The BBM equation is a nonlinear dispersive scalar PDE related to the KdV equation. However, it has a non-convex dispersion relation that introduces a variety of novel wave structures. These waves are highlighted by considering numerical solutions of Riemann problems, in which a smoothed step function initial condition u(x,0) exhibits long-time behavior that is a…

In this talk we review some classical algorithms for solving structured convex optimization problems, passing from gradient descent to proximal iterations and going further to modern proximal primal-dual splitting algorithms in the case of more complicated objective functions. We put special attention to constrained convex optimization, in which we accelerate the performance of the algorithms…

Fluid-structure interaction (FSI) problems describe the dynamics of multi-physics systems that involve fluid and solid components. These are everyday phenomena in nature, and arise in various applications ranging from biomedicine to engineering. Mathematically, FSI problems are typically non-linear systems of partial differential equations (PDEs) of mixed hyperbolic-parabolic type, defined on time-changing domains. In this lecture…

I will discuss some aspects of two recent results on singularity of Bernoulli matrices. I will emphasize the use of new Littlewood-Offord-type inequalities in the proofs of the results. Partially based on a joint work with A.Litvak.   Zoom meeting Link

We consider a non-local optimization problem, which is motivated by a simple model for swarming and other self-assembly/aggregation models, and prove the existence of different phases. In particular, we show that in the large mass regime the ground state density profile is the characteristic function of a round ball. An essential ingredient in our proof…

The emergence of nonlocal theories as promising models in different areas of science (continuum mechanics, biology, image processing) has led the mathematical community to conduct varied investigations of systems of integro-differential equations. In this talk I will present some recent results on systems of integral equations with weakly singular kernels, flux-type boundary conditions, as well…

We consider the mixed ray transform of tensor fields on a three-dimensional compact simple Riemannian manifold with boundary. We prove the injectivity of the transform, up to natural obstructions, and establish stability estimates for the normal operator on generic three dimensional simple manifold in the case of 1+1 and 2+2 tensors fields. We show how the…

The study of evolution equations in spaces of measures has seen tremendous growth in the last decades, of which resulted in general metric space theories for analyzing variational evolutions—evolutions driven by one or more energies/entropies. On the other hand, physics and large-deviation theory suggest the study of generalized gradient flows—gradient flows with non-homogeneous dissipation potentials—which…

Mean Field Games (MFGs) have been introduced independently by Lasry-Lions and Huang, Malhamé and Caines in 2006. The main purpose of this theory is to simplify the analysis of stochastic differential games with a large number of small and indistinguishable players. Applications of MFGs include models in Economics, Mathematical Finance, Social Sciences and Engineering. In…

Archetypal analysis is an unsupervised learning method that uses a convex polytope to summarize multivariate data. For fixed k, the method finds a convex polytope with k vertices, called archetype points, such that the polytope is contained in the convex hull of the data and the mean squared distance between the data and the polytope…

Quasiconvex functions, a major generalization of convex functions, naturally arise in calculus of variations, optimal control and differential games in L-infinity. This connection with HJ equations, representation formulas, obstacle problems, and reach-avoid problems will be discussed. Zoom meeting ID: 802 764 2791

In this talk, we will discuss a recent global existence result on the Poiseuille flow of nematic liquid crystals via full Ericksen-Leslie model. The existing results on the Ericksen-Leslie model for the liquid crystals mainly focused on the parabolic and elliptic type models by omitting the kinetic energy term. In this recent progress, we established…

A nonlinear analogue of the Rademacher type of a Banach space was introduced in classical work of Enflo. The key feature of Enflo type is that its definition uses only the metric structure of the Banach space, while the definition of Rademacher type relies on its linear structure. I will speak about the joint work with…

This work explores the recovery stochastic volatility models (SVMs) from market models for the VIX futures term structure. Market models have more flexibility for fitting of curves than do SVMs, and therefore they are better-suited for pricing VIX futures and derivatives. But the VIX itself is a derivative of the S&P500 (SPX) and it is…

In my talk, I will explain the approach of finding solutions to nonlinear PDEs via tug-of-war games. I will focus on the context of p-Laplacian and the non-local geometric p-Laplacian. Zoom meeting: Link

In a Hilbert space, for convex optimization, we report on recent advances regarding the acceleration of first-order algorithms. We rely on inertial dynamics with damping driven by the Hessian, and the link between continuous dynamic systems and algorithms obtained by temporal discretization. We first review the classical results, from Polyak's heavy ball with friction method…

We survey the known results concerning the minimal Riesz energy on sufficiently smooth sets, and present some new results on fractal sets, which are the key examples of non-rectifiable sets. In particular, we will talk about the connection of these problems to ergodic theory and probability theory: the key step in our proofs is applying…

A popular method to approximate a fixed point of a non-expansive map is C is the Krasnoselskii-Mann iteration. This covers a wide range of iterative methods in convex minimization, equilibria, and beyond. In the Euclidean setting, a flexible method to obtain convergence rates for this iteration is the PEP methodology introduced by Drori and Teboulle…

Mesh-free methods for boundary value problems (BVPs) can be convenient on manifolds where generating a mesh may be difficult or when the manifold is not known explicitly but is determined by data.  Moreover, BVPs are important in machine learning since they provide a rigorous method of regularization for many regression problems.  In this talk we…

Mean field game PDE systems were introduced by J-M. Lasry and P.-L. Lions to describe Nash equilibria in multi-agents dynamic optimization. In the simplest model, those are forward-backward systems coupling Hamilton-Jacobi with Fokker-Planck equations. In this talk I will discuss the long time behavior of second order systems in the periodic case under suitable stability…