We study expectations of powers and correlations for characteristic polynomials of N x N non-Hermitian random matrices. This problem is related to the analysis of planar models (log-gases) where a Gaussian (or other) background measure is perturbed by a finite number of point charges in the plane. I will discuss the critical asymptotics, for example when a point charge collides with the boundary of the support, or when two point charges collide with each other (coalesce) in the bulk. In many of these situations, we are able to express the results in terms of Painlevé transcendents. The application to certain d-fold rotationally invariant models will be discussed. This is joint work with Alfredo Deaño (University of Kent).

Fyodorov-Hiary-Keating established a series of conjectures concerning the large values of the Riemann zeta function in a random short interval. After reviewing the origins of these predictions through the random matrix analogy, I will explain recent work with Louis-Pierre Arguin and Maksym Radziwill, which proves a strong form of the upper bound for the maximum.

I introduce a recently developed variational approach for hyperbolic PDE's. The method allows to show the existence of weak solutions to fluid-structure interactions where a visco-elastic bulk solid is interacting with an incompressible fluid governed by the unsteady Navier Stokes equations. This is a joint work with M. Kampschulte and B. Benesova.

In the talk, we study the Navier–Stokes-like problems for the flows of homogeneous incompressible fluids. We introduce a new type of boundary condition for the shear stress tensor, which includes an auxiliary stress function and the time derivative of the velocity. The auxiliary stress function serves to relate the normal stress to the slip velocity via rather general maximal monotone graph. In such way, we are able to capture the dynamic response of the fluid on the boundary. Also, the constitutive relation inside the domain is formulated implicitly. The main result is the existence analysis for these problems.

Given a curve , what is the surface  that has smallest area among all surfaces spanning ? This classical problem and its generalizations are called Plateau's problem. In this talk we consider area minimizers among the class of integral currents, or roughly speaking, orientable manifolds. Since the 1960s a lot of work has been done by De Giorgi, Almgren, et al to study the interior regularity of these minimizers. Much less is known about the boundary regularity, in the case of codimension greater than 1. I will speak about some recent progress in this direction.

Much progress in the study of 3-manifolds has been made by considering the geometric structures they admit. This is nowhere more true than for 3-manifolds which admit a hyperbolic structure. However, in the land of algorithms a more combinatorial approach is necessary, replacing our charts and isometries with finite simplicial complexes that are defined by a finite amount of data. 

In this talk we'll have a look at how in fact one can combine the two approaches, using the geometry of hyperbolic 3-manifolds to assist in this more combinatorial approach. To do so we'll combine tools from Hyperbolic Geometry, Triangulations, and perhaps suprisingly Polynomial Algebra to find explicit bounds on the runtime of an algorithm for comparing Hyperbolic manifolds.

We study a continuous time equilibrium model of limit order book (LOB) in which the liquidity dynamics follows a non-local, reflected mean-field stochastic differential equation (SDE) with evolving intensity. We will see that the frontier of the LOB (e.g., the best ask price) is the value function of a mean-field stochastic control problem, as the limiting version of a Bertrand-type competition among the liquidity providers.
With a detailed analysis on the N-seller static Bertrand game, we formulate a continuous time limiting mean-field control problem of the representative seller.
We then validate the dynamic programming principle (DPP) and show that the value function is a viscosity solution of the corresponding Hamilton-Jacobi-Bellman (HJB) equation.
We argue that the value function can be used to obtain the equilibrium density function of the LOB. (Joint work with Jin Ma)

We consider sensitivity of a generic stochastic optimization problem to model uncertainty. We take a non-parametric approach and capture model uncertainty using Wasserstein balls around the postulated model. We provide explicit formulae for the first order correction to both the value function and the optimizer and further extend our results to optimization under linear constraints.  We present applications to statistics, machine learning, mathematical finance and uncertainty quantification. In particular, we prove that LASSO leads to parameter shrinkage, propose measures to quantify robustness of neural networks to adversarial examples and compute sensitivities of optimised certainty equivalents in finance. We also propose extensions of this framework to a multiperiod setting. This talk is based on joint work with Daniel Bartl, Samuel Drapeau and Jan Obloj.

Modern cellular radio systems such as 4G and 5G use antennas with multiple elements, a technique known as MIMO, and the intention is to increase the capacity of the radio channel.  5G allows even more possibilities, such as massive MIMO, where there can be hundreds of elements in the transmit antenna, and beam-forming (or beam-steering), where the phase of the signals fed to the antenna elements is adjusted to focus the signal energy in the direction of the receivers.  However, this technology poses some difficult optimization problems, and here mathematicians can contribute.   In this talk I will explain the background, and then look at questions such as: what is an appropriate objective function?; what constraints are there?; are any problems of this type convex (or quasi-convex, or difference-of-convex)?; and, can big problems of this type be solved in real time?