Spring 2019 Seminars

I will present certain developments related to the macroscopic limit of the Cucker-Smale flocking model. The main focus will be on the multi-D euclidean space results from the recent joint work with Raphael Danchin, Piotr Mucha and Bartosz Wróblewski.

Several important integral operators in harmonic analysis will be presented in an organized way. The fundamental question about these operators is the boundedness on Lebesgue spaces. We will show that the difficulty to establish the boundedness will change dramatically once we move from the classic type of operators to their curved and discrete analogs. It is very interesting to see that harmonic analysis can interact with many other fields of mathematics such as PDE, ergodic theory, number theory, combinatorics, and even algebraic geometry.

Self-interacting systems of particles/agents arise in many areas of science, such as particle systems in physics, flocking and swarming models in biology, and opinion dynamics in social science. A natural question is to learn the laws of interaction between the particles/agents from data consisting of trajectories. In the case of distance-based interaction laws, we present efficient regression algorithms to estimate the interaction kernels, and we develop a nonparametric statistical learning theory addressing identifiability, consistency and optimal rate of convergence of the estimators. Especially, we show that despite the high-dimensionality of the systems, optimal learning rates can still be achieved. (Joint work with Mauro Maggioni, Sui Tang and Ming Zhong).

Visual objects are often known up to some ambiguity, depending on the methods used to acquire them. The first-order approximation to any transformation is, by definition, affine, and the affine approximation to changes between images has been used often in computer vision. Thus it is beneficial to deal with objects known only up to an affine transformation. For example, feature points on a planar transform projectively between different views, and the projective transformation can in many cases be approximated by an affine transformation. More generally, given two visual objects in a containing Euclidean space R^k, one may study vision group actions between these two objects often with an underlying signature which are equivalent under some symmetry or minimal distortion action with respect to a suitable metric inherited by this action. For example, Euclidean groups, similarity, Equi-Affine, projections, camera rotations and video groups. The study of the space of ordered configurations of n distinct points in R^k up to similarity transformations was pioneered by Kendall who coined the name shape space. For different groups of transformations (rigid, similarity, linear, affine, projective for example) one obtains different shape spaces. Moreover, while these formulations allow often global optimal optimization, e.g. using convex objectives , many of the problems above require efficient approximation methods which work locally. This framework has applications to biological structural molecule reconstruction problems, to recognition tasks and to matching features across images with minimal distortion” This talk will discuss various work with collaborators around this circle of ideas.

A PDE/ODE system can evolve in 3 time scales when the fast scales are associated with 2 small parameters that tend to zero at different rates. We investigate the limiting dynamics when the fast dynamics is generated by 2 skew-seft-adjoint operators and the initial time derivative is uniformly bounded regardless of the small parameters. To find the subspace that the limiting dynamics resides, we rely on matrix perturbation theory.

Yves Couder and coworkers in Paris discovered that droplets walking on a vibrating fluid bath exhibit several features previously thought to be exclusive to the microscopic, quantum realm. These walking droplets propel themselves by virtue of a resonant interaction with their own wavefield, and so represent the first macroscopic realization of a pilot-wave system of the form proposed for microscopic quantum dynamics by Louis de Broglie in the 1920s. New experimental and theoretical results allow us to rationalize the emergence of quantum-like behavior in this hydrodynamic pilot-wave system in a number of settings, and explore its potential and limitations as a quantum analog.

A key question in population biology is understanding the conditions under which the species of an ecosystem persist or go extinct. Theoretical and empirical studies have shown that persistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of n interacting species that live in a stochastic environment. Our models are described by n dimensional piecewise deterministic Markov processes. These are processes (X(t), r(t)) where the vector X denotes the density of the n species and r(t) is a finite state space process which keeps track of the environment. In any fixed environment the process follows the flow given by a system of ordinary differential equations. The randomness comes from the changes or switches in the environment, which happen at random times. We give sharp conditions under which the populations persist as well as conditions under which some populations go extinct exponentially fast. As an example we look at the competitive exclusion principle from ecology and show how the random switching can `rescue' species from extinction. The talk is based on joint work with Dang H. Nguyen (University of Alabama).

Crumples in a sheet of paper, wrinkles on curtains, cracks in metallic alloys, and defects in superconductors are examples of patterns in materials. A thorough understanding of the underlying phenomenon behind the pattern formation provides a different prospective on the properties of the existing materials and contributes to the development of new ones. In my talk I will address the issue of modelling pattern formation via nonconvex energy minimization problems, regularized by higher order terms. Two particular examples of such models will be described in greater details: formation of vortices in Ginzburg-Landau model of superconductors, as well as emergence of patterns in phase transitions in shape-memory alloys. I will discuss the issue of well-posedness of such modelling, which reduces to the question of the existence of minimizers in certain functional classes. I will also provide some examples qualitative properties of minimizers via sharp energy bounds.

In a recent paper, Bedrossian, Masmoudi and Mouhot proved the stability of equilibria satisfying the Penrose condition for the Vlasov-Poisson equation (with screened potential) on the whole space. We shall discuss a joint work with Nguyen and Rousset where we propose a new proof of this result, based on a lagrangian approach.

Initial boundary value problems (and also transmission problems) in dimension d= 1 for 2X2 hyperbolic systems are well understood. However, for many applications, and especially for the description of surface water waves, dispersive perturbations of hyperbolic systems must be considered. We shall describe here a transmission problem for the Boussinesq dispersive approximation arising in the description of wave-structure interactions. We shall insist on the differences and similarities with respect to the standard hyperbolic case, and focus our attention on a new phenomenon, namely, the apparition of a dispersive boundary layer. This is a joint work with D. Lannes and G. Métivier.

Computational modeling can be used to reveal insights into the mechanisms and potential side effects of a new drug. Here we will focus on two major diseases: diabetes, which affects 1 in 10 people in North America, and hypertension, which affects 1 in 3 adults. For diabetes, we are interested in a class of relatively novel drug treatment, the SGLT2 inhibitors (sodium-glucose co-transporter 2 inhibitors). E.g., Dapagliflozin, Canagliflozin, and Empagliflozin. We conduct simulations to better understand any side effect these drugs may have on our kidneys (which are the targets of SGLT2 inhibitors). Interestingly, these drugs may have both positive and negative side effects. For hypertension, we want to better understand the sex differences in the efficacy of some of the drug treatments. Women generally respond better to ARBs (angiotensin receptor blockers) than ACE inhibitors (angiontensin converting enzyme inhibitors), whereas the opposite is true for men. We have developed the first sex-specific computational model of blood pressure regulation, and applied that model to assess whether the "one-size-fits-all" approach to blood pressure control is appropriate with regards to sex.

Random matrix statistics emerge in a broad class of strongly correlated systems, with evidence suggesting they can play a universal role comparable to the one Gaussian and Poisson distributions do classically. Indeed, observational studies have identified these statistics among heavy nucleii, Riemann zeta zeros, pedestrians, land divisions, parked cars, perched birds, and other forms of traffic. Noticing that these latter real-world systems all operate in a decentralized manner, we investigate the simplest possible games that admit Coulomb gas dynamics as a Nash equilibrium and investigate their basic features, many of which are atypical or even new for the literature on many player games. For example, there is a nonlocal-to-local transition in the population argument of the N-Nash system of PDEs and, perhaps most significantly, there is a sensitivity of local limit behavior to the chosen model of player information. If there is time, this talk will also discuss some future research directions based on the many questions these results raise.

In this talk, I will discuss a nonlocal aggregation equation with degenerate diffusion, which describes the mean-field limit of interacting particles driven by nonlocal interactions and localized repulsion. When the interaction potential is attractive, it is previously known that all stationary solutions must be radially decreasing up to a translation, but uniqueness (for a given mass) within this class was open, except for some special interaction potentials. For general attractive potentials, we show that the uniqueness/non-uniqueness criteria are determined by the power of the degenerate diffusion, with the critical power being $m=2$. Namely, for $m\geq2$, we show the stationary solution for any given mass is unique for any attractive potential, by tracking the associated energy functional along a novel interpolation curve. And for $1< m< 2$, we construct examples of smooth attractive potentials, such that there are infinitely many radially decreasing stationary solutions of the same mass. This is a joint work with Matias Delgadino and Xukai Yan.