In practice, it is standard to initialize Artificial Neural Networks (ANN) with random parameters. We will see that this allows to describe, in the functional space, the limit of the evolution of (fully connected) ANN when their width tends towards infinity. Within this limit, an ANN is initially a Gaussian process and follows, during learning, a gradient descent convoluted by a kernel called the Neural Tangent Kernel. 

This description allows a better understanding of the convergence properties of neural networks, of how they generalize to examples during learning and has 

practical implications on the training of wide ANNs. 

The study of random matrix moments of moments has connections to number theory, combinatorics, and log-correlated fields. Our results give the leading order of these functions for integer moment parameters by exploiting connections with Gelfand-Tsetlin patterns and counts of lattice points in convex sets. This is joint work with Jon Keating and Theo Assiotis.

Mittag-Leffler condition ensures the exactness of the inverse limit of short exact sequences indexed on a partially ordered set admitting a countable cofinal subset. We extend Mittag-Leffler condition by relatively relaxing the countability assumption. As an application we prove an exactness result about the completion functor in the category of ultrametric locally convex vector spaces, and in particular we prove that a strict morphism between these spaces has closed image if its kernel is Fréchet.

We "review" how one can expand a filtration by continuously adding a stochastic process. The new results (obtained with Léo Neufcourt) relate to the seimartingale decompositions after the expansion. We give some possible applications. 

I will present numerical methods for low-rank matrix and tensor problems that explicitly make use of the geometry of rank constrained matrix and tensor spaces. We focus on two types of problems: The first are optimization problems, like matrix and tensor completion, solving linear systems and eigenvalue problems. Such problems can be solved by numerical optimization for manifolds, called Riemannian optimization methods. We will explain the basic elements of differential geometry in order to apply such methods efficiently to rank constrained matrix and tensor spaces. The second type of problem is ordinary differential equations, defined on matrix and tensor spaces. We show how their solution can be approximated by the dynamical low-rank principle, and discuss several numerical integrators that rely in an essential way on geometric properties that are characteristic to sets of low rank matrices and tensors. Based on joint work with André Uschmajew (MPI MiS Leipzig).

On any hyperbolic surface, the number of curves of length at most L is finite. However, it is not immediately clear how quickly this number grows with L. We will discuss Mirzakhani’s breakthrough result regarding the asymptotic behaviour of this number, along with recent efforts to generalise her result using currents.

The genus of a knot in a 3-manifold is defined to be the minimum genus of a compact, orientable surface bounding that knot, if such a surface exists. We consider the computational complexity of determining knot genus. Such problems have been studied by several mathematicians; among them are the works of Hass--Lagarias--Pippenger, Agol--Hass--Thurston, Agol and Lackenby. For a fixed 3-manifold the knot genus problem asks, given a knot K and an integer g, whether the genus of K is equal to g. In joint work with Lackenby, we prove that for any fixed, compact, orientable 3-manifold, the knot genus problem lies inNP, answering a question of Agol--Hass--Thurston from 2002. Previously this was known for rational homology 3-spheres by the work of Lackenby.

Numerous mathematical models have been proposed for modelling cancerous tumour invasion (Gatenby and Gawlinski 1996), angiogenesis (Owen et al 2008), growth kinetics (Wang et al 2009), response to irradiation (Gao et al 2013) and metastasis (Qiam and Akcay 2018). In this study, we attempt to model the qualitative behavior of growth, invasion, angiogenesis and fragmentation of tumours at the tissue level in an explicitly spatial and continuous manner in two dimensions. We simulate the effectiveness of radiation therapy on a growing tumour in comparison with immunotherapy and propose a novel framework based on vector fields for modelling the impact of interstitial flow on tumour morphology. The results of this model demonstrate the effectiveness of employing a system of partial differential equations along with vector fields for simulating tumour fragmentation and that immunotherapy, when applicable, is substantially more effective than radiation therapy.

Limit groups are a powerful tool in the study of free and hyperbolic groups (and even broader classes of groups). I will define limit groups in various ways: algebraic, logical and topological, and draw connections between the different definitions. We will also see how one can equip a limit group with an action on a real tree, and analyze this action using the Rips machine, a generalization of Bass-Serre theory to real trees. As a conclusion, we will obtain that hyperbolic groups whose outer automorphism group is infinite, split non-trivially as graphs of groups.