Past

The Stein Variational Gradient Descent method is a variational inference method in statistics that has recently received a lot of attention. The method provides a deterministic approximation of the target distribution, by introducing a nonlocal interaction with a kernel. Despite the significant interest, the exponential rate of convergence for the continuous method has remained an open problem, due to the difficulty of establishing the related so-called Stein-log-Sobolev inequality. Here, we prove that the inequality is satisfied for each space dimension and every kernel whose Fourier transform has a quadratic decay at infinity and is locally bounded away from zero and infinity. Moreover, we construct weak solutions to the related PDE satisfying exponential rate of decay towards the equilibrium. The main novelty in our approach is to interpret the Stein-Fisher information as a duality pairing between $H^{-1}$ and $H^{1}$, which allows us to employ the Fourier transform. We also provide several examples of kernels for which the Stein-log-Sobolev inequality fails, partially showing the necessity of our assumptions. This is a joint work with J. A. Carrillo and J. Warnett. 

The large sieve inequality for Dirichlet characters is a central result in analytic number theory, which encodes a strong orthogonality property between primitive characters of varying conductors. This can be viewed as a statement about $\mathrm{GL}_1$ automorphic representations, and it is a key open problem to prove similar results in the higher $\mathrm{GL}_m$ setting; for $m \ge 2$, our best bounds are far from optimal. We'll outline two approaches to such results (sketching them first in the elementary case of Dirichlet characters), and discuss work-in-progress of Thorner and the author on an improved $\mathrm{GL}_m$ large sieve. No prior knowledge of automorphic representations will be assumed.

Spin glasses are models of statistical mechanics in which a large number of elementary units interact with each other in a disordered manner. In the simplest case, there are direct interactions between any two units in the system, and I will start by reviewing some of the key mathematical results in this context. For modelling purposes, it is also desirable to consider models with more structure, such as when the units are split into two groups, and the interactions only go from one group to the other one. I will then discuss some of the technical challenges that arise in this case, as well as recent progress.

I will describe a generalisation of the Seiberg-Witten equations to a Spin-c manifold of any dimension. The equations are for a U(1) connection A and spinor \phi and also an odd-degree differential form b (of inhomogeneous degree). Clifford action of the form is used to perturb the Dirac operator D_A. The first equation says that (D_A+b)(\phi)=0. The second equation involves the Weitzenböck remainder for D_A+b, setting it equal to q(\phi), where q(\phi) is the same quadratic term which appears in the usual Seiberg-Witten equations. This system is elliptic modulo gauge in dimensions congruent to 0,1 or 3 mod 4. In dimensions congruent to 2 mod 4 one needs to take two copies of the system, coupled via b. I will also describe a variant of these equations which make sense on manifolds with a Spin(7) structure. The most important difference with the familiar 3 and 4 dimensional stories is that compactness of the space of solutions is, for now at least, unclear. This is joint work with Partha Ghosh and, in the Spin(7) setting, Ragini Singhal.

This brief pedagogical talk introduces key concepts of Carroll geometries, which arise as the limit of relativistic spacetimes in the vanishing speed of light regime. In this limit, light cones collapse along a timelike direction, resulting in a manifold equipped with a degenerate metric. Consequently, physics in such spacetimes exhibits peculiar properties. Despite this, the Carroll contraction is relevant to a wide range of applications, from flat-space holography to condensed matter physics. To complement this introduction, and depending on the audience’s interests, I can discuss Carroll affine connections, symmetry groups, conservation laws, and Carroll-invariant field theories.

We present results (joint with Hiroshi isozaki, Matti lassas, and Teemu Tyni) on reconstruction of certain nonlinear wave operators from knowledge of their far field effect on incoming waves. The result depends on the reformulation of the problem as a non-linear Goursat problem in the Penrose conformal compactification, for suitably small incoming waves. The non-linearity is exploited to generate secondary waves, which eventually probe the geometry of the space-time. Some extensions to cosmological space-times will also be discussed.  Time permitting, we will contrast these results with near-field inverse scattering obtained for only linear waves, where no non-linearity can be exploited, and the methods depend instead on unique continuation. (The latter joint with Ali Feizmohammadi and Lauri Oksanen). 

In this talk, I will discuss Figalli and Gigli’s formulation of optimal transport between non-negative Radon measures in the setting of metric pairs. This framework allows for the comparison of measures with different total masses by introducing an auxiliary set that compensates for mass discrepancies. Within this setting, classical characterisations of optimal transport plans extend naturally, and the resulting spaces of measures are shown to be complete, separable, geodesic, and non-branching, provided the underlying space possesses these properties. Moreover, we prove that the spaces of measures 
equipped with the $L^2$-optimal partial transport metric inherit non-negative curvature in the sense of Alexandrov. Finally, generalised spaces of persistence diagrams embed naturally into these spaces of measures, leading to a unified perspective from which several known geometric properties of generalised persistence diagram spaces follow. These results build on recent work by Divol and Lacombe and generalise classical results in optimal transport.

The main challenges in constructing a holographic correspondence for asymptotically flat spacetimes lie in the null nature of the conformal boundary and the non-conservation of gravitational charges in the presence of bulk radiation. In this talk, I shall demonstrate that there exists a systematic and mathematically robust approach to understanding and deriving the associated flux-balance laws from intrinsic boundary geometric considerations — an aspect of crucial importance for flat-space holography, as I shall argue during the presentation. 

For self-containment, I shall begin by reviewing key aspects of the geometry at null infinity, which has been termed conformal Carroll geometry. Reviving Ashtekar’s old statement, I shall emphasise that boundary affine connections possess degrees of freedom that precisely serve as the sources encoding radiation from a holographic perspective. I shall conclude by deriving flux-balance laws in an effective field theory framework at the boundary, employing novel techniques that introduce “hypermomenta” as responses to fluctuations in the boundary connection. The strength of our formalism lies in its ability to perform all computations in a manifestly coordinate- and Weyl-invariant manner within the framework of Sir Penrose’s conformal compactification.

Many justifications can be offered for the study of the history of mathematics. Here we focus on three, each of them illustrated by a specific historical example: it can aid in the learning of mathematics; it can prompt the development of new mathematics; and last but certainly not least – it's fun and interesting!

The category PAb of profinite abelian groups is an abelian category with many nice properties, which allows us to do most of standard homological algebra. The category PAb naturally embeds into the category TAb of topological abelian groups, but TAb is not abelian, nor does it have a satisfactory theory of tensor products. On the other hand, PAb also naturally embeds into the category CondAb of "condensed abelian groups", which is an abelian category with nice properties. We will show that the embedding of profinite modules into condensed modules (actually, into "solid modules") preserves usual homological notions such Ext and Tor, so that the condensed world might be a better place to study profinite modules than the topological world.

Recent progress in the development of equivariant neural network architectures predominantly used for machine learning interatomic potentials (MLIPs) has opened new possibilities in the development of data-driven coarse-graining strategies. In this talk, I will first present our work on the development of learning potential energy surfaces and other physical quantities, namely the Hyperactive Learning framework[1], a Bayesian active learning strategy for automatic efficient assembly of training data in MLIP and ACEfriction [2], a framework for equivariant model construction based on the Atomic Cluster Expansion (ACE) for learning of configuration-dependent friction tensors in the dynamic equations of molecule surface interactions and Dissipative Particle Dynamics (DPD). In the second part of my talk, I will provide an overview of our work on the simulation and analysis of Generalized Langevin Equations [3,4] as obtained from systematic coarse-graining of Hamiltonian Systems via a Mori-Zwanzig projection and present an outlook on our ongoing work on developing data-driven approaches for the construction of dynamics-preserving coarse-grained representations.

References:

[1] van der Oord, C., Sachs, M., Kovács, D.P., Ortner, C. and Csányi, G., 2023. Hyperactive learning for data-driven interatomic potentials. npj Computational Materials

[2] Sachs, M., Stark, W.G., Maurer, R.J. and Ortner, C., 2024. Equivariant Representation of Configuration-Dependent Friction Tensors in Langevin Heatbaths. to appear in Machine Learning: Science & Technology

[3] Leimkuhler, B. and Sachs, M., 2022. Efficient numerical algorithms for the generalized Langevin equation. SIAM Journal on Scientific Computing

[4] Leimkuhler, B. and Sachs, M., 2019. Ergodic properties of quasi-Markovian generalized Langevin equations with configuration-dependent noise and non-conservative force. In Stochastic Dynamics Out of Equilibrium: Institut Henri Poincaré, 2017 

Due to connections to flat space holography, Carrollian geometry, physics and fluid dynamics have received an explosion of interest over the last two decades. In the Carrollian limit of vanishing speed of light c, relativistic fluids reduce to a set of PDEs called the Carrollian fluid equations. Although in general these equations are not well understood, and their PDE theory does not appear to have been studied, in dimensions 1+1 it turns out that there is a duality with the Galilean compressible Euler equations in 1+1 dimensions inherited from the isomorphism of the Carrollian (c to 0) and Galilean (c to infinity) contractions of the Poincar\'e algebra. Under this duality time and space are interchanged, leading to different dynamics in evolution. I will discuss recent work with N. Athanasiou (Thessaloniki), M. Petropoulos (Paris) and S. Schulz (Pisa) in which we establish the first rigorous PDE results for these equations by introducing a notion of Carrollian isentropy and studying the equations using Lax’s method and compensated compactness. In particular, I will explain that there is global existence in rough norms but finite-time blow-up in smoother norms.

Galilean and Carrollian algebras are dual contractions of the Poincaré algebra. They act on two-dimensional Newton--Cartan and Carrollian manifolds and are isomorphic. A consequence of this property is a duality correspondence between one-dimensional Galilean and Carrollian fluids. I will describe the algebras and the dynamics of these systems as they emerge from the relevant  limits of Lorentzian hydrodynamics, and explore the advertised duality relationship. This interchanges longitudinal and transverse directions with respect to the flow velocity, and permutes equilibrium and out-of-equilibrium observables, unveiling specific features of Carrollian physics. I will also discuss the hydrodynamic-frame invariance in Lorentzian systems and its fate in the Galilean and Carrollian avatars.

Rank-based models for equity markets are reduced-form models where the asset dynamics depend on the rank that asset occupies in the investment universe. Such models are able to capture certain stylized macroscopic properties of equity markets, such as stability of the capital distribution curve and collision rates of stock rank switches. However, when calibrated to real equity data the models possess undesirable features such as an "Atlas stock" effect; namely the smallest security has an unrealistically large drift. Recently, Campbell and Wong (2024) identified that listings and delistings (i.e. entrances and exists) of securities in the market are important drivers for the stability of the capital distribution curve. In this work we develop a framework for ranked-based models with listings and delistings and calibrate them to data. By incorporating listings and delistings the calibration procedure no longer leads to an "Atlas stock" behaviour. Moreover, by studying an appropriate "local model", focusing on a specific target rank, we are able to connect collisions rates with a notion of particle density, which is more stable and easier to estimate from data than the collision rates. The calibration results are supported by novel theoretical developments such as a new master formula for functional generation of portfolios in this setting. This talk is based on joint work in progress with Martin Larsson and Licheng Zhang.  

A way to study rational points on a variety is by looking at their image in the p-adic points. Some natural questions that arise are the following: is there any obstruction to weak approximation on the variety? Which primes might be involved in it? I will explain how primes of good reduction can play a role in the Brauer-Manin obstruction to weak approximation, with particular emphasis on the case of K3 surfaces.

The Landau equation stands as one of the fundamental equations in kinetic theory and plays a key role in plasma physics. However, computing it presents significant challenges due to the complexity of the Landau operator,  the dimensionality, and the need to preserve the physical properties of the solution. In this presentation, I will introduce deep learning assisted particle methods aimed at addressing some of these challenges. These methods combine the benefits of traditional structure-preserving techniques with the approximation power of neural networks, aiming to handle high dimensional problems with minimal training. 

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

On this talk we will focus on the family of aggregation-diffusion equations

$$\frac{\partial \rho}{\partial t} = \mathrm{div}\left(\mathrm{m}(\rho)\nabla (U'(\rho) + V) \right).$$

Here, $\mathrm{m}(s)$ represents a continuous and compactly supported nonlinear mobility (saturation) not necessarily concave. $U$ corresponds to the diffusive potential and includes all the porous medium cases, i.e. $U(s) = \frac{1}{m-1} s^m$ for $m > 0$ or $U(s) = s \log (s)$ if $m = 1$. $V$ corresponds to the attractive potential and it is such that $V \geq 0$, $V \in W^{2, \infty}$.

Taking advantage of a family of approximating problems, we show the existence of $C_0$-semigroups of $L^1$ contractions. We study the $\omega$-limit of the problem, its most relevant properties, and the appearance of free boundaries in the long-time behaviour. Furthermore, since this problem has a formal gradient-flow structure, we discuss the local/global minimisers of the corresponding free energy in the natural topology related to the set of initial data for the $L^\infty$-constrained gradient flow of probability densities. Finally, we explore the properties of a corresponding implicit finite volume scheme introduced by Bailo, Carrillo and Hu.

The talk presents joint work with Prof. J.A. Carrillo and Prof. D.  Gómez-Castro.

Inversion problems, such as full waveform inversion (FWI), based on wave propagation, are computationally costly optimization processes used in many applications, ranging from seismic imaging to brain tomography. In most of these uses, high-order methods are required for both accuracy and computational efficiency. Within finite element methods (FEM), using high(er)-order can provide accuracy and the usage of flexible meshes. However, FEM are rarely employed in connection with unstructured simplicial meshes because of the computational cost and complexity of code implementation. They are used frequently with quadrilateral or hexahedral spectral finite elements, but the mesh adaptivity on those elements has not yet been fully explored. In this work, we address these challenges by developing software that leverages accurate higher-order mass-lumped simplicial elements with a mesh-adaption parameter, allowing us to take advantage of the computational efficiency of newer mass-lumped simplicial elements together with waveform-adapted meshes and the accuracy of higher-order function spaces. We also calculate these mesh-related parameters and develop software for high-order spectral element methods, allowing mesh flexibility. We will also discuss future developments. The open-source code was implemented using the Firedrake framework and the Unified Form Language (UFL), a mathematical-based domain specific language, allowing flexibility in a wide range of wave-based problems. 

The beautiful displays exhibited by fish schools and bird flocks have long fascinated scientists, but the role of their complex behavior remains largely unknown. In particular, the influence of hydrodynamic interactions on schooling and flocking has been the subject of debate in the scientific literature. I will present a model for flapping wings that interact hydrodynamically in an inviscid fluid, wherein each wing is represented as a plate that executes a prescribed time-periodic kinematics. The model generalizes and extends thin-airfoil theory by assuming that the flapping amplitude is small, and permits consideration of multiple wings through the use of conformal maps and multiply-connected function theory. We find that the model predictions agree well with experimental data on freely-translating, flapping wings in a water tank. The results are then used to motivate a reduced-order model for the temporally nonlocal interactions between schooling wings, which consists of a system of nonlinear delay-differential equations. We obtain a PDE as the mean-field limit of these equations, which we find supports traveling wave solutions. Generally, our results indicate how hydrodynamics may mediate schooling and flocking behavior in biological contexts.