Past

This talk, based on joint work with Alexander Zlokapa, concerns Bayesian inference with neural networks. 

I will begin by presenting a result giving exact non-asymptotic formulas for Bayesian posteriors in deep linear networks. A key takeaway is the appearance of a novel scaling parameter, given by # data * depth / width, which controls the effective depth of the posterior in the limit of large model and dataset size. 

Additionally, I will explain some quite recent results on the role of this effective depth parameter in Bayesian inference with deep non-linear neural networks that have shaped activations.

Speaker: Mattia Magnabosco (Newton Fellow, Maths)
Title: Synthetic Ricci curvature bounds in sub-Riemannian manifolds
Abstract: In Riemannian manifolds, a uniform bound on the Ricci curvature tensor allows to control the volume growth along the geodesic flow. Building upon this observation, Lott, Sturm and Villani introduced a synthetic notion of curvature-dimension bounds in the non-smooth setting of metric measure spaces. This condition, called CD(K,N), is formulated in terms of the optimal transport interpolation of measures and consists in a convexity property of the Rényi entropy functionals along Wasserstein geodesics. The CD(K,N) condition represents a lower Ricci curvature bound by K and an upper bound on the dimension by N, and it is coherent with the smooth setting, as in a Riemannian manifold it is equivalent to a lower bound on the Ricci curvature tensor. However, the same relation between curvature and CD(K,N) condition does not hold for sub-Riemannian (and sub-Finsler) manifolds. 

Speaker: Rebecca Lewis (Florence Nightingale Bicentenary Fellow, Stats)
Title: High-dimensional statistics
Abstract: Due to the increasing ease with which we collect and store information, modern data sets have grown in size. Whilst these datasets have the potential to yield new insights in a variety of areas, extracting useful information from them can be difficult. In this talk, we will discuss these challenges.

Periodic point clouds naturally arise when modelling large homogenous structures like crystals. They are naturally attributed with a map to a d-dimensional torus given by the quotient of translational symmetries, however there are many surprisingly subtle problems one encounters when studying their (persistent) homology. It turns out that bisheaves are a useful tool to study periodic data sets, as they unify several different approaches to study such spaces. The theory of bisheaves and persistent local systems was recently introduced by MacPherson and Patel as a method to study data with an attributed map to a manifold through the fibres of this map. The theory allows one to study the data locally, while also naturally being able to appeal to local systems of (co)sheaves to study the global behaviour of this data. It is particularly useful, as it permits a persistence theory which generalises the notion of persistent homology. In this talk I will present recent work on the theory and implementation of bisheaves and local systems to study 1-periodic simplicial complexes. Finally, I will outline current work on generalising this theory to study more general periodic systems for d-periodic simplicial complexes for d>1. 

Associated to a complex admissible representation of a p-adic group is an invariant known is the "canonical dimension". It is closely related to the more well-studied invariant called the "wavefront set". The advantage of the canonical dimension over the wavefront set is that it allows for a completely different approach in computing it compared to the known computational methods for the wavefront set. In this talk we illustrate this point by finding a lower bound for the canonical dimension of any depth-zero supercuspidal representation, which depends only on the group and so is independent of the representation itself. To compute this lower bound, we consider the geometry of the associated Bruhat-Tits building.

(Joint work with S. Montenegro)

The striking resemblance between the behaviour of pseudo-algebraically closed, pseudo real closed and pseudo p-adically fields has lead to numerous attempts at describing their properties in a unified manner. In this talk I will present another of these attempts: the class of pseudo-T-closed fields, where T is an enriched theory of fields. These fields verify a « local-global » principle with respect to models of T for the existence of points on varieties. Although it very much resembles previous such attempts, our approach is more model theoretic in flavour, both in its presentation and in the results we aim for.

The first result I would like to present is an approximation result, generalising a result of Kollar on PAC fields, respectively Johnson on henselian fields. This result can be rephrased as the fact that existential closeness in certain topological enrichments come for free from existential closeness as a field. The second result is a (model theoretic) classification result for bounded pseudo-T-closed fields, in the guise of the computation of their burden. One of the striking consequence of these two results is that a bounded perfect PAC field with n independent valuations has burden n and, in particular, is NTP2.

The talk is based on joint work with Lasse Grimmelt. We prove a theorem that allows one to count solutions to determinant equations twisted by a periodic weight with high uniformity in the modulus. It is obtained by using spectral methods of $\operatorname{SL}_2(\mathbb{R})$ automorphic forms to study Poincaré series over congruence subgroups while keeping track of interactions between multiple orbits. This approach offers increased flexibility over the widely used sums of Kloosterman sums techniques. We give applications to correlations of the divisor function twisted by periodic functions and the fourth moment of Dirichlet $L$-functions on the critical line.

We show the super-hedging duality for multi-action options which generalise American options to a larger space of actions (possibly uncountable) than {stop, continue}. We put ourselves in the framework of Bouchard & Nutz model relying on analytic measurable selection theorem. Finally we consider information delay on the action component of the product space. Information delay is expressed as a possibility to look into the future in the dual formulation. This is a joint work with Ivan Guo, Shidan Liu and Zhou Zhou.

Artificial Intelligence (AI) will strongly determine our future prosperity and well-being. Due to its generic nature, AI will have an impact on all sciences and business sectors, our private lives and society as a whole. AI is pre-eminently a multidisciplinary technology that connects scientists from a wide variety of research areas, from behavioural science and ethics to mathematics and computer science.

Without downplaying the importance of that variety, it is apparent that mathematics can and should play an active role. All the more so as, alongside the successes of AI, also critical voices are increasingly heard. As Robert Dijkgraaf (former director of the Princeton Institute of Advanced Studies) observed in May 2019: ”Artificial intelligence is in its adolescent phase, characterised by trial and error, self-aggrandisement, credulity and lack of systematic understanding.” Mathematics can contribute to the much-needed systematic understanding of AI, for example, greatly improving reliability and robustness of AI algorithms, understanding the operation and sensitivity of networks, reducing the need for abundant data sets, or incorporating physical properties into neural networks needed for super-fast and accurate simulations in the context of digital twinning.

Mathematicians absolutely recognize the potential of artificial intelligence, machine learning and (deep) neural networks for future developments in science, technology and industry. At the same time, a sound mathematical treatment is essential for all aspects of artificial intelligence, including imaging, speech recognition, analysis of texts or autonomous driving, implying it is essential to involve mathematicians in all these areas. In this presentation, we highlight the role of mathematics as a key enabling technology within the emerging field of scientific machine learning. Or, as I always say: ”Real intelligence is needed to make artificial intelligence work.”

Minimal surfaces, which are critical points of the area functional, have long been a source of fruitful problems in geometry. In this talk, I will introduce a new approach, primarily coming from a recent paper of M. Struwe, to constructing free boundary minimal discs using a gradient flow of a suitable energy functional. I will discuss the uniqueness of solutions to the gradient flow, including recent work on the uniqueness of weak solutions, and also what is known about the qualitative behaviour of the flow, especially regarding the interpretation of singularities which arise. Time permitting, I will also mention ongoing joint work with M. Rupflin and M. Struwe on extending this theory to general surfaces with boundary.

Traditionally, stochastic processes are modelled one of two ways: a continuum Fokker-Planck approach, where a PDE is solved to determine the time evolution of the probability density, or a Langevin approach, where the SDE describing the system is sampled, and multiple simulations are used to collect statistics. There is also a third way: the functional or path integral. Originally developed by Wiener in the 1920s to model Brownian motion, path integrals were famously applied to quantum mechanics by Feynman in the 1950s. However, they also have much to offer classical stochastic processes (and statistical physics).  

In this talk I will introduce the formalism at a physicist’s level of rigour, and focus on determining the dominant contribution to the path integral when the noise is weak. There exists a remarkable correspondence between the most-probable stochastic paths and Hamiltonian dynamics in an effective potential [1,2,3]. I will then discuss some applications, including reaction pathways conditioned on finite time [2]. We demonstrate that the most probable pathway at a finite time may be very different from the usual minimum energy path used to calculate the average reaction rate. If time permits, I will also discuss the extremely nonlinear crystal dislocation response to applied stress [4].  

[1] Ge, Hao, and Hong Qian. Int. J. Mod. Phys. B 26.24 1230012 (2012)     

[2] Fitzgerald, Steve, et al. J. Chem. Phys. 158.12 (2023).

[3] Honour, Tom and Fitzgerald, Steve. in press J. Phys. A (2024)

[4] Fitzgerald, Steve. Sci. Rep. 6 (1) 39708 (2016)

Motivated by work of Hrushovski on pseudofinite fields with an additive character we investigate the theory ACFA+ which is the model companion of the theory of difference fields with an additive character on the fixed field. Building on results by Hrushovski we can recover it as the characteristic 0-asymptotic theory of the algebraic closure of finite fields with the Frobenius-automorphism and the standard character on the fixed field. We characterise 3-amalgamation in ACFA+. As cosequences we obtain that ACFA+ is a simple theory, an explicit description of the connected component of the Kim-Pillay group and (weak) elimination of imaginaries. If time permits we present some results on higher amalgamation.

RFRS groups were introduced by Ian Agol in connection with virtual fibering of 3-manifolds. Notably, the class of RFRS groups contains all compact special groups, which are groups with particularly nice cocompact actions on cube complexes. In this talk, I will give an introduction to ℓ²-Betti numbers from an algebraic perspective and discuss what group theoretic properties we can conclude from the (non)vanishing of the ℓ²-Betti numbers of a RFRS group.

One possible version of the Kirchberg—Phillips theorem states that simple, separable, nuclear, purely infinite C*-algebras are classified by KK-theory. In order to generalize this result to non-simple C*-algebras, Kirchberg first restricted his attention to those that absorb the Cuntz algebra O2 tensorially. C*-algebras in this class carry no KK-theoretical information in a strong sense, and they are classified by their ideal structure alone. It should be mentioned that, although this result is in Kirchberg’s work, its full proof was first published by Gabe. In joint work with Gábor Szabó, we showed a generalization of Kirchberg's O2-stable theorem that classifies G-C*-algebras up to cocycle conjugacy, where G is any second-countable, locally compact group. In our main result, we assume that actions are amenable, sufficiently outer, and absorb the trivial action on O2 up to cocycle conjugacy. In very recent work, I moreover show that the range of the classification invariant, consisting of a topological dynamical system over primitive ideals, is exhausted for any second-countable, locally compact group.

In this talk, I will recall the classification of O2-stable C*-algebras, and describe their classification invariant. Subsequently, I will give a short introduction to the C*-dynamical working framework and present the classification result for equivariant O2-stable actions. Time permitting, I will give an idea of how one can build a C*-dynamical system in the scope of our classification with a prescribed invariant. 

Let X and H be large, and consider n ranging from 1 to X. For an arithmetic function f(n), what is the best mean square approximation of f(n) by a restricted divisor sum (a function of the sort sum_{d|n, d < H} a_d)? I hope to explain how for a wide variety of arithmetic functions, when X grows and H grows like a power of X, a solution of this problem is connected to the evaluation of random matrix integrals. The problem is connected to some combinatorial formula for computing high moments of traces of random unitary matrices and I hope to discuss this also.

Measure equivalence was introduced by Gromov as a measure-theoretic analogue to quasi-isometry between finitely generated groups. In this talk I will present measure equivalence classification results for right-angled Artin groups, and more generally graph products. This is based on joint works with Jingyin Huang and with Amandine Escalier. 

The saturation number of a graph is a famous and well-studied counterpoint to the Turán number, and the rainbow saturation number is a generalisation of the saturation number to the setting of coloured graphs. Specifically, for a given graph $F$, an edge-coloured graph is $F$-rainbow saturated if it does not contain a rainbow copy of $F$, but the addition of any non-edge in any colour creates a rainbow copy of $F$. The rainbow saturation number of $F$ is the minimum number of edges in an $F$-rainbow saturated graph on $n$ vertices. Girão, Lewis, and Popielarz conjectured that, like the saturation number, for all $F$ the rainbow saturation number is linear in $n$. I will present our attractive and elementary proof of this conjecture, and finish with a discussion of related results and open questions.

One of the oldest open problems in representation theory is to classify the irreducible unitary representations of a semisimple Lie group G_R. Such representations play a fundamental role in harmonic analysis and the Langlands program and arise in physics as the state space of quantum mechanical systems in the presence of G_R-symmetry. Most unitary representations of G_R are realized, via some kind of induction, from unitary representations of proper Levi subgroups. Thus, the major obstacle to understanding the unitary dual of G_R is identifying the "non-induced" unitary representations of G_R. In previous joint work with Losev and Matvieievskyi, we have proposed a general construction of these non-induced representations, which we call "unipotent" representations of G_R. Unfortunately, the methods we employ do not provide a proof that these representations are unitary. In this talk, I will explain how one can apply Saito's theory of mixed Hodge modules to overcome this difficulty, giving a uniform proof of the unitarity of all unipotent representations. This is joint work in progress with Dougal Davis

We study the stochastic quantisation for the fractional $\varphi^4$ theory. The model has been studied by Brydges, Mitter and Scopola in 2003 as a natural extension of $\phi^4$ theories to fractional sub-critical dimensions. The stochastic quantisation equation is given by the (formal) SPDE 

\[

(\partial_t + (-\Delta)^{s}) \varphi = - \lambda \varphi^3 + \xi\]

where $\xi$ is a space-time white noise over the three dimensional torus. The equation is sub-critical for $s > \frac{3}{4}$.

We derive a priori estimates in the full sub-critical regime $s>\frac{3}{4}$. These estimates rule out explosion in finite time and they imply the existence of an invariant measure with a standard Krylov-Bogoliubov argument. 

Our proof is based on the strategy developed for the parabolic case $s=1$ in [Chandra, Moinat, Weber, ARMA 2023]. In order to implement this strategy here, a new Schauder estimate for the fractional heat operator is developed. Additionally, several algebraic arguments from [Chandra, Moinat, Weber, ARMA 2023] are streamlined significantly. 

This is joint work with Hendrik Weber (Münster). 

It is a classical fact of rational homotopy theory that the $E_\infty$-algebra of rational cochains on a sphere is formal, i.e., quasi-isomorphic to the cohomology of the sphere. In other words, this algebra is square-zero. This statement fails with integer or mod p coefficients. We show, however, that the cochains of the n-sphere are still $E_n$-trivial with coefficients in arbitrary cohomology theories. This is a consequence of a more general statement on (iterated) loops and suspensions of $E_n$-algebras, closely related to Koszul duality for the $E_n$-operads. We will also see that these results are essentially sharp: if the R-valued cochains of $S^n$ have square-zero $E_{n+1}$-structure (for some rather general ring spectrum R), then R must be rational. This is joint work with Markus Land.