Conférence à venir :

The area of stochastic partial differential equations has seen rapid progress over the past decade, spurred by the introduction of the theory of regularity structures and of para-controlled calculus. Despite the close connections of singular SPDEs to physical phenomena, the theory of singular SPDEs has largely been developed homogeneous settings, involving constant-coefficient operators.
First I shall motivate the study of SPDEs in heterogeneous environments using the examples of Φ^4 as a model of ferromagnetism and the parabolic Anderson model as it relates to branching processes.
After recalling the renormalisation of the classical homogeneous versions of these equations, I will describe what changes in the variable-coefficient setting. In particular, I will present a natural choice of renormalisation functions that is local and, for sufficiently covariant regularisations of the noise, explicit.
The talk will be non-technical and focused on the main ideas.

The problem of estimating the maximal number H(m) of limit cycles that planar polynomial vector fields of degree m can exhibit has long been a central question in the qualitative theory of planar dynamical systems. A natural extension to the threedimensional space is to study the maximum number N(m) of limit tori that can occur in spatial polynomial vector fields of degree m. In this work, we focus on normally hyperbolic limit tori and show that the corresponding maximum number N_h(m), if finite, increases strictly with m. More precisely, we prove that N_h(m+1) ⩾ N_h(m)+ 1. Our proof relies on two central results established in this paper. The first is that the normal
hyperbolicity of compact invariant manifolds is preserved under time reparametrizations.

Despite the fundamental nature of this statement, a complete proof has, surprisingly, not previously appeared in the literature, except under rather restrictive assumptions on the flow restricted to the invariant manifold. The second result concerns the torus bifurcation phenomenon near Hopf–Zero equilibria in spatial vector fields. While the conditions for such bifurcations are typically expressed in terms of higher-order normal form coefficients, we derive explicit and verifiable criteria for the occurrence of torus bifurcation, assuming only that the linear part of the unperturbed vector field is in Jordan normal form.

This approach not only circumvents intricate computations involving higher-order normal forms but also ensures the normal hyperbolicity of the bifurcated torus.