I will present recent results concerning a class of nonlinear parabolic systems of partial differential equations with small cross-diffusion (see doi.org/10.1051/m2an/2018036 and arXiv:1906.08060). Such systems can be interpreted as a perturbation of a linear problem and they have been proposed to describe the dynamics of a variety of large systems of interacting particles. I will discuss well-posedness, regularity, stability and convergence to the stationary state for (strong) solutions in an appropriate Banach space. I will also present some applications and refinements of the above-mentioned results for specific models.

In this talk, I will report about a joint work with D. Ciubotaru, in which we investigate the Dunkl version of the classical Howe-duality (O(k),spo(2|2)). Similar Fischer-type decompositions were studied before in the works of Ben-Said, Brackx, De Bie, De Schepper, Eelbode, Orsted, Soucek and Somberg for other Howe-dual pairs. Our work builds on the notion of a Dirac operator for Drinfeld algebras introduced by Ciubotaru, which was inspired by the analogous theory for Lie algebras, as well as the work of Cheng and Wang on classical Howe dualities.

I will talk about a way of building graded Lie algebras from certain Heisenberg groups. The input for this construction arises naturally when studying families of algebraic curves, and we'll look at some examples in which Lie theory interacts with number theory in an illuminating way. 

We present several Itô-Wentzell formulae on Wiener spaces for real-valued functionals random field of Itô type depending on measures. We distinguish the full- and marginal-measure flow cases. Derivatives with respect to the measure components are understood in the sense of Lions.
This talk is based on joint work with V. Platonov (U. of Edinburgh), see https://arxiv.org/abs/1910.01892.
 

Yang-Mills theory plays an important role in the Standard Model and is behind many mathematical developments in geometric analysis. In this talk, I will present several recent results on the problem of constructing quantum Yang-Mills measures in 2 and 3 dimensions. I will particularly speak about a representation of the 2D measure as a random distributional connection and as the invariant measure of a Markov process arising from stochastic quantisation. I will also discuss the relationship with previous constructions of Driver, Sengupta, and Lévy based on random holonomies, and the difficulties in passing from 2 to 3 dimensions. Partly based on joint work with Ajay Chandra, Martin Hairer, and Hao Shen.

This paper evaluates the effect of market integration on prices and welfare, in a model where two Lucas trees grow in separate regions with similar investors. We find equilibrium asset price dynamics and welfare both in segmentation, when each region holds its own asset and consumes its dividend, and in integration, when both regions trade both assets and consume both dividends. Integration always increases welfare. Asset prices may increase or decrease, depending on the time of integration, but decrease on average. Correlation in assets' returns is zero or negative before integration, but significantly positive afterwards, explaining some effects commonly associated with financialization.

We investigate properties of minimisers of a variational model describing the shape of charged liquid droplets. Roughly speaking, the shape of a charged liquid droplet is determined by the competition between an ”aggerating” term, due to surface tension forces, and to a ”disaggergating” term due to the repulsive effect between charged particles.

In my talk I want to present our ”first” analysis of the so called Deby-Hückel-type free energy. In particular we show that minimisers satisfy a partial regularity result, a first step of understanding the further properties of a minimiser. The presented results are joint work with Guido De Philippis and Giulia Vescovo.

The Gibbons-Hawking ansatz is a powerful method for constructing a large family of hyperkaehler 4-manifolds (which are thus Ricci-flat), which appears in a variety of contexts in mathematics and theoretical physics. I will describe work in progress to understand the theory of minimal surfaces and mean curvature flow in these 4-manifolds. In particular, I will explain a proof of a version of the Thomas-Yau Conjecture in Lagrangian mean curvature flow in this setting. This is joint work with G. Oliveira.

The translation method for constructing quasiconvex lower bound of a given function in the calculus of variations and the notion of compensated convex transforms for tightly approximate functions in Euclidean spaces will be briefly reviewed. By applying the upper compensated convex transform to the finite maximum function we will construct computable quasiconvex functions with finitely many point wells contained in a subspace with rank-one matrices. The complexity for evaluating the constructed quasiconvex functions is O(k log k) with k the number of wells involved. If time allows, some new applications of compensated convexity will be briefly discussed.