitation, a fundamental mechanism in glacial sliding where the ice detaches from the bedrock along the downstream area of an obstacle. Such problems are generally modelled as a viscous Stokes flow with a free boundary and contact boundary conditions. In this talk, I will present a framework for solving such problems numerically. I will start by introducing the mathematical formulation of these viscous contact problems and the challenges that arise when trying to approximate them numerically. I will then show how, given a stable scheme for the free boundary equation, one can build a penalty formulation for the viscous contact problem in such a way that the resulting algorithm remains stable and robust.

Title: Measuring association with Wasserstein distances

Abstract: Let π ∈ Π(μ, ν) be a coupling between two probability measures μ and ν on a Polish space. In this talk we propose and study a class of nonparametric measures of association between μ and ν, which we call Wasserstein correlation coefficients. These coefficients are based on the Wasserstein distance between ν and the disintegration of π with respect to the first coordinate. We also establish basic statistical properties of this new class of measures: we develop a statistical theory for strongly consistent estimators and determine their convergence rate in the case of compactly supported measures μ and ν. Throughout our analysis we make use of the so-called adapted/bicausal Wasserstein distance, in particular we rely on results established in [Backhoff, Bartl, Beiglböck, Wiesel. Estimating processes in adapted Wasserstein distance. 2020]. Our approach applies to probability laws on general Polish spaces.

In a world of polynomial Pell’s equations, where the integers are replaced by polynomials with complex coefficients, and its smallest solution is used to generate all other solutions $(u_{n},v_{n})$, $n\in\mathbb{Z}$. One junior number theory group will embark on a journey in search of the properties of the factors of $v_{n}(t)$. There will be Galois extensions, there will be estimations and of course there will be loglogs.

We show that the existence of hyperbolic manifolds fibering over the circle is not a phenomenon confined to dimension 3 by exhibiting some examples in dimension 5. More generally, there are hyperbolic manifolds with perfect circle-valued Morse functions in all dimensions $n\le 5$. As a consequence, there are hyperbolic groups with finite-type subgroups that are not hyperbolic.

The main tool is Bestvina - Brady theory enriched with a combinatorial game recently introduced by Jankiewicz, Norin and Wise. These are joint works with Battista, Italiano, and Migliorini.

TBA

It is well-known that the spectral data of the Gaudin model associated to a finite semisimple Lie algebra is encoded by the differential data of certain flat connections associated to the Langlands dual Lie algebra on the projective line with regular singularities, known as oper/Gaudin correspondence. Recently, some progress has been made in understanding the correspondence associated with affine Lie algebras. I will present a physical perspective from Kondo line defects, physically describing a local impurity chirally coupled to the bulk 2d conformal field theory. The Kondo line defects exhibit interesting integrability properties and wall-crossing behaviors, which are encoded by the generalized monodromy data of affine opers. In the physics literature, this reproduces the known ODE/IM correspondence. I will explain how the recently proposed 4d Chern Simons theory provides a new perspective which suggests the possibility of a physicists’ proof. 

This session will take place live in L1 and online. A Teams link will be shared 30 minutes before the session begins.