We study the regularity properties of viscosity solutions to a class of degenerate normalized $p$-laplacian equations. In particular, we prove that the gradient of viscosity solutions are Hölder continuous, and we give the optimal exponent. Moreover, we also show that viscosity solutions to equations with very general degeneracy laws are differentiable.

Using a fluctuation dissipation method, we construct an invariant measure for the surface quasi-geostrophic equation (SQG) and 3D Euler equation. Since the support of the measure contains entire solutions, we obtain a manifold containing solutions that do not blow-up. This complements results in which a blow-up solutions for SQG and grow up solutions for Euler are constructed. The method of the proof relies on an addition of a stochastic forcing and a small dissipation to the equation. For such stochastic equation, one can construct an invariant measure and by passing the strength of the forcing and the dissipation to zero, we obtain the desired invariant measure. We also discuss the size of the support of the measure, which relies on the number of conservation laws for the particular equation.

This is a joint project with Mouhamadou Sy.

One of the most successful lemmas of modern Harmonic Analysis is that "curvature induces decay of the Fourier transform". For example, if sources of electromagnetic waves are distributed on a sphere, the resulting waves will point in all possible directions and thus interact destructively; the same would not happen if those sources were distributed on a plane. In the 1970s, leading analysts such as E.M.Stein proposed to investigate and quantify this kind of phenomena. Since the main mathematical tool involved is the Fourier transform, this gave birth to a field of Harmonic Analysis known as "Fourier restriction theory".

This talk is aimed at non-specialists. We will study the following problem: how to optimally distribute wave sources on the sphere, maximizing the size of the corresponding superposition of waves? We will give a fully detailed solution of the 3-d case, then give a brief explanation of why this problem is still open in higher dimension.

The purpose of this talk is to explore how heterogeneity in the connectivity of an epidemiological network impacts the interaction between strains in a co-infection SIS framework.

In the first part, we will present an epidemiological model of co-infection with multiple strains under the assumption of homogeneous connectivity among hosts. We will show that, under the assumption of neutrality, this model satisfies a neutral null property, an ecologically important concept. Mathematically, this neutrality means that the $\omega$-limit set of any trajectory is a central manifold that can be parameterized by the proportion $(z_i)$ of each strain $i$.

This neutrality can be relaxed using a slow-fast argument, leading to an equation for $(z_i)$ that describes the slow dynamics on the central manifold. This equation is the well-known replicator equation, whose parameters are explicitly linked to macro-level ecological parameters. This connection allows us to understand pathogen ecology through the epidemiological dynamics at the host scale.

In the second part, we will explain how this approach can be extended when considering a heterogeneous network of host connectivity. After precisely describing this new model, we will show that the final replicator equation retains traces of this heterogeneity, providing a direct way to model how the complexity of host interactions affects the dynamics of a pathogen and its multiple variants.

In this talk I will review the notion of unbalanced optimal transport, introduced $\sim10$ years ago to handle mass variarions betwenn nonnegative measures. I will discuss the induced pseudo-Riemannian structure and gradient-flow evolution, corresponding to some class of parabolic reaction-diffusion equations. If time permits I will present two applications: a fitness-driven model from population dynamics, and a Hele-Shaw free boundary problem for tumor growth.