We consider scalar conservation laws on networks.
We show that we can glue together two Riemann solvers to get a third Riemann solver.
On networks, we also show that two different good PDE theories do exist: a Kruzkov theory and a Hamilton-Jacobi theory. For solvers outside these two theories, we have basically an infinite number of open problems.

We want our statistical tools to be as accurate as possible. One way of measuring how accurate statistical methods are is from the frequentist interpretation of probability theory. In practise, people want to use Bayesian methods. Bayesian methods work by specifying model, and a prior on the model parameters. The posterior distribution in the leading object in the inference, which is the conditional probability distribution of the prior given the data. We can view these as just another method and study their performance from a frequentist point of view. In my talk I will go over some of the main topics in Frequentist Bayes. In particular, I will talk about consistency, contraction rates and the Bernstein-von Mises property. In the end I will briefly talk about my research.

Recently there is a rising interest in the research of mean-field optimization, in particular because of its role in analyzing the training of neural networks. In this talk, by adding the Fisher information (in other word, the Schrodinger kinetic energy) as the regularizer, we relate the mean-field optimization problem with a so-called mean field Schrodinger (MFS) dynamics. We develop a free energy method to show that the marginal distributions of the MFS dynamics converge exponentially quickly towards the unique minimizer of the regularized optimization problem. We shall see that the MFS is a gradient flow on the probability measure space with respect to the relative entropy. Finally we propose a Monte Carlo method to sample the marginal distributions of the MFS dynamics. This is a joint work with Giovanni Conforti, Zhenjie Ren and Songbo Wang.

Zero-sum stochastic games model the dynamic interactions between two adversary players. At each stage, the players choose actions in order to maximize their payoff determined by the actions and the current state. The state variable follows a Markov chain that depends on the players’ actions.
We consider zero-sum games with stage duration. In stochastic games with stage duration h, players act at times 0, h, 2h, and so on. The payoff and leaving probabilities are proportional to h. When h tends to 0, such discrete-time games approximate games played in continuous time.
We are interested in the behavior of the value when h tends to 0. While the case of full observation of the state is well-studied, the situation is much less clear when players cannot fully observe the state. We provide examples that demonstrate this.

This talk is devoted to studying propagation phenomena in reaction diffusion and integro-differential equations with a weakly degenerate non-linearity. The reaction term can be seen as an intermediate between the classical logistic (or Fisher-KPP) non-linearity and the standard weak Allee effect one. We study the effect of the tails of the dispersal kernel on the rate of expansion. When the tail of the kernel is sub-exponential, the exact separation between existence and nonexistence of travelling waves is exhibited. This, in turn, provides the exact separation between finite speed propagation and acceleration in the Cauchy problem. Moreover, the exact rates of acceleration for dispersal kernels with sub-exponential and algebraic tails are provided. Our approach is generic and covers a large variety of dispersal kernels including those leading to convolution and fractional Laplace operators. Numerical simulations are provided to illustrate our results. This comes from joint works with Professor Jérôme Coville and Xi Zhang.

Phase transitions displaying a mixed first order/critical character emerge in different contexts. 
Let us name a few examples ranging from physics, mathematics and computer science: the dynamic structure factor of liquids near the glass transition; mean field spin glasses with a one step replica symmetry transition; bootstrap percolation on regular trees; random XORSAT and other combinatorial optimization problems. 
The common feature is the emergence of a multistep relaxation when we approach criticality and a key issue is to characterize how dynamical correlation functions  relax to and depart from a plateau.
In all these cases physicists apply the so-called mode coupling theory, which provide an intergro-differential equation for correlations which, modulo a power law ansatz, yields closed formulas for the (non integer) exponents determining the asymptotic power law approach and depart from the plateau. The agreement of numerical results with these formulas is striking.  But...what can we say rigorously on these equations?