Hilbert's 19th problem asks if minimizers of "natural" variational integrals are smooth. For the past century, this problem inspired fundamental regularity results for elliptic and parabolic PDES. It also led to the construction of several beautiful counterexamples to regularity. The dichotomy of regularity vs. singularity is related to that of single PDE (the scalar case) vs. system of PDEs (the vectorial case), and low dimension vs. high dimension. I will discuss some interesting recent counterexamples to regularity in low-dimensional vectorial cases, and outstanding open problems. Parts of this are joint works with A. Figalli and O. Savin.

# Forthcoming Seminars

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

I will present a procedure for perturbatively constructing the field content of gravitational theories from a convolutive product of two Yang-Mills theories. A dictionary "gravity=YM * YM" is developed, reproducing the symmetries and dynamics of the gravity theory from those of the YM theories. I will explain the unexpected, yet crucial role played by the BRST ghosts of the YM system in the construction of gravitational fields. The dictionary is expected to develop into a solution-generating technique for gravity.

We prove that the dimension of every poset whose comparability graph has maximum degree $\Delta$ is at most $\Delta\log^{1+o(1)} \Delta$. This result improves on a 30-year old bound of F\"uredi and Kahn, and is within a $\log^{o(1)}\Delta$ factor of optimal. We prove this result via the notion of boxicity. The \emph{boxicity} of a graph $G$ is the minimum integer $d$ such that $G$ is the intersection graph of $d$-dimensional axis-aligned boxes. We prove that every graph with maximum degree $\Delta$ has boxicity at most $\Delta\log^{1+o(1)} \Delta$, which is also within a $\log^{o(1)}\Delta$ factor of optimal. We also show that the maximum boxicity of graphs with Euler genus $g$ is $\Theta(\sqrt{g \log g})$, which solves an open problem of Esperet and Joret and is tight up to a $O(1)$ factor. This is joint work with Alex Scott (arXiv:1804.03271).

I will present a variation of *positive model theory* which addresses the issues of approximations of conventional geometric structures by sequences of Zariski structures as well as approximation by sequences of finite structures. In particular I am interested in applications to quantum mechanics.

I will report on a progress in defining and calculating oscillating in- tegrals of importance in quantum physics. This is based on calculating Gauss sums of order higher or equal to 2 over rings **Z**/*m***Z **for very specific *m*.

Part of the series 'What do historians of mathematics do?'

In this talk, we will survey the movement of mathematical ideas in the 17th century. We will explore, in particular, the mathematical cultures of Paris, Amsterdam, Rome, Cape Town, Goa, Kyoto, Beijing, and London, as well as the journey of mathematical knowledge on a global scale. As it will be an ambitious task to complete a round-the-world history tour in an hour, the focus will be on East Asia. By employing the digital humanities technique, this presentation will use digital media to effectively show historical sources and help the audience imagine the world as a “round” entity when we discuss a global history of mathematics.

In this talk we will introduce and analyse a class of robust numerical methods for nonlocal possibly nonlinear diffusion and convection-diffusion equations. Diffusion and convection-diffusion models are popular in Physics, Chemistry, Engineering, and Economics, and in many models the diffusion is anomalous or nonlocal. This means that the underlying “particle" distributions are not Gaussian, but rather follow more general Levy distributions, distributions that need not have second moments and can satisfy (generalised) central limit theorems. We will focus on models with nonlinear possibly degenerate diffusions like fractional Porous Medium Equations, Fast Diffusion Equations, and Stefan (phase transition) Problems, with or without convection. The solutions of these problems can be very irregular and even possess shock discontinuities. The combination of nonlinear problems and irregular solutions makes these problems challenging to solve numerically.

The methods we will discuss are monotone finite difference quadrature methods that are robust in the sense that they “always” converge. By that we mean that under very weak assumptions, they converge to the correct generalised possibly discontinuous generalised solution. In some cases we can also obtain error estimates. The plan of the talk is: 1. to give a short introduction to the models, 2. explain the numerical methods, 3. give results and elements of the analysis for pure diffusion equations, and 4. give results and ideas of the analysis for convection-diffusion equations.

Title: Generalized McKean-Vlasov stochastic control problems.

Abstract: I will consider McKean-Vlasov stochastic control problems

where the cost functions and the state dynamics depend upon the joint

distribution of the controlled state and the control process. First, I

will provide a suitable version of the Pontryagin stochastic maximum

principle, showing that, in the present general framework, pointwise

minimization of the Hamiltonian with respect to the control is not a

necessary optimality condition. Then I will take a different

perspective, and present a variational approach to study a weak

formulation of such control problems, thereby establishing a new

connection between those and optimal transport problems on path space.

The talk is based on a joint project with J. Backhoff-Veraguas and R. Carmona.