# Abstracts

Dieter Armbruster, Arizona State University

*Swarms in bounded domains*

We numerically study the Vicsek swarm model in a circular, a rectangular and a channel domain as a function of the parameters governing the individual particles and their interaction and the geometry of the domain. Scale free parameters characterizing the observed swarm behavior are presented. We find different regimes and characterize them through experimental observables like the trajectory of the center of mass, the mean density profile of a swarming motion and the connectivity to the swarm. While individual particles are reflected at a boundary the swarm as a whole interacts in much more complicated ways with the boundary. We discuss how the interaction of the particles with the boundary creates friction and possible boundary layers for the swarm flows.

Henri Berestycki, EHESS (Ecole des hautes études en sciences sociales) – Paris

*Propagation and blocking for reaction-diffusion equations in non homogeneous media*

I will discuss bi-stable reaction-diffusion equations in cylinders with varying cross-sections motivated by biology and medicine. The aim is to understand the effect of the non-homogeneous medium on propagation or blocking of advancing waves. The role played by the geometry of the domain of propagation is of particular interest for these models. I will report on joint work with Juliette Bouhours and Guillemette Chapuisat.

Filippo Cagnetti, University of Sussex

*The rigidity problem for symmetrization inequalities*

Steiner symmetrization is a very useful tool in the study of isoperimetric inequality. This is also due to the fact that the perimeter of a set is less or equal than the perimeter of its Steiner symmetral. In the same way, in the Gaussian setting, it is well known that Ehrhard symmetrization does not increase the Gaussian perimeter. We will show characterization results for equality cases in both Steiner and Ehrhard perimeter inequalities. We will also characterize rigidity of equality cases. By rigidity, we mean the situation when all equality cases are trivially obtained by a translation of the Steiner symmetral (or, in the Gaussian setting, by a reflection of the Ehrhard symmetral). We will achieve this through the introduction of a suitable measure-theoretic notion of connectedness, and through a fine analysis of the barycenter function for a special class of sets. These results are obtained in collaboration with Maria Colombo, Guido De Philippis, and Francesco Maggi.

Fabio Chalub, U. Nova de Lisboa, Portugal

*The Forward Generalized Kimura Equation*

We consider the Forward Generalized Kimura Equation (FGKE):

∂tp = κ∂x2 (x(1 − x)p) − ∂x (x(1 − x)ψ(x)p) , 2

where p(x, t) is the probability to find x mutants at time t in a population consisting of two different genotypes: the wild-type and the mutant. The fitness difference between the mutant and the wild type is given by ψ : [0, 1] → R and κ is the “intensity of selection”.

We will show that this equation, when supplemented by two appropri- ate conservation laws, approximates the evolution given by certain Markov processes (e.g, the Moran process or the Wright-Fisher process). These conservation laws can be obtained from the discrete processes. We prove existence and uniqueness of solutions in the sense of measures. Furthermore, we prove that the first time scale of the FGKE is compatible with the solutions of the celebrated Replicator Equation (RE) in evolutionary dynamics, proving that the initial behavior of the Moran of the Wright Fisher can be modeled by the RE. We will obtain expressions for the fixation probability of the mutant and also for the expected time for fixation of any type. These equations are of no practical use and will be simplified using asymptotic expansions, given different expressions depending on the sign of the function ψ on the interval [0, 1]. We also show existence and uniqueness of solution in measure sense of the Forward Generalized Kimura equation. Finally, we will study the same equation for time dependent fitness.

This is a joint work with Max Souza (Brazil) and Olga Danilkina (Russia).

Fernando Charro, The University of Texas at Austin

*On a Fractional Monge–Ampère Operator*

In this talk we consider a fractional analogue of the Monge- Ampère operator. Our operator is a concave envelope of fractional linear operators of the form infA∈A LAu, where the set of operators corresponds to all affine transformations of determinant one of a given multiple of the fractional Laplacian.

We set up a relatively simple framework of global solutions prescribing data at infinity and global barriers. In our key estimate, we show that the operator that realizes the infimum remains strictly elliptic, which allows to deduce an Evans-Krylov regularity result and therefore that solutions are classical.

Maria Colombo, Scuola Normale Superior, Pisa, Italy

*Regularity results for a very degenerate elliptic equation with applications to traffic dynamic*

We consider a family of equations which naturally arises in the context of traffic congestion. They are very degenerate elliptic equations, where the coefficients are identically zero in the region where the gradient of the solution is smaller than a certain constant. We discuss the regularity properties of solutions, extending a previous result by Santambrogio and Vespri valid only in dimension 2. (Joint work with Alessio Figalli)

Guido De Philippis, Universität Zürich

*Faber-Krahn Inequalities in Sharp Quantitative Form*

In this talk we present a sharp quantitative improvement of the celebrated Faber- Krahn inequality. The latter asserts that balls uniquely minimize the first eigenvalue of the Dirichlet-Laplacian, among sets with given volume. We prove that indeed more can be said: the difference between the first eigenvalue λ(Ω) of a set Ω and that of a ball of the same volume controls the deviation from spherical symmetry of Ω. Moreover, such a control is the sharpest possible. This settles a conjecture by Bhattacharya, Nadirashvili and Weitsman. The result is valid for more general geometric quantities, like

λ2,q(Ω) = min |∇u|2 : ∥u∥Lq(Ω) = 1 . u∈W 1,2 (Ω) Ω

The proof is based on various reduction steps: among these, a central role is played by a Selection Principle for the torsional rigidity functional, which essentially permits to reduce the task to prove the desired result for small smooth deformations of a ball.

The result here presented is contained in a recent joint paper with Lorenzo Brasco (Marseille) and Bozhidar Velichkov (Pisa).

Alessio Figalli, The University of Texas at Austin

*Mini-course: Regularity for the Monge-Ampère equation, with applications to the semigeostrophic equations*

The Monge-Ampère equation arises in connections with several problems from geometry and analysis (regularity for optimal transport maps, the Minkowski problem, the affine sphere problem, etc.), and its regularity theory has been widely studied. However, a natural question that remained open for long in the theory was the Sobolev regularity of solutions when the right hand sided is merely bounded away from zero and infinity. Apart from its own interest, this question naturally arise in the theory of existence of global solutions to the semigeostrophic equations. The latter are a simple model used in meteorology to describe large scale atmospheric flows, and they can be derived from the 3-d incompressible Euler equations, with Boussinesq and hydrostatic approximations, subject to a strong Coriolis force. All previous known results on the semigeostrophic equations where of two types: either they were proving existence of Lagrangian solutions, or they were showing existence of solutions in some “dual variables”. However, mainly because of lack of regularity results for the Monge-Ampère equation, no global existence results for the “real” semigeostrophic system was known for long time. The aim of this course is first to introduce the semigeostrophic equations and discuss their relation with Monge-Ampere, then we will show how to obtain Sobolev regularity estimates for Monge-Ampere, and finally we will apply them to prove global existence of solutions to the semigeostrophic equations.

Nestor Guillen, UCLA

*Homogenization of free boundary problems in randomly perforated media*

Consider the evolution of a free boundary (e.g. a Hele-Shaw cell) within a randomly perforated domain, obtained from $\mathbb{R}^d$ by making random perforations of size $\epsilon$ which are $\epsilon$ apart from each other. Our main result is that with probability one the free boundaries converge uniformly to the free boundary of a deterministic, homogeneous problem set in an unperforated domain. A by product of the proof are a Harnack inequality for (divergence) linear elliptic equations and the homogenization of the obstacle problem in perforated domains. Joint work with Inwon Kim.

Jeff Haack, Mathematics & ICES, UT Austin

*Deterministic spectral method for the Boltzmann collision operator with angularly dependent cross section, with examples in Coulomb scattering*

In this talk, I will present recent work on deterministic computation of the Boltzmann collision operator with angularly dependent cross section. Previous deterministic methods for solving the Boltzmann operator typically use short range ‘billiard ball’ like collisions, which greatly simplify the Jeff Haack (continued)

scattering cross section, but a more physical model is to assume that particles interact via two-body potentials. Under this assumption the collision operator can be formulated in a very similar manner, however the scattering cross section is highly anisotropic and possibly singular in the angular variable. The most classic example of this is the case of Coulombic interactions which was first studied by Landau, who derived the eponymous Landau collision operator for this case, where ‘grazing’ interactions dominate. Building on the work of Gamba and Tharkabhushanam (2009), we derive a numerical method of the Boltzmann operator with angularly dependent cross section in which the cumbersome cross section integrals are precomputed in parallel, resulting in a weighted convolution of the distribution function when evaluating the collision operator. We numerically test this method by looking at screened Coulomb interactions and other related cross sections and compare the results to the limiting Landau operator. I will also relate these results to some subtleties of the models in relation to the BBGKY hierarchy.

Shi Jin, University of Wisconsin-Madison

*Uncertainty quantification for multiscale hyperbolic and kinetic equations with random coefficients and diffusion limits*

In this talk we will study generalized polynomial chaos (gPC) approach to transport equation with uncertain cross-sections and show that they can be made asymptotic-preserving, in the sense that in the diffusion limit the gPC scheme for the transport equation approaches to the gPC scheme for the diffusion equation with random diffusion coefficient. This allows the implementation of the gPC method without numerically resolving (by space, time, and gPC modes) the small mean free path for transport equation in the diffusive regime.

Moon-Jin Kang, University of Texas at Austin

*Dynamics of the hydrodynamic Cucker-Smale model and the system coupled with incompressible fluid*

We first introduce a hydrodynamic Cucker-Smale (in short, HCS) model, which is of the pressureless Euler system with non-local flocking dissipation, and formally derived from kinetic Cucker-Smale model via a mono-kinetic ansatz. We mainly present the existence and large time behavior of classical solution to the HCS model with initial compact support. We also study the collective behavior of two phase flows that is described as the coupled system of the HCS and the incompressible Navier-Stokes equations.

Li Li, Rutgers University

*Trend to equilibrium of the Vlasov-Poisson-Boltzmann system*

Li Li (continued) The motion of dilute charged particles can be modeled by Vlasov-Poisson-Boltzmann system. We study the large time stability of the VPB system. To be precise, we prove that when time goes to infinity, the solution of VPB system tends to the global Maxwellian state in a rate faster than power functions, by using a method developed for Boltzmann equation without force in the work of Desvillettes and Villani (2005).

Yanyan Li, Rutgers University

*On a fractional Nirenberg problem*

We present some results on the existence and compactness of solutions of a fractional Nirenberg problem. This is an integral equation of critical exponent. We will discuss various analytic ingredients and outline the proof. This is a joint work with Tianling Jin and Jingang Xiong.

Fanghua Lin, Courant Institute of Mathematical Sciences, NYU

*Mini-course**: Optimal partitions of Dirichelet eigenvalues*

In these two lectures, we will discuss the following problem: Given a bounded domain $\Omega$ in R^n, and a positive energy N, one divides $\Omega$ into N subdomains, $\Omega_j, j= 1, 2,…, N$. We consider the so-called optimal partitions that give the least possible value for the sum of the first Dirichelet eigenvalues on these sumdomains among all admissible partitions of $\Omega$. After a brief survey on what had been done over last 20+ years on this and some related problems, in particular, my joint work with Luis Caffarelli, I shall discuss some recent progress and conjectures on the analysis on asymptotic behavior these optimal partitions as N tends to infinite.

Jian-Guo Liu, Duke University

*An analysis of merging-splitting group dynamics by Bernstein function theory*

We study coagulation-fragmentation equations inspired by a simple model derived in fisheries science to explain data on the size distribution of schools of pelagic fish. Although the equations lack detailed balance and admit no H-theorem, we are able to develop a rather complete description of equilibrium profiles and large-time behavior, based on complex function theory for Bernstein and Pick functions. The generating function for discrete equilibrium profiles also generates the Fuss-Catalan numbers (derived by Lambert in 1758) that count all ternary trees with $n$ nodes. The structure of equilibrium profiles and other related sequences is explained through a new and elegant characterization of the generating functions of completely monotone sequences as those Pick functions analytic and nonnegative on (-infty,1). This is joint work with Bob Pego and Pierre Degond.

Francesco Maggi, The University of Texas at Austin

*Mini-course**: Perimeter minimizing bubble clusters*

We review the state of the art concerning the theory of partitioning problems initiated by Almgren in 1976, addressing issues like existence, regularity, characterization results and qualitative properties for perimeter minimizing bubble clusters. In particular, we shall describe some recent results we have obtained in collaboration with Marco Cicalese (TU Munich) and Gian Paolo Leonardi (U. Modena-Reggio Emilia) concerning global and local stability properties of bubble clusters.

Antoine Mellet, University of Maryland

*Transport equations and anomalous diffusion*

We will review old and new results concerning the derivation of fractional diffusion equations from kinetic equations. One of the focus will be on the role of boundary conditions.

Sebastien Motsch, Arizona State University

*Emergence of macroscopic behavior in complex systems*

In a human crowd or in a shoal of fish, thousands of individuals interact and form large scale structures. Although the interaction among individuals might be simple, the resulting dynamics is quite complex. Modeling is an essential tool to understand such dynamics. For instance, agent-based models, also referred to as microscopic models, are widely developed to analyze various dynamics such as swarming and opinion formations. In this talk, we investigate the emergence of macroscopic behavior for such models. The challenge is to link “microscopic models” describing each agent with “macroscopic models” describing the evolution of a fluid. To achieve this transition, we present a novel approach based on kinetic theory and asymptotic analysis. Numerical simulations are also presented to illustrate the results.

Xavier Ros-Oton, Universitat Politècnica de Catalunya, Spain

*Boundary regularity for elliptic integro-differential equations*

We study the boundary regularity of solutions to elliptic integro-differential equations. First we prove that, for the fractional Laplacian $(-\Delta)^s$ with $s\in (0,1)$, solutions $u$ satisfy that $u/d^s$ is H\”older continuous up to the boundary, where $d(x)$ is the distance to the boundary of the domain $\Omega$. We will show that, in this fractional context, the quantity $u/d^s|_{\partial\Omega}$ plays the role that the normal derivative plays in second order equations. Finally, we present new boundary regularity results for fully nonlinear integrodifferential equations.

Russell Schwab, Michigan State University

*Neumann Homogenization via Integro-Differential Operators*

We use a recent result about the representation of the Dirichlet-to-Neumann operator for fully nonlinear equations as an integro-differential operator on the boundary of the domain to guide the analysis of the homogenization problem with oscillatory Neumann data. This allows us to attack the homogenization problem as a nonlocal homogenization on the boundary, which is amenable to methods already established for integro-differential equations. We will present the case of a infinite strip domain with almost periodic Neumann data. The emphasis will be on the method of converting the Neumann analysis into an auxiliary nonlocal problem which lives only on the boundary. This is joint work with Nestor Guillen.

Luis Silvestre, University of Chicago

*Regularity estimates for non local equations and applications*

In this talk we will survey some of the regularity results available for elliptic and parabolic integro-differential equations and we will discuss some of their applications.

Dejan Slepcev, Carnegie Mellon University

*Continuum limit of total variation on point clouds*

I will discuss variational problems arising in machine learning and their consistency as the number of data points goes to infinity. It turns out that techniques of calculus of variation are particularly suitable to address these questions. Consider point clouds obtained as random samples of an underlying measure on a Euclidean domain. Graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points. I will discuss when is the graph cut, and more generally, total variation, on such graph a good approximation of the perimeter (total variation) in the continuum setting. The question will be considered in the setting of Gamma convergence. The Gamma limit, and associated compactness property, are considered with respect to a new topology which uses optimal transportation to suitably compare functions defined on different sets of points. Taking the Gamma limit is enabled by connecting the graph cuts with the nonlocal continuum perimeter. The talk is based on joint work with Nicolas Garcia Trillos.

Charles Smart, MIT

*Stochastic homogenization of nonlinear energy functionals*

I will discuss joint work with Scott Armstrong in which we prove quantitative stochastic homogenization results for random nonlinear energy functionals in the finite range of dependence regime. Our techniques are novel in that they are purely variational and apply to systems.

Pablo Stinga, The University of Texas at Austin

*Regularity for fractional nonlocal equations*

We prove Schauder estimates for solutions to fractional nonlocal equations. The operators we consider are fractional powers of divergence form elliptic operators in bounded domains under Dirichlet boundary condition. The basic example here is the fractional Dirichlet Laplacian, which is defined in a spectral way by using the Dirichlet eigenfunctions of the Laplacian. Our methods involve the semigroup language and the extension characterization. Joint work with Luis Caffarelli.

Lan Tang, Academia Sinica, Taipei, Taiwan

*Partial Regularity of Suitable Weak Solutions to Fractional Navier-Stokes Equations*

In this talk, we consider the fractional Navier-Stokes equations in $ R^3 _\times (0,\infty)$ with the power s of the negative Laplacian satisfying 3/4<s<1. We show that the suitable weak solution is regular away from a singular set whose (5-4s)-dimensional Hausdorff_ measure is zero. The result is a generalization of the classical result given by Caffarelli, Kohn and Nirenberg for Navier-Stokes system.

Enrico Valdinoci, Weierstrass Institute, Germany

*Concentrating solutions for a nonlocal Schroedinger equation*

We consider a nonlocal nonlinear Schroedinger equation in a bounded domain with zero Dirichlet datum. The Planck constant plays the role of a smooth parameter for a singularly perturbed fractional problem. We construct a family of solutions that concentrate at an interior point of the domain, by bifurcating from the ground state solution in the entire space. Unlike the classical case, the leading order of the associated reduced energy functional in a variational reduction procedure is of polynomial instead of exponential order on the distance from the boundary, due to the nonlocal effect.

The exponent of the energy asymptotics is n+4s, which is different from the decay of the fundamental solution of the fractional Laplacian (that is n-2s), from the one of the linearized operator (that is n+2s), and from the one of the ground state of the associated Schroedinger equation (that is again n+2s). A nondegeneracy argument and an appropriate Lyapunov-Schmidt reduction are also used.

This result was obtained in collaboration with J. Davila, M. del Pino and S. Dipierro.

Alexis Vasseur, The University of Texas at Austin

*De Giorgi regularity method applied to Hamilton Jacobi*

We provide a new proof of the Holder continuity of bounded solutions to Hamilton-Jacobi equations with rough coefficients, first showed by Cardaliaguet using probability methods, and by Cardaliaguet and Silvestre using explicit super and sub solutions. Our proof uses the De Giorgi method first applied to regularity for elliptic equations with rough coefficients. The method allows to obtain a sort of Harnack inequality. This is a joint work with Chi Hin Chan.

Bo Yang, Rutgers University

*On a rotationally symmetric Ricci flow from complex geometry*

I will talk about a rotationally symmetric Ricci flow from complex geometry. It is a degenerate parabolic equation. It might be interesting to connect some geometric and PDE aspects of this equation.