L3

During this talk we will be interested in a one-dimensional exclusion process subject to strong kinetic constraints, which belongs to the class of cooperative kinetically constrained lattice gases. More precisely, its stochastic short range interaction exhibits a continuous phase transition to an absorbing state at a critical value of the particle density. We will see that the macroscopic behavior of this microscopic dynamics, under periodic boundary conditions and diffusive time scaling, is ruled by a non-linear PDE belonging to free boundary problems (or Stefan problems). One of the ingredients is to show that the system typically reaches an ergodic component in subdiffusive time.

Based on joint works with O. Blondel, C. Erignoux and M. Sasada

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. 

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. 

Abstract:

Quasicrystals have been observed recently in soft condensed mater, providing new insight into the ongoing quest to understand their formation and thermodynamic stability. I shall explain the stability of certain soft-matter quasicrystals, using surprisingly simple classical field theories, by making an analogy to Faraday waves. This will provide a recipe for designing pair potentials that yield crystals with (almost) any given symmetry.

In the process of studying the zeta-function for one parameter families of Calabi-Yau manifolds we have been led to a manifold, for which the quartic numerator of the zeta-function factorises into two quadrics remarkably often. Among these factorisations, we find persistent factorisations; these are determined by a parameter that satisfies an algebraic equation with coefficients in Q, so independent of any particular prime.  We note that these factorisations are due a splitting of Hodge structure and that these special values of the parameter are rank two attractor points in the sense of IIB supergravity. To our knowledge, these points provide the first explicit examples of non-singular, non-rigid rank two attractor points for Calabi-Yau manifolds of full SU(3) holonomy. Modular groups and modular forms arise in relation to these attractor points in a way that, to a physicist, is unexpected. This is a report on joint work with Xenia de la Ossa, Mohamed Elmi and Duco van Straten.

RF-engineering defines the “perfect” parabolic shape a foldable reflector antenna (e.g. the membrane) should have. In practice it is virtually impossible to design a deployable backing structure that can meet all RF-imposed requirements. Inevitably the shape of the membrane will deviate from its ideal parabolic shape when material properties and pragmatic mechanical design are considered. There is therefore a challenge to model such membranes in order to find the form they take and then use the model as a design tool and perhaps in an optimisation objective function, if tractable. 

The variables we deal with are:
Elasticity of the membrane (anisotropic or orthotropic typ)
Boundary forces (by virtue of the interaction between the membrane and it’s attachment)
Elasticity of the backing structure (e.g. the elasticity properties of the attachment)
Number, location and elasticity of the membrane fixing points

There are also in-orbit environmental effects on such structures for which modelling could also be of value. For example, the structure can undergo thermal shocks and oscillations can occur that are un-dampened by the usual atmospheric interactions at ground level etc. There are many other such points to be considered and allowed for.

We discuss properties of the cosmological horizon of a de Sitter universe, and compare to those of ordinary black holes. We consider both the Lorentzian and Euclidean picture. We discuss the relation to the sphere partition function and give a group-theoretic picture in terms of the de Sitter group. Time permitting we discuss some properties of three-dimensional de Sitter theories with higher spin particles. 

The duality between color and kinematics present in scattering amplitudes of Yang-Mills theory strongly suggest the existence of a hidden kinematic Lie algebra that controls the gauge theory. While associated BCJ numerators are known on closed forms to any multiplicity at tree level, the kinematic algebra has only been partially explored for the simplest of four-dimensional amplitudes: up to the MHV sector. In this paper we introduce a framework that allows us to characterize the algebra beyond the MHV sector. This allows us to both constrain some of the ambiguities of the kinematic algebra, and better control the generalized gauge freedom that is associated with the BCJ numerators. Specifically, in this paper, we work in dimension-agnostic notation and determine the kinematic algebra valid up to certain O((εi⋅εj)2) terms that in four dimensions compute the next-to-MHV sector involving two scalars. The kinematic algebra in this sector is simple, given that we introduce tensor currents that generalize standard Yang-Mills vector currents. These tensor currents controls the generalized gauge freedom, allowing us to generate multiple different versions of BCJ numerators from the same kinematic algebra. The framework should generalize to other sectors in Yang-Mills theory.

The optimal matching problem is about the rate of convergence
in Wasserstein distance of the empirical measure of iid uniform points
to the Lebesgue measure. We will start by reviewing the macroscopic
behaviour of the matching problem and will then report on recent results
on the mesoscopic behaviour in the thermodynamic regime. These results
rely on a quantitative large-scale linearization of the Monge-Ampere
equation through the Poisson equation. This is based on joint work with
Michael Goldman and Felix Otto.
 

The deep neural network has achieved impressive results in various applications, and is involved in more and more branches of science. However, there are still few theories supporting its empirical success. In particular, we miss the mathematical tool to explain the advantage of certain structures of the network, and to have quantitive error bounds. In our recent work, we used a regularised relaxed control problem to model the deep neural network.  We managed to characterise its optimal control by the invariant measure of a mean-field Langevin system, which can be approximated by the marginal laws. Through this study we understand the importance of the pooling for the deep nets, and are capable of computing an exponential convergence rate for the (stochastic) gradient descent algorithm.