For some nonlocal PDEs, its steady states can be seen as critical points of an associated energy functional. Therefore, if one can construct perturbations around a function such that the energy decreases to first order along the perturbation, this function cannot be a steady state. In this talk, I will discuss how this simple variational approach has led to some recent progresses in the following equations, where the key is to carefully construct a suitable perturbation.

I will start with the aggregation-diffusion equation, which is a nonlocal PDE driven by two competing effects: nonlinear diffusion and long-range attraction. We show that all steady states are radially symmetric up to a translation (joint with Carrillo, Hittmeir and Volzone), and give some criteria on the uniqueness/non-uniqueness of steady states within the radial class (joint with Delgadino and Yan).

I will also discuss the 2D Euler equation, where we aim to understand under what condition must a stationary/uniformly-rotating solution be radially symmetric. Using a variational approach, we settle some open questions on the radial symmetry of rotating patches, and also show that any smooth stationary solution with compactly supported and nonnegative vorticity must be radial (joint with Gómez-Serrano, Park and Shi).

Symbol alphabets of n-particle amplitudes in N=4 super-Yang-Mills theory are known to contain certain cluster variables of Gr(4,n) as well as certain algebraic functions of cluster variables. In this talk we suggest an algorithm for computing these symbol alphabets from plabic graphs by solving matrix equations of the form C.Z = 0 to associate functions on Gr(m,n) to parameterizations of certain cells of Gr_+ (k,n) indexed by plabic graphs. For m=4 and n=8 we show that this association precisely reproduces the 18 algebraic symbol letters of the two-loop NMHV eight-point amplitude from four plabic graphs. We further show that it is possible to obtain all rational symbol letters (in fact all cluster variables) by solving C.Z = 0 if one allows C to be an arbitrary cluster parameterization of the top cell of Gr_+ (n-4,n).

There is a connection between certain smooth representations of a reductive p-adic group and the representations of the Iwahori-Hecke algebra of this p-adic group. This Iwahori-Hecke algebra is a specialisation of a more general affine Hecke algebra. In this talk, we will discuss affine Hecke algebras and graded Hecke algebras. We will state a result from Lusztig (1989) that relates the representation theory of an affine Hecke algebra and a particular graded Hecke algebra and we will present a simple example of this relation.

Nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations This talk is about joint work with Viorel Barbu. We consider a class of nonlinear Fokker-Planck (- Kolmogorov) equations of type \begin{equation*} \frac{\partial}{\partial t} u(t,x) - \Delta_x\beta(u(t,x)) + \mathrm{div} \big(D(x)b(u(t,x))u(t,x)\big) = 0,\quad u(0,\cdot)=\mu, \end{equation*} where $(t,x) \in [0,\infty) \times \mathbb{R}^d$, $d \geq 3$ and $\mu$ is a signed Borel measure on $\mathbb{R}^d$ of bounded variation. In the first part of the talk we shall explain how to construct a solution to the above PDE based on classical nonlinear operator semigroup theory on $L^1(\mathbb{R}^d)$ and new results on $L^1- L^\infty$ regularization of the solution semigroups in our case. In the second part of the talk we shall present a general result about the correspondence of nonlinear Fokker-Planck equations (FPEs) and McKean-Vlasov type SDEs. In particular, it is shown that if one can solve the nonlinear FPE, then one can always construct a weak solution to the corresponding McKean-Vlasov SDE. We would like to emphasize that this, in particular, applies to the singular case, where the coefficients depend "Nemytski-type" on the time-marginal law of the solution process, hence the coefficients are not continuous in the measure-variable with respect to the weak topology on probability measures. This is in contrast to the literature in which the latter is standardly assumed. Hence we can cover nonlinear FPEs as the ones above, which are PDEs for the marginal law densities, realizing an old vision of McKean.

References V. Barbu, M. Röckner: From nonlinear Fokker-Planck equations to solutions of distribution dependent SDE, Ann. Prob. 48 (2020), no. 4, 1902-1920. V. Barbu, M. Röckner: Solutions for nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations, J. Funct. Anal. 280 (2021), no. 7, 108926.