Past

Let $X$ denote an open subset of $\mathbb{C}^d$, and $\mathcal{O}$ its sheaf of holomorphic functions. In the 1970’s, Ishimura studied the morphisms of sheaves $P\colon\mathcal{O}\to\mathcal{O}$ of $\mathbb{C}$-vector spaces which are continuous, that is the maps $P(U)\colon\mathcal{O}(U)\to\mathcal{O}(U)$ on the sections are continuous. In this talk, we explain his result, and explore its analogues in the non-Archimedean world.

Modern Data Assimilation (DA) can be traced back to the sixties and owes a lot to earlier developments in linear filtering theory. Since then, DA has evolved independently of Filtering Theory. To-date it is a massively important area of research due to its many applications in meteorology, ocean prediction, hydrology, oil reservoir exploration, etc. The field has been largely driven by practitioners, however in recent years an increasing body of theoretical work has been devoted to it. In this talk, In my talk, I will advocate the interpretation of DA through the language of stochastic filtering. This interpretation allows us to make use of advanced particle filters to produce rigorously validated DA methodologies. I will present a particle filter that incorporates three additional add-on procedures: nudging, tempering and jittering. The particle filter is tested on a two-layer quasi-geostrophic model with O(10^6) degrees of freedom out of which only a minute fraction are noisily observed.

We study the global linear convergence of policy gradient (PG) methods for finite-horizon exploratory linear-quadratic control (LQC) problems. The setting includes stochastic LQC problems with indefinite costs and allows additional entropy regularisers in the objective. We consider a continuous-time Gaussian policy whose mean is linear in the state variable and whose covariance is state-independent. Contrary to discrete-time problems, the cost is noncoercive in the policy and not all descent directions lead to bounded iterates. We propose geometry-aware gradient descents for the mean and covariance of the policy using the Fisher geometry and the Bures-Wasserstein geometry, respectively. The policy iterates are shown to obey an a-priori bound, and converge globally to the optimal policy with a linear rate. We further propose a novel PG method with discrete-time policies. The algorithm leverages the continuous-time analysis, and achieves a robust linear convergence across different action frequencies. A numerical experiment confirms the convergence and robustness of the proposed algorithm.

This is joint work with Yufei Zhang and Christoph Reisinger.

I will talk about work in progress with Ana Caraiani aimed at proving Ihara’s lemma for quaternionic Shimura varieties, generalising the strategy of Manning-Shotton for Shimura curves. As an arithmetic motivation, in the first part of the talk I will recall an approach to the Birch and Swinnerton-Dyer conjecture based on congruences between modular forms, relying crucially on Ihara’s lemma.

TBA

The Camassa--Holm (CH) equation is a nonlocal equation that manifests supercritical behaviour in ``wave-breaking" and non-uniqueness. In this talk, I will discuss the existence of global (dissipative weak martingale) solutions to the CH equation with multiplicative, gradient type noise, derived as an inviscid limit. The goal of the talk is twofold. The stochastic CH equation will be used to illustrate aspects of a stochastic compactness and renormalisation method which is popularly used to derive well-posedness and continuous dependence results in SPDEs. I shall also discuss how a lack of temporal compactness introduces fundamental difficulties in the case of the stochastic CH equation.

This talk is based on joint works with L. Galimbert and H. Holden, both at NTNU, and with K.H. Karlsen at the University of Oslo. 

The handlebody group is defined as the mapping class group of a three-dimensional handlebody. We will survey some geometric and algebraic properties of the handlebody groups and compare them to those of two of the most studied (classes of) groups in geometric group theory, namely mapping class groups of surfaces, and ${\rm Out}(F_n)$. We will also introduce the disk graph, the handlebody-analogon of the curve graph of a surface, and discuss some of its properties.

Since Segal's formulation of axioms for 2d CFTs in the 80s, it has remained a major problem to construct examples of CFTs that satisfy the axioms.

I will report on ongoing joint work with James Tener in that direction.

I discuss the set of rates of growth of a finitely generated 
group with respect to all its finite generating sets. In a joint work 
with Sela, for a hyperbolic group, we showed that the set is 
well-ordered, and that each number can be the rate of growth of at most 
finitely many generating sets up to automorphism of the group. I may 
discuss its generalization to acylindrically hyperbolic groups.

We prove that every finite colouring of the plane contains a monochromatic pair of points at an odd (integral) distance from each other. We will also discuss some further results with Rose McCarty and Michal Pilipczuk concerning prime and polynomial distances.

Fibres coming from the Springer resolution on the nilpotent cone are incredibly rich algebraic varieties that have many applications in representation theory and combinatorics. Though their geometry can be very difficult to describe in general, in type A at least, their irreducible components can be described using standard Young tableaux, and this can help describe their geometry in small dimensions. In this talk, I will report on recent and ongoing work with Lewis Topley and separately Daniele Rosso on geometrical and combinatorial applications of the classical ‘type A’ Springer fibres and the ‘exotic’ type C Springer fibres coming from Kato’s exotic Springer correspondence.

Baseline T cell infiltration and the spatial distribution of T cells within a tumour has been found to be a significant indicator of patient outcomes. This observation, coupled with the increasing availability of spatially-resolved imaging data of individual cells within the tumour tissue, motivates the development of mathematical models which capture the spatial dynamics of T cells. Agent-based models allow the simulation of complex biological systems with detailed spatial resolution, and generate rich spatio-temporal datasets. In order to fully leverage the information contained within these simulated datasets, spatial statistics provide methods of analysis and insight into the biological system modelled, by quantifying inherent spatial heterogeneity within the system. We present a cellular automaton model of interactions between tumour cells and cytotoxic T cells, and an analysis of the model dynamics, considering both the temporal and spatial evolution of the system. We use the model to investigate some of the standard assumptions made in these models, to assess the suitability of the models to accurately describe tumour-immune dynamics.

We present a new approach to the gluing problem in General Relativity, that is, the problem of matching two solutions of the Einstein equations along a spacelike or characteristic (null) hypersurface. In contrast to previous constructions, the new perspective actively utilizes the nonlinearity of the constraint equations. As a result, we are able to remove the 10-dimensional spaces of obstructions to gluing present in the literature. As application, we show that any asymptotically flat spacelike initial data set can be glued to Schwarzschild initial data of sufficiently large mass. This is joint work with I. Rodnianski.

A modular functor is defined as a system of mapping class group representations on vector spaces (the so-called conformal blocks) that is compatible with the gluing of surfaces. The notion plays an important role in the representation theory of quantum groups and conformal field theory. In my talk, I will give an introduction to the theory of modular functors and recall some classical constructions. Afterwards, I will explain the approach to modular functors via cyclic and modular operads and their bicategorical algebras. This will allow us to extend the known constructions of modular functors and to classify modular functors by certain cyclic algebras over the little disk operad for which an obstruction formulated in terms of factorization homology vanishes. (The talk is based to a different extent on different joint works with Adrien Brochier, Lukas Müller and Christoph Schweigert.)

We introduce so-called functional input neural networks defined on infinite dimensional weighted spaces, where we use an additive family as hidden layer maps and a non-linear activation function applied to each hidden layer. Relying on approximation theory based on Stone-Weierstrass and Nachbin type theorems on weighted spaces, we can prove global universal approximation results for (differentiable and) continuous functions going beyond approximation on compact sets. This applies in particular to approximation of (non-anticipative) path space functionals via functional input neural networks but also via linear maps of the signature of the respective paths. We apply these results in the context of stochastic portfolio theory to generate path dependent portfolios that are trained to outperform the market portfolio. The talk is based on joint works with Philipp Schmocker and Josef Teichmann.

I will present two new results. The first concerns minimal surfaces of the hyperbolic space and is a relation between their renormalised area (in the sense of Graham and Witten) and the length of their ideal boundary measured in different metrics of the conformal infinity. The second result concerns minimal submanifolds of the sphere and is a relation between their volume and antipodal-ness. Both results were obtained from the same framework, which involves new monotonicity theorems and a comparison principle for them. If time permits, I will discuss how to use these to answer questions about uniqueness and non-existence of minimal surfaces.

Conformal Field Theories (CFT) are believed to be exactly solvable once their primary scaling fields and their 3-point functions are known. This input is called the spectrum and structure constants of the CFT respectively. I will review recent work where this conformal bootstrap program can be rigorously carried out for the case of Liouville CFT, a theory that plays a fundamental role in 2d random surface theory and many other fields in physics and mathematics. Liouville CFT has a probabilistic formulation on an arbitrary Riemann surface and the bootstrap formula can be seen as a "quantization" of the plumbing construction of surfaces with marked points axiomatically discussed earlier by Graeme Segal. Joint work with Colin Guillarmou, Remi Rhodes and Vincent Vargas

Conformal Field Theories (CFT) are believed to be exactly solvable once their primary scaling fields and their 3-point functions are known. This input is called the spectrum and structure constants of the CFT respectively. I will review recent work where this conformal bootstrap program can be rigorously carried out for the case of Liouville CFT, a theory that plays a fundamental role in 2d random surface theory and many other fields in physics and mathematics. Liouville CFT has a probabilistic formulation on an arbitrary Riemann surface and the bootstrap formula can be seen as a "quantization" of the plumbing construction of surfaces with marked points axiomatically discussed earlier by Graeme Segal. Joint work with Colin Guillarmou, Remi Rhodes and Vincent Vargas.

Matthew Buckland 
Branching Interval Partition Diffusions

We construct an interval-partition-valued diffusion from a collection of excursions sampled from the excursion measure of a real-valued diffusion, and we use a spectrally positive Lévy process to order both these excursions and their start times. At any point in time, the interval partition generated is the concatenation of intervals where each excursion alive at that point contributes an interval of size given by its value. Previous work by Forman, Pal, Rizzolo and Winkel considers self-similar interval partition diffusions – and the key aim of this work is to generalise these results by dropping the self-similarity condition. The interval partition can be interpreted as an ordered collection of individuals (intervals) alive that have varying characteristics and generate new intervals during their finite lifetimes, and hence can be viewed as a class of Crump-Mode-Jagers-type processes.

Ofir Gorodetsky
Smooth and rough numbers

We all know and love prime numbers, but what about smooth and rough numbers?
We'll define y-smooth numbers -- numbers whose prime factors are all less than y. We'll explain their application in cryptography, specifically to factorization of integers.
We'll shed light on their density, which is modelled using a peculiar differential equation. This equation appears naturally in probability theory.
We'll also explain the dual notion to smooth numbers, that of y-rough numbers: numbers whose prime factors are all bigger than y, and in some sense generalize primes.
We'll explain their importance in sieve theory. Like smooth numbers, their density has interesting properties and will be surveyed.

At the intersection of Topological Data Analysis and machine learning, the field of cellular signal processing has advanced rapidly in recent years. In this context, each signal on the cells of a complex is processed using the combinatorial Laplacian and the resulting Hodge decomposition. Meanwhile, discrete Morse theory has been widely used to speed up computations by reducing the size of complexes while preserving their global topological properties. In this talk, we introduce an approach to signal compression and reconstruction on complexes that leverages the tools of discrete Morse theory. The main goal is to reduce and reconstruct a cell complex together with a set of signals on its cells while preserving their global topological structure as much as possible. This is joint work with Stefania Ebli and Kelly Maggs.