# Rivière-Fabes: Past symposia

## Speakers, abstracts, and materials

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2021

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April 16-17 and 23-24, 2021

#### Pierre Germain

Courant Institute, NYU

##### Derivation of the kinetic wave equation

The kinetic wave equation (KWE) is expected to describe nonlinear dispersive equations in the regime known as weak turbulence: interactions are chaotic, and weakly nonlinear. Weak turbulence is important because of its wide range of applications (plasmas, atmosphere and ocean science, elasticity, nonlinear optics...), but also because it is possibly an entry point into the mysterious world of turbulence. During these lectures, I will present the KWE, its significance, and recent progress on its rigorous derivation.

#### Rowan Killip

University of California, Los Angeles

##### Well-posedness for integrable PDE

I will describe a suite of techniques developed recently to prove well-posedness of completely integrable Hamiltonian PDE. These have lead to sharp results for a number of well-know problems: KdV, mKdV, and cubic NLS in one dimension. This touches on joint work with Bjoern Bringmann, Benjamin Harrop-Griffiths, Monica Visan, and Xiaoyi Zhang.

#### Ovidiu Savin

Columbia University

##### Free boundary regularity for the N membranes problem

Abstract: For a positive integer N, the N-membranes problem describes the equilibrium position of N ordered elastic membranes subject to forcing and boundary conditions. If the heights of the membranes are described by real functions u_1, u_2,...,u_N, then the problem can be understood as a system of N-1 coupled obstacle problems with interacting free boundaries which can cross each other. When N=2 there is only one free boundary and the problem is equivalent to the classical obstacle problem. In my first lecture I will review some of the regularity theory for the standard obstacle problem, and in my second lecture I will discuss some recent work in collaboration with Hui Yu about the case when there are two or more interacting free boundaries.

#### Lu Wang

California Institute of Technology

##### Entropy in Mean Curvature Flow

Mean curvature flow is the gradient flow for the area functional, and it is a basic example of extrinsic curvature flows. In their study of singularity analysis of mean curvature flow, Colding and Minicozzi introduced an important notion of entropy for hypersurfaces, which is given by the supremum over all Gaussian integrals with varying centers and scales. And the entropy is a geometric invariant that measures geometric complexity. In the two talks, I will survey some recent results about some sharp lower bounds on entropy, and geometric and analytic properties of mean curvature flow of low entropy.

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2020

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April 24-26, 2020

The symposium did not meet in 2020; it is the only time the symposium was canceled.

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2019

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April 12-14, 2019

#### Guy David

University Paris XI

**Elliptic measure with a lower dimensional boundary.**

We'll try to discuss work in progress with Max Engelstein, Joseph Feneuil, and Svitlana Mayboroda. We consider a domain in $R^n$ bounded by an Ahlfors regular set $E$, typically of dimension smaller than $n$, and want to study the elliptic measure associated to some degenerate elliptic operators $L$. Under rather weak size conditions on the coefficients of $L$ (how fast do they tend to infinity near $E$), there is a reasonable elliptic measure. Then we study the regularity of this measure (typically, its absolute continuity with respect to Hausdorff measure), in terms of the geometry of $E$ and the regularity of the coefficients. So far we have some extensions of the celebrated Dahlberg theorem, and hope to study the converse.

#### Jonathan Luk

Stanford University

##### The strong cosmic censorship conjecture in general relativity

The strong cosmic censorship conjecture is a fundamental conjecture in general relativity. It posits global uniqueness for solutions to the Einstein equations. In particular, certain failure of determinism exhibited in some explicit black hole solutions are expected to be non-generic. In the first lecture, I will explain the motivation and the formulation of the conjecture, as well as some recent progress. In the second lecture, I will discuss some techniques to understand singular solutions to the Einstein equations, and explain their relevance to the strong cosmic censorship conjecture.

#### Eugenia Malinnikova

Norwegian University of Science and Technology

##### Two questions of Landis and their applications

We discuss two old questions of Landis concerning behavior of solutions of second order elliptic equations. The first one is on propagation of smallness from sets of positive measure, we answer this question and as a corollary prove an estimate for eigenfunctions of Laplace operator conjectured by Donnelly and Fefferman. The second question is on the decay rate of solutions of the Schrödinger equation, we give the answer in dimension two. The talk is based on joint works with A. Logunov, N. Nadirashvili, and F. Nazarov. --

#### Juncheng Wei

University of British Columbia, Vancouver

##### Talk I: On Gluing Method I: Type II Blow-up for Fujita Equation in Matano-Merle Regime

##### Talk II: On Gluing Method II: Second Order Estimates of Allen-Cahn Equation

These talks are concerned with applications of recently developed gluing methods. Gluing methods have been used widely in nonlinear elliptic equation, e.g. counterexamples of De Giorgi's Conjecture. In these series of talks, I will discuss new developments in the gluing methods. In the first talk, I will use the parabolic gluing method to construct various Type II blow-up solutions for nonlinear Fujita heat equation with exponent belonging to the Matano-Merle regime. Matano-Merle have shown that in the radially symmetric case all blow-ups are Type I. In the first example we prove a geometry-driven Type II blow-up in a non-convex domain. The second example will be Type II blow-up with thin cylinderical tubes with self-similar size when the exponent is 3 and dimension is greater than 5.

In the second talk, I will show how to use the reverse process of infinite dimensional gluing method to analyze the collapsing interfaces for Allen-Cahn equation. This corresponds to minimal surfaces with higher multiplicity. We will prove that curvature decaying and second order estimates and multiplicity one for stable solutions of Allen-Cahn in dimensions less than 11. To this end, we use the reverse gluing method to show that the obstruction of these estimates is the existence of Toda system. As a consequence we prove that finite Morse index solutions of two-dimensional Allen-Cahn implies finite ends.

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2018

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April 27-29, 2018

#### Ciprian Demeter

Indiana University

##### Decouplings and Applications

Lecture slides: Ciprian Demeter

In the first lecture I will introduce a Fourier analytic tool called decoupling, and will present a few applications to PDEs and number theory. In the second lecture, I will attempt to sketch the proof of the main theorem in the simplest possible case.

#### Charles Epstein

University of Pennsylvania

##### The Amazing Kimura Operator

Lecture slides: The Amazing Kimura Operator

The Kimura Operator is a second order operator defined on the simplex in any dimension. It is the basic building block of diffusion models used in Population Genetics. This operator is, in many sense, the Laplace operator of the simplex. In this talk we will introduce diffusion models in population genetics and explore some remarkable features of the Kimura operator itself.

##### Analytic Foundations for Diffusion Models in Population Genetics

Lecture slides: Analytic Foundations for Diffusion Models in Population Genetics

Diffusion models in Population Genetics are usually described in terms of second order PDEs defined on manifolds-with-corners. We call these generalized Kimura operators. The principal symbol of these operators degenerate in a very particular way along the boundary of this space, rendering them beyond standard elliptic/parabolic theory, even for previous analyzed classes of degenerate operators. In this talk I describe recent

analytic work (joint with Rafe Mazzeo and Camelia Pop) establishing existence, uniqueness and regularity of solutions to the elliptic and parabolic problems defined by generalized Kimura operators. I will also describe properties of the stochastic processes they define.

#### Joachim Krieger

Ecole Polytechnique Federale de Lausanne

##### Dynamics of Critical Geometric Wave Equations

I will discuss some recent developments in the theory of energy-critical nonlinear wave equations of geometric character, such as the Wave Maps, Maxwell-Klein-Gordon, and Yang-Mills equation. In particular, large data dynamics of soliton type which figure in the so-called soliton resolution conjecture will be discussed, as well as cases where all solutions eventually scatter.

#### Tatiana Toro

University of Washington

##### Elliptic measure and rectifiability I & II

In these talks we will describe some recent results concerning the relationship between the behavior of the elliptic measure for certain divergence form elliptic operators and the geometry of the boundary of the domain where the operators are defined. The results bear a strong resemblance to those obtained for the harmonic measure. One of the main difference between these two cases is the use of compactness techniques, which play a central role. These will be presented in some detail. This is joint work with S. Hofmann, J.M. Martell, S. Mayboroda and Z. Zhao.

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2017

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April 15-17, 2017

#### Philip Isett

University of Texas, Austin

##### A Proof of Onsager’s Conjecture for the Incompressible Euler Equations

In an effort to explain how anomalous dissipation of energy occurs in hydrodynamic turbulence, Onsager conjectured in 1949 that weak solutions to the incompressible Euler equations may fail to exhibit conservation of energy if their spatial regularity is below 1/3-Hölder. I will discuss a proof of this conjecture that shows that there are nonzero, (1/3-\epsilon)-Hölder Euler flows in 3D that have compact support in time. The construction is based on a method known as "convex integration," which has its origins in the work of Nash on isometric embeddings with low codimension and low regularity. A version of this method was first developed for the incompressible Euler equations by De Lellis and Székelyhidi to build Hölder-continuous Euler flows that fail to conserve energy, and was later improved by Isett and by Buckmaster-De Lellis-Székelyhidi to obtain further partial results towards Onsager's conjecture. The proof of the full conjecture combines convex integration using the “Mikado flows” introduced by Daneri-Székelyhidi with a new “gluing approximation” technique. The latter technique exploits a special structure in the linearization of the incompressible Euler equations.

#### Sylvia Serfaty

Courant Institute, NYU

##### Mean-Field Limits for Ginzburg-Landau vortices

Lecture slides: Sylvia Serfaty

Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular I will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation. I will also discuss the situation with disorder and related homogenization questions.

#### Luis Silvestre

University of Chicago

##### Integro-differential equations and their applications.

Lecture 1 slides: Luis Silvestre | Lecture 2 slides: Luis Silvestre

Integro-differential equations have been a very active research topic in recent years. In these talks we will start by explaining what they are and reviewing some basic results related to general problems. Then we will move on to some applications of these results to problems of mathematical physics. In particular, we will discuss some recent regularity estimates for the Boltzmann equation.

#### Manuel del Pino

Universidad de Chile

##### Singularity formation in critical parabolic problems

Lecture 1 slides: Manuel del Pino | Lecture 2 slides: Manuel del Pino

We deal with construction and stability analysis of blow-up of solutions for parabolic equations that involve bubbling phenomena, corresponding to gradient flows of variational energies. The term bubbling refers to the presence of families of solutions which at main order look like scalings of a single stationary solution which in the limit become singular but at the same time have an approximately constant energy level. This phenomenon arises in various problems where critical loss of compactness for the underlying energy appears. Specifically, we present construction of threshold-dynamic solutions with infinite time blow-up in the Sobolev critical semilinear heat equation in $\R^n$, and finite time blow up for the harmonic map flow from a two-dimensional domain into $S^2$.

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2016

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April 15-17, 2016

#### Andrea Bertozzi

UCLA

##### Aggregation equations - well posedness, self-similarity, and collapse

Lecture slides (aggregation equations): Andrea Bertozzi

This talk reviews a body of work on multidimensional aggregation equations that arise in models of biological swarms. The basic problem is a nonlocal density model in which the velocity field is the gradient of a nonlocal interaction kernel convolved with the density. This model has parallels to classical problems in 2D fluid dynamics - it is the analogue of the vorticity equation from incompressible flow, only with gradient structure rather than incompressibility and with more diverse interaction kernels than the Biot-Savart kernel. In the past ten years there has been a large body of work on aggregation equations. I will review basic well-posedness results in all dimensions including a sharp condition on the interaction kernel for global well-posedness of smooth solutions. I will also discuss results for weak solutions include patch-like solutions in L-infinity and solutions in Lp spaces. We also discuss results involving nonlinear diffusion that arises from local anticrowding behavior. The talk will include both analysis and results from numerical simulations.

See references below. This work includes collaborations of the speaker with her former students and postdocs and visitors to her group at UCLA: Thomas Laurent (Loyola Marymount), Theodore Kolokolnikov (Dalhousie), Dejan Slepcev (Carnegie Mellon), Jose Antonio Carrillo (Imperial College London), Nancy Rodriguez (UNC Chapel Hill), Jacob Bedrossian (U Maryland), John Garnett (UCLA), Jesus Rosado Linares (U Buenos Aires), Yanghong Huang (U Manchester), Chad Topaz (Macalester), Mark Lewis (Alberta), Yao Yao (Georgia Tech), James von Brecht (CSULB), Katy Craig (UCSB), David Uminsky (USF), Tom Witelski (Duke), Hui Sun (UCSD), Joan Verdera (Barcelona), and Flavien Leger (NYU).

##### Geometric graph-based methods for high dimensional data

Lecture slides (Geometric methods): Andrea Bertozzi

We present new methods for segmentation of large datasets with graph based structure. The method combines ideas from classical nonlinear PDE-based image segmentation with fast and accessible linear algebra methods for computing information about the spectrum of the graph Laplacian. We review the connection between motion by mean curvature in Euclidean space, total variation minimization, and the Ginzburg-Landau functional for diffuse interface approximations of TV - leading to the Allen - Cahn equation as an L2 gradient descent. We will discuss the analogue problem on a discrete graph and in the context of modern machine learning applications such as semi-supervised learning and unsupervised data classification and clustering. We will review recent results involving Gamma convergence of the graph Ginzburg-Landau energy and the MBO scheme on graphs. We will show examples for binary and multiclass data clustering problems with application to image labeling and hyperspectral video segmentation, and community detection in social networks, including modularity optimization posed as a graph total variation minimization problem.

Some References:

* refers to Thomson Reuters ESI highly cited paper 2015

Mar 27 (5 days ago)

- A. L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in R^n (springer link), Comm. Math. Phys., 274, p. 717-735, 2007
- *Andrea L. Bertozzi, Jose A. Carrillo, and Thomas Laurent Blowup in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009) 683-710.
- Andrea L. Bertozzi and Jeremy Brandman, Finite-time blow-up of L-infinity-weak solutions of an aggregation equation Communications in the Mathematical Sciences, Vol. 8, No. 1, pp. 45-65, 2010
- Yanghong Huang and A. L. Bertozzi, Self-similar blowup solutions to an aggregation equation in RN, SIAM. J. Appl. Math.,Volume 70, Issue 7, pp. 2582-2603 (2010)
- Andrea Bertozzi and Dejan Slepcev, Existence and Uniqueness of Solutions to an Aggregation Equation with Degenerate Diffusion, Comm. Pur. Appl. Anal., 9(6), 2010, pp. 1617-1637.
- *Andrea L. Bertozzi, Thomas Laurent, and Jesus Rosado, Lp theory for the multidimensional aggregation equation, Comm. Pur. Appl. Math., Vol. 64, No. 1, pages 45-83, January 2011.
- *J. Bedrossian, Nancy Rodriguez, and Andrea Bertozzi, Local and Global Well-Posedness for Aggregation Equations and Patlak-Keller-Segel Models with Degenerate Diffusion, Nonlinearity 24 (2011) 1683-1714.
- Theodore Kolokolnikov, Hui Sun, David Uminsky, Andrea L. Bertozzi, Stability of ring patterns arising from 2D particle interactions, Phys. Rev. E, Rapid Communications, 84(1), 015203(R), 2011.
- Hui Sun, David Uminsky, and Andrea L. Bertozzi, A generalized Birkhoff-Rott Equation for 2D Active Scalar Problems, SIAM J. Appl. Math, 72(1), pp. 382-404, 2012.
- Yanghong Huang and Andrea L. Bertozzi, Asymptotics of blowup solutions for the aggregation equation, Discrete and Continuous Dynamical Systems Series B, pages 1309 - 1331, Volume 17, Issue 4, June 2012.
- Andrea L. Bertozzi, Thomas Laurent, and Flavien Leger, Aggregation and Spreading via the Newtonian Potential: The Dynamics of Patch Solutions M3AS,vol. 22, Supp. 1, 2012
- *James von Brecht, David Uminsky, Theodore Kolokolnikov, and Andrea L. Bertozzi, Predicting pattern formation in particle interactions M3AS, vol. 22, Supp. 1, 1140002, 2012.
- Andrea L. Bertozzi, John B. Garnett, and Thomas Laurent, Characterization of radially symmetric finite time blowup in multidimensional aggregation equations, SIAM J. Math. Anal., 44(2), pp. 651-681, 2012.
- Yanghong Huang, Thomas P. Witelski, and Andrea L. Bertozzi, Anomalous exponents of self-similar blow-up solutions to an aggregation equation in odd dimensions, Applied Mathematics Letters, 25(12), p. 2317-2321, 2012.
- K. Craig and A. L. Bertozzi, A blob method for the aggregation equation, ArXiv:1405.6424, published electronically in Math. Comp., Dec. 4, 2015.
- Yao Yao and Andrea L. Bertozzi, Blow-up dynamics for the aggregation equation with degenerate diffusion, Physica D, 260, pp. 77-89, 2013, special issue on Emergent Behaviour in Multi-particle Systems with Non-local Interactions.
- James von Brecht and Andrea L. Bertozzi, Well-posedness theory for aggregation sheets, Comm. Math. Phys., 319(2), pp. 451-477, 2013.
- *C.M. Topaz, A.L. Bertozzi, and M.A. Lewis. A nonlocal continuum model for biological aggregation. Bulletin of Mathematical Biology, 68(7), pages 1601-1623, 2006
- Andrea Bertozzi, John Garnett, Thomas Laurent, Joan Verdera, The regularity of the boundary of a multidimensional aggregation patch, Preprint available on the arxiv, 2015.

#### Roman Vershynin

University of Michigan

##### Concentration of random matrices and applications

The concentration phenomenon in probability theory quantifies the tendency of various random objects to stay close to their expectations. The law of large numbers, for example, explains the concentration for random sums. In these two lectures, we will explore the concentration phenomenon for random matrices. Applications will be given to random graphs and algorithmic problems on networks, signal recovery and regression problems in statistics.

#### Jonathan Mattingly

Duke University

##### Ellipticity, Hypoellipticity, and Smoothing for SPDEs or Why the ergodic theory of Markov processes in infinite dimensions is different

I will start by recalling ideas of smoothing for the PDE governing the evolution of the density of a finite dimensional Markov. Then I will show how smoothing is an important component of proving a system is uniquely ergodic (and a single unique invariant measure). I will then show how things are more complicated and sublet if the Markov process is generated by a stochastically forced PDE. This will lead us to change our expectations and think about but what kind of smoothing one might expect in infinite dimensions. What does it mean to be elliptic or hypo-elliptic. I will discuss a view of hypoellipticity for SPDEs at the end.

#### Fanghua Lin

Courant Institute, NYU

##### Superfluids passing an obstacle---Nucleation of Vortices

The problem of classical (compressible) fluids passing an obstacle was well-known and studied by many. Roughly speaking, when the velocity of the fluid is suitably small, the flow would be smooth and nothing much would occur(subsonic region). When the fluid velocity is very large, there are shocks and the fluids become rather turbulent and mathematically hard to describe (supersonic). In between, there is a critical speed at which the fluid reach the maximum speed (sound speed for the fluid) at the boundary of the obstacle.

Since a superfluid by definition is frictionless, hence it would not develop shocks. On the other hand, formal arguments imply that the long-wave approximations of superfluid flows (semiclassical limits) would be a classical flow described by the compressible Euler equations. The latter may develop shocks however. A natural question is: how would one explain superfluid flows then? Reasonable arguments from physics which are also supported by various numerical simulations lead to the so-called vortex nucleations. The aim of this talk is to present some recent work on this problem.

##### Topological Defects of Liquid Crystals--Ericksen Model

A fundamental and challenging issue in the theory of liquid crystals is to understand the topological defects in both static and dynamic cases. For the classical model of Oseen-Frank, one has a good theory for stable topological point defects. It has been well-known that the line defects or disclinations are out of reach by such a model. Though it may be possible (and various recent works indicated that) that a general theory of order-parameters(Q-tensors) would describe most of observed defects, a mathematically (and physically) much simpler and consistent model due Ericksen (1990) for uniaxiall liquid crystals were proven to be sufficient to describe domain walls, disclinations and point defects.In this lecture, I shall discuss a recent joint work with Onur Alper and Robert Hardt concerning disclinations for energy minimizing configurations in the Ericksen's model. We can show a global topological strcture of such defects. Moreover, we can also prove a bound on the one-dimensional Hausdorff measure of the defect sets.

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2015

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April 17-19, 2015

#### Alessio Figalli

University of Texas

##### Transport theory: from functional inequalities to random matrices

The optimal transport problem consists in finding the cheapest way to transport a distribution of mass from one place to another. Apart from its applications to economics, optimal transport theory is an efficient tool to construct change of variables between probability densities, and this fact can be applied for instance to prove stability of minimizers of several geometric/functional inequalities.

More recently, motivated by problems arising in random matrix theory, people have tried to apply these methods in very large dimensions. However the regularity of optimal maps seem to play an important role in this context, and unfortunately one cannot hope in general to obtain regularity estimates that are uniform with respect to the dimension. Based on these considerations, it seems hopeless to apply optimal transport theory in this context. Still, ideas coming from optimal transport can be used to construct approximate transport maps (i.e., maps which send a density onto another up to a small error) which enjoy regularity estimates that are uniform in the dimension, and such maps can then be used to show universality results for the distribution of eigenvalues in random matrices.

The aim of these lectures is to give a self-contained presentation of all these results.

#### Nader Masmoudi

Courant Institute, NYU

##### Inviscid damping and enhanced dissipation in 2D Euler and Navier-Stokes.

Lecture slides: Nader Masmoudi

I will review a few aspects of the inviscid damping in 2D Euler near the 2D Couette flow (Joint with J. Bedrossian) and then will talk about Enhanced dissipation and inviscid damping in the Navier-Stokes equations near the 2D Couette flow and study the stability in the inviscid limit.

The main goal is the study of the long time, inviscid limit of the 2D Navier-Stokes equations near the periodic Couette flow, and in particular, we confirm at the nonlinear level the qualitative behavior predicted by Kelvin's 1887 linear analysis. (joint with J. Bedrossian and V. Vicol).

#### Vitaly Milman

University of Tel Aviv

##### Analytic and algebraic related structures on the families of convex sets and log-concave functions. (Based on Joint works with Shiri Artstein and Liran Rotem)

##### Elementary operator equations and classical constructions in analysis. (Based on joint works with Hermann Koenig)

The main goal of the talks is to show how some classical constructions in Geometry and Analysis appear (and in a unique way) from elementary and very simplest properties. For example, the polarity relation and support functions are very important and well known constructions in Convex Geometry, but what would be their functional version, say, in the class of log-concave functions? And yes, they are uniquely defined also for this class, as well as for many other classes of functions.

Another example: How can one identify the "square root of a convex body"? Yes, it is possible (however, "the square of a convex body" does not exist in general).

In the first talk we will mostly deal with Geometric results of this nature. We also construct summation operation on the class of log-concave functions which polarizes the Lebegue Integral and introduces the notion of a mixed integral parallelly to mixed volumes for convex bodies. We will show some inequalities coming from Convex geometry, but which are presented already on the level of log-concave (or even more generally quasi-concave) functions.

In the preparation to the second talk we will characterize the Fourier transform (on the Schwartz class in R^n) as, essentially, the only map which transforms the product to the convolution.

The talks will be (obviously) accessible to graduate students.

#### Laure Saint-Raymond

École Normale Supérieure - Université Pierre et Marie Curie

##### Lecture 1: From molecular dynamics to kinetic theory and fluid mechanics

Lecture slides, part 1: Laure Saint-Raymond

Lecture slides, part 2: Laure Saint-Raymond

Abstract 1: In his sixth problem, Hilbert asked for an axiomatization of gas dynamics, and he suggested to use the Boltzmann equation as an intermediate description between the (microscopic) atomic dynamics and (macroscopic) fluid models. The main difficulty to achieve this program is to prove the asymptotic decorrelation between the local microscopic interactions (referred to as propagation of chaos) on a time scale much larger than the mean free time. This is indeed the key property to observe some relaxation towards local thermodynamic equilibrium.

This control of the collision process can be obtained in fluctuation regimes [1, 2]. In [2], we have established a long time convergence result to the linearized Boltzmann equation, and eventually derived the acoustic and incompressible Stokes-Fourier equations in dimension 2. The proof relies crucially on symmetry arguments, combined with a suitable pruning procedure to discard super exponential collision trees.

Keywords: system of particles, low density, Boltzmann-Grad limit, kinetic equation, fluid models

[1] T. Bodineau, I. Gallagher, L. Saint-Raymond. The Brownian motion as the limit of a deterministic system of hard-spheres, to appear in Invent. Math. (2015).

[2] T. Bodineau, I. Gallagher, L. Saint-Raymond. From hard spheres to the linearized Boltzmann equation: an L2 analysis of the Boltzmann-Grad limit, in preparation.

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2014

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April 25-27, 2014

#### Alice Chang

Princeton University

##### Boundary value problems on conformal compact Einstein manifolds

Given a class of conformally compact Einstein manifolds with boundary, we are interested to study the boundary behavior of their compactified metrics. In the 3+1 case, I will report on some joint work with Yuxin Ge and Paul Yang on a compactness result in such setting, which is a study of some 4th order elliptic system with some matching 3rd order boundary conditions. In the general n+1 case, . I will report on some recent joint work with Jeffrey Case in which we study the positivity of a class of non-local conformal covariant operators, which are fractional GJMS operators on the boundary defined via scattering theory on the interior, and which includes the Dirichlet-Neumann operator as a special case.

#### Alexandru Ionescu

Princeton University

##### On the long-term dynamics of solutions of certain fluid models

I will present several recent theorems on the long-term behavior of solutions of certain equations describing fluid dynamics. The main models to be discussed are (1) two-fluid interface models in 2 dimensions, and (2) the Euler--Maxwell two-fluid system in 3 dimensions. The main questions we are considering concern the long-term dynamics of solutions, i.e. global existence and formation of singularities.

I will present joint work with F. Pusateri, C. Fefferman, V. Lie, B. Pausader, and Y. Guo.

#### Frank Merle

Universite de Cergy-Pontoise and IHES

##### Blow-up for mass critical KdV and universality

We give a complete description of the dynamical behavior (including blow-up) of solutions with initial data close to ground state.

##### On soliton resolution for the radial critical wave equation

In this joint work with Duyckaerts and Kenig we describe the asymptotic behavior of global solutions in a decoupled sum of soliton and a radiation.

#### Maciej Zworski

University of California, Berkeley

##### Decay of correlations for classical and quantum hyperbolic systems

The talk will based on joint work with Stéphane Nonnenmacher on the distribution of decay rates for systems with thin hyperbolic trapped sets: either filamentary, or smooth and normally hyperbolic. The results will be illustrated by recent numerical and experimental data (Borthwick, Barkhofen et al) and by the case of constant negative curvature (Dyatlov—Faure—Guillarmou).

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2013

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April 19-21, 2013

#### Marianna Csörnyei

University of Chicago

##### Geometry of Null Sets

We will show how elementary product decompositions of measures can detect directionality in sets. In order to prove this we will need to prove results about the geometry of sets of small Lebesgue measure: we show that sets of small measure are always contained in a "small" collection of Lipschitz surfaces.

The talk is based on a joint work with G. Alberti, P. Jones and D. Preiss.

#### Mihalis Dafermos

Princeton University

##### The Black Hole Stability Problem in General Relativity

A celebrated open problem in classical general relativity is that of black hole stability. I will introduce the background necessary to understand the formulation of this problem, describe recent mathematical results that have been obtained, and end with a discussion of several surprises that have emerged in the context of the extremal and negative cosmological constant cases, with ramifications to various questions in high energy physics

#### Andrea Nahmod

University of Massachusetts

##### Randomization in Nonlinear PDE and the Supercritical Periodic Quintic NLS in 3D

In the last two decades significant progress has been made in the study of nonlinear dispersive and wave equations, settling questions about existence of solutions, their long time behavior, and singularity formation. The thrust of this body of work has focused primarily on deterministic aspects of wave phenomena, where sophisticated tools from nonlinear Fourier analysis, geometry, and also analytic number theory have played a crucial role in the underlying methods. Yet some important obstacles and open questions remain. A natural approach to overcome these, and one which has recently seen a growing interest, is to consider certain evolution equations from a non-deterministic point of view (e.g. the random data Cauchy problem ) and incorporate powerful tools from probability as well. In this talk we will explain some of these ideas and describe recent joint work with G. Staffilani for the 3D periodic quintic nonlinear Schrodinger equation below the critical energy space.

#### Assaf Naor

Courant Institute, NYU

##### Ultrametric Skeletons

Let (X,d) be a compact metric space, and let mu be a Borel probability measure on X. We will show that any such metric measure space (X,d,mu) admits an “ultrametric skeleton”: a compact subset S of X on which the metric inherited from X is approximately an ultrametric, equipped with a probability measure nu supported on S such that the metric measure space (S,d,nu) mimics useful geometric properties of the initial space (X,d,mu). We will make this geometric picture precise, and explain a variety of applications of ultrametric skeletons in analysis, geometry, computer science, and probability theory. Based on joint work with Manor Mendel.

#### Fedor Nazarov

Kent State University

##### The Number of Real Zeroes of Random Polynomials with I.I.D. Coefficients

We will show that the mean number of real zeroes of a random polynomial of degree $n$ with non-degenerate (say, without point masses) i.i.d. real coefficients is bounded from above by C log(n+1) where the constant C>0 is independent of the distribution. This is a joint work with M. Krishnapur, M. Sodin, and O. Zeitouni.

#### Peter Topalov

Northeastern University

##### On the Spectral Rigidity of a Class of Integrable Billiards

The Radon transform of an integrable billiard is an averaged quantity associated to the Liouville tori in the phase space of the billiard. I will discuss the relation of the Radon transform to some new iso-spectral invariants of the Laplace-Beltrami operator associated to smooth deformations of the Riemannian metric of the billiard table. In several cases the injectivity of the Radon transform implies spectral rigidity.

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2012

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April 20-22, 2012

#### Antonio Còrdoba

Universidad Autónoma de Madrid

##### Singular Integrals in Fluid Mechanics: Blow up of solutions for a transport equation; Interface evolution: The Muskat and Hele-Shaw problem

Some new estimates for classical Singular Integrals will be introduced, discussing their applications to several problems in Fluid Mechanics.

#### Panagiotis Souganidis

University of Chicago

##### Stochastic homogenization

In these talks I will describe recent advances to the theory of homogenization of first- and second-order partial differential equations set in general stationary ergodic environments.

#### Thomas Alazard

CNRS and Ecole Normale Supérieure, Paris

##### On the Cauchy problem for the water-waves equations

The water-waves problem consists in describing the motion, under the influence of gravity, of a fluid occupying a domain delimited below by a fixed bottom and above by a free surface. We consider the Cauchy theory for low regularity solutions. In terms of Sobolev embeddings, the initial surfaces we consider turn out to be only of C3/2 class and consequently have unbounded curvature. Furthermore, no regularity assumption is assumed on the bottom. We also take benefit from an elementary observation to solve a question raised by Boussinesq on the water-wave equations in a canal.

#### Giuseppe Mingione

Universitá degli Studi di Parma

##### Linear and nonlinear Calderon-Zygmund theories

Calderon-Zygmund theory deals with a fundamental problem in the theory of partial differential equations of elliptic and parabolic type: given a certain PDE, can we determine, in a possibly sharp way, the regularity and, especially, the integrability properties of the solution in terms of those of the assigned datum? In the linear case sharp answers are related to the theory of singular integrals, whose fundamentals have been established in the multidimensional case by Calderon and Zygmund more that fifty years ago. Recent years have witnessed a considerable activity towards establishing a series of analogous results for nonlinear equations, up to the stage that it appears to be possible to think about a nonlinear Calderon-Zygmund theory. I will give a survey of such results up to a few recent developments.

#### Gabriella Tarantello

Universitá di Roma 'Tor Vergata'

##### Liouville–type systems in the study of non-topological solutions in Chern Simons theory

We discuss elliptic systems of Liouville type in presence of singular sources, as derived from the study of non-abelian (selfdual) Chern-Simons vortices. We shall focus on the search of the so called non-topological vortex configurations. We present some known results and discuss many of the still open questions.

#### Rachel Ward

University of Texas at Austin

##### Strengthened Sobolev inequalities for a random subspace of functions

We introduce some Sobolev inequalities for functions on the unit cube satisfying a random collection of linear constraints. We then explain how these inequalities provide near-optimal guarantees for accurate image recovery from under-sampled measurements using total variation minimization, with applications to medical imaging. We finish by discussing several open problems.

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2011

####
April 15-17, 2011

#### Michael Taylor

University of North Carolina

##### Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains.

These talks discuss results on layer potentials for elliptic boundary problems, with emphasis on the Dirichlet problem for the Laplace operator. They start by reviewing results for domains with moderately smooth boundary, then for Lipschitz domains, and proceed to discuss results obtained in joint work of the speaker with S. Hofmann and M. Mitrea, for a class of domains we call regular Semmes-Kenig-Toro (SKT) domains, often called chord-arc domains with vanishing constant, and for delta-regular SKT domains, often called chord-arc domains with small constant. Lecture notes pertaining to the talk can be found on the speaker's web page.

#### Hitoshi Ishii

Waseda University

##### Lecture 1: Long-time behavior of solutions of Hamilton-Jacobi equations with Neumann type boundary conditions

We discuss the long-time behavior of solutions of the convex Hamilton-Jacobi equation ut + H(x,Du) = 0 in a bounded domain Ω of Rn with the Neumann type boundary condition Dγu = g, where γ is a vector field on the boundary ∂Ω pointing a direction oblique to ∂Ω. We explain a convergence result to asymptotic solutions together with some related results concerning is the stationary problem associated with ut +H(x,Du) = 0 is: H(x,Dv) = c in Ω and Dγv = 0 on ∂Ω, where the pair, v ∈ C(Ω ̄) and c ∈ R, is the unknown.

Abstract PDF: Hitoshi Ishii lecture 1

Slides: Hitoshi Ishii lecture 1

##### Lecture 2: Stochastic perturbations of Hamiltonian flows: a PDE approach

We present a pde approach to the study of averaging principles for small stochastic perturbations of Hamiltonian flows in 2D, which is based on a recent joint work with P. E. Souganidis of the University of Chicago. Freidlin and Wentzel initiated the study of such problems and there have been an extensive study in this field in the last several years. Asymptotically the slow (averaged) motion has 1D character and takes place on a graph, and the question is to identify the limit motion in terms of pde problems. Our approach is based on pde techniques and applies to general degenerate elliptic operators while previous work has relied on the probabilistic techniques.

Abstract PDF: Hitoshi Ishii lecture 2

Slides: Hitoshi Ishii lecture 2

#### Xiaochun Li

University of Illinois at Urbana-Champaign

##### Recent progress on discrete restriction

#### Yvan Martel

University of Versailles Saint-Quentin-en-Yvelines

##### Inelastic interaction of solitons for the quartic gKdV equation

We present two recent works in collaboration with Frank Merle concerning the interaction of two solitons for the (nonintegrable) quartic (gKdV) equation. In two specific asymptotic cases (almost equal speeds / very different speeds), we can describe the collision in details. In particular, we prove that at the main order, the two solitons are preserved by the interaction as in the integrable case. However, unlike in the integrable case, we prove that the collision is inelastic.

References:

Y. Martel et F. Merle, Description of two soliton collision for the quartic gKdV equation, to appear in Annals of Math.

Y. Martel et F. Merle, Inelastic interaction of nearly equal solitons for the quartic gKdV equation, to appear in Inventiones Mathematicae.

#### Robert McCann

University of Toronto

##### Optimal multidimensional pricing facing informational asymmetry

The monopolist's problem of deciding what types of products to manufacture and how much to charge for each of them, knowing only statistical information about the preferences of an anonymous field of potential buyers, is one of the basic problems analyzed in economic theory. The solution to this problem when the space of products and of buyers can each be parameterized by a single variable (say quality X, and income Y) garnered Mirrlees (1971) and Spence (1974) their Nobel prizes in 1996 and 2001. The multidimensional version of this question is a largely open problem in the calculus of variations (see Basov's book "Multidimensional Screening".) I plan to describe recent work with A Figalli and Y-H Kim, identifying structural conditions on the value b(X,Y) of product X to buyer Y which reduce this problem to a convex program in a Banach space--- leading to uniqueness and stability results for its solution, confirming robustness of certain economic phenomena observed by Armstrong (1996) such as the desirability for the monopolist to raise prices enough to drive a positive fraction of buyers out of the market, and yielding conjectures about the robustness of other phenomena observed Rochet and Chone (1998), such as the clumping together of products marketed into subsets of various dimension. The passage to several dimensions relies on ideas from differential geometry / general relativity, optimal transportation, and nonlinear PDE.

#### Susanna Terracini

University of Milano-Bicocca

##### Lecture 1 - Partitions and strongly competiting systems

We deal with the free boundary problem associated with optimal partitions related with linear and nonlinear eigenvalues. We are concerned with extremality conditions and the regularity of the interfaces. These properties are then linked with extremality conditions of the nodal set of eigenfunctions and the number of their nodal components.

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2010

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April 23-25, 2010

#### Tristan Rivière

ETH Zürich

##### Lecture 1: Conservation laws for conformally invariant problems and the Noether theorem in the absence of symmetry

We will review applications of Noether's Theorem for conformally invariant Lagrangians of maps from a Riemann surface into a symmetric manifold. We will explain how the existence of conserved quantities, issued from the symmetry of the target, - the so called Noether Currents - play a decisive role in the analysis of critical points to 2-dimensional conformally invariant Lagrangians. Once the symmetry assumption is dropped, Noether Theorems a-priori does not apply anymore. However we will exhibit the survival of Noether Currents for general target, beyond the symmetry assumption. These generalized Noether Currents will play again a central role in the analysis of 2-dimensional conformally invariant Lagrangians of maps into general manifold and will permit us in particular to prove the Heinz-Hildebrandt regularity conjecture.

Slides: Tristan Rivière lecture 1

##### Lecture 2: A PDE version of the constant variation method and the sub-criticality of Schroedinger Systems with antisymmetric potentials

Abstract 2: We will explain how the approach we developed in the first talk in order to find conservation laws for critical points to conformally invariant problems can be systematized and applied to a large family of equations : Linear Schroedinger Systems with antisymmetric potentials of various orders. This leads to a series of new compactness and regularity results for PDEs which are a-priori critical but happen in fact to have a subcritical behaviors. We will present some applications of these results to problems from geometric non-linear analysis.

Slides: Tristan Rivière lecture 2

#### Daniel Tataru

University of California, Berkeley

##### Large data wave maps

I will describe recent work, joint with Jacob Sterbenz, on the wave map equation in 2+1 dimensions. This is an energy critical problem, i.e. the energy of the wave map is invariant with respect to the natural scaling of the problem. For initial data with large energy we establish a dichotomy between global existence and scattering, on one hand, and soliton-like concentration on the other hand.

Slides: Daniel Tataru lecture 1

Slides: Daniel Tataru lecture 2

#### Inwon Kim

University of California, Los Angeles

##### Homogenization of interface velocities in periodic and random media

We will discuss several free boundary problems where the free boundary moves in heterogeneous environment with oscillatory boundary velocities. The stability of the interface in the homogeniation limit, as well as the properties of the limiting interface will be discussed. The key tool is maximum principle-type arguments as well as strong averaging properties of the media. We will also discuss similarities and differences with viscosity solutions method used to homogenization of nonilnear PDES.

#### Alexander Olevskii

Tel Aviv University

##### Wiener's "closer of translates" problem and Piatetskii-Shapiro uniqueness phenomenon.

Wiener characterized cyclic vectors (with respect to translations) in lp(Z)and Lp(R) (p=1,2) in terms of zero sets of Fourier transform. He conjectured that a similar characterization should be true for 1 < p < 2. I will discuss this conjecture.

Joint work with Nir Lev.

#### Natasa Sesum

University of Pennsylvania

##### Ancient solutions to the Ricci flow and Ricci solitons

We will give a classification of ancient solutions to the Ricci flow on compact surfaces. We show the contracting spheres and Angenant ovals are the only possibilities. We will also discuss some classsification results for Ricci solitons and some geometric properties of those.

Paper: Natasa Sesum (joint with P. Daskalopoulos and R. Hamilton)

#### Andrej Zlatoš

University of Chicago

##### Traveling Fronts in Combustible Media

Traveling fronts are special solutions of reaction-diffusion equations which model phenomena such as propagation of species in an environment or spreading of flames in combustible media. In this talk we will address questions of existence, uniqueness, and stability of traveling fronts in general inhomogeneous media. We will show that in certain circumstances they are global attractors of the corresponding parabolic evolution, thus describing long time dynamics for very general solutions of the PDE. We will also present examples of media in which no traveling fronts exist.

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2009

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April 17-19, 2009

#### Thierry Gallay

Universite de Grenoble

##### Interaction of Vortices in Viscous Planar Flows

It is a well-established fact that the long-time behavior of two-dimensional decaying turbulence is essentially governed by a few basic mechanisms, such as vortex interactions and, especially, vortex merging. The aim of this talk is to describe in a rigorous way the interaction of widely separated viscous vortices. To this end, we consider the inviscid limit of the solution of the two-dimensional incompressible Navier-Stokes equation in the particular case where the initial vorticity is a finite collection of point vortices. Assuming that vortex collisions do not occur, we obtain to leading order a superposition of Lamb-Oseen vortices whose centers evolve according to a viscous regularization of the Helmholtz-Kirchhoff system. Our approach also gives an accurate description of the asymptotic profile of each individual vortex, and this allows to estimate the self-interactions which play a crucial role in the convergence proof.

#### Hongjie Dong

Brown University

##### Regularity of Elliptic and Parabolic Equations with Rough Coefficients

I will present some recent results about the regularity and solvability of elliptic and parabolic equations in divergence and non-divergence forms. The leading coefficients are assumed to be measurable in one or two directions and have vanishing mean oscillation in the orthogonal directions. Applications of the results and extensions to higher order (fully couples) systems will also be discussed. Most part of the talk is based on joint work with Nicolai Krylov and with Doyoon Kim.

#### Ermanno Lanconelli

##### A Lie Groups Approach to the Analysis of Kolmogorov-Fokker-Planck Equations

Let L be a Hormander-type operator, sum of squares of vector fields+drift. We show sufficient conditions on the vector fields and on the drift term for the existence of a Lie group structure G such that L is left invarient on G. We also investigate the existence of a global fundamental solution for L, providing results that ensure a suitable left invariance property. We will show several examples of operators to which our results apply: some are new, some appear in recent literature usually quoted as Kolmogorov-Fokker-Planck operators. Our examples arise in several theoretical and applied settings, such as diffusion theory, computer and human vision, phasr noise Fokker-Planck equations, curvature Brownian motion.

#### Cedric Villani

##### Landau Damping: Relaxation Without Dissipation

Abstract 1: The Landau damping may be the single most famous and paradoxical phenomenon in classical plasma physics, predicting relaxation without any irreversibility. While it has been treated by various authors at the linear level, its nonlinear version has remained elusive so far. In a joint work with Clement Mouhot, we develop a new theory for Landau damping, for the full (not linearized) model. I will describe some of the physical and mathematical advances uncovered (to our own surprise) in this study.

#### Ioan Bejenaru

University of Chicago

##### Global Schrodinger Maps in Dimensions *d* ≥ 2: Small Data in the Critical Sobolev Spaces

#### Alexis Vasseur

University of Texas

##### Recent Results in Fluid Mechanics

We will present, in this talk, new applications of De Giorgi's methods and blow-up techniques to fluid mechanics problems. Those techniques have been successfully applied to show full regularity of the solutions to the surface quasi-geostrophic equation in the critical case.

We will present, also, a new nonlinear family of spaces allowing to control higher derivatives of solutions to the 3D Navier-Stokes equation. Finally, we will present a regularity result for a reaction-diffusion system which has almost the same supercriticality than the 3D Navier-Stokes equation.

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2008

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April 11-13, 2008

#### Alberto Bressan

Penn State University

##### Impulsive Control and Swim-Like Motion in a Perfect Fluid

Consider a body (or a chain of bodies) with variable shape, immersed in an incompressible, non-viscous fluid. Given an initial configuration, our main goal is to understand how the body can ``swim", i.e. reach other positions by changing its shape and its internal mass distribution.

The problem can be geometrically reformulated in terms of a foliation in a finite dimensional Riemann manifold, where the metric is given by the kinetic energy. At each given time, the controller forces the system to lie on a particular leaf of the foliation, by means of frictionless constraints.

Depending on the geometric structure of the problem, the equations of motion can contain the time derivative of the control function in a linear, or in a quadratic way. The spaces of admissible control functions, and the techniques used to achieve controllability can be very different in these two cases.

The linear case typically arises when the controller can completely determine the shape of the body immersed in the fluid. In this case, non-trivial displacements are achieved by generating Lie brackets of non-commuting vector fields.

On the other hand, the quadratic case occurs when the controller does not entirely determine the shape of the body, i.e. in the presence of some freely flapping parts. In this case, already the vibration of one single component of the control can produce motion.

Background Reference Material:

[1] A. Bressan, Impulsive control of Lagrangian systems and locomotion in fluids, {\it Discr. Cont. Dynam. Syst.} {\bf 20} (2008), 1-35

##### A New Class of Variational Problems Related to Optimal Confinement of Forest Fires

Abstract 2: The area burned by the fire (or contaminated by a spreading chemical agent) at time $t>0$ is modelled as the reachable set for a differential inclusion $\dot x\in F(x)$, starting from an initial set $R_0$. We assume that the spreading of the contamination can be controlled by constructing walls. In the case of a forest fire, one may think of a thin strip of land which is either soaked with water poured from above (by airplane or helicopter),or cleared from all vegetation using a bulldozer.

The first part of the talk will examine which conditions guarantee the existence of a strategy that completely blocks the fire within a bounded domain.

Next, we consider functions $\alpha(x)$ describing the unit value of the land at the location $x$, and $\beta(x)$ accounting for the cost of building a unit length of wall near $x$. This leads to an optimization problem, where one seeks to minimize the total value of the burned region, plus the cost of building the barrier.

A general theorem on the existence of optimal strategies will be presented, together with various necessary conditions for optimality.

REFERENCES:

[2] A.Bressan, Differential inclusions and the control of forest fires, {\it J. Differential Equations} (special volume in honor of A. Cellina and J. Yorke), {\bf 243} (2007), 179-207.

[3] A. Bressan, M. Burago, A. Friend, and J.Jou, Blocking Strategies for a Fire Control Problem, {\it Analysis and Applications}, to appear.

[4] A. Bressan and C. De Lellis, Existence of optimal strategies for a fire confinement problem. Preprint 2008.

#### Philippe Souplet

Université Paris-13

##### Grow-up Rate and Refined Asymptotics for a Two-Dimensional Keller-Segel Model in Chemotaxis

This is joint work with Nikos Kavallaris, U. of the Aegean, Greece. We consider a special case of the Keller-Segel system in a disc, which arises in the modelling of chemotaxis phenomena. For a critical value of the total mass, the solutions are known to be global in time but with density becoming unbounded. We establish the precise grow-up rate and obtain refined asymptotic estimates of the solutions. Unlike in most of the similar, recently studied, grow-up problems, the rate is neither linear nor exponential. In fact, the maximum of the density behaves like $e^{\sqrt{2t}}$ for large time. In particular, our study provides a rigorous proof of a behavior suggested by Chavanis and Sire [Phys. Rev. E, 2002] on the basis of formal arguments.

#### Monica Visan

Institute for Advanced Study

##### The Energy-Critical Nonlinear Schrodinger Equation

We survey old and new well-posedness results for the energy-critical NLS and the techniques used to prove them. Recent progress includes treatment of the focusing equation for radial and subsequently non-radial initial data.

#### Camillo de Lellis

Universität Zürich

##### Almgren's $Q$-Valued Functions Revisited

Multiple (or $Q$-) valued functions have been introduced by Almgren at the end of the seventies in order to study branching phenomena in minimal surfaces of codimension higher than $1$. These phenomena are tightly linked with the branching of holomorphic varieties around singular points.

The theory of multiple valued functions occupies almost $1/5$ of Almgren's 1000 pages proof of his big regularity theorem for mass-minimizing currents. It deals with minimizers of a suitable generalization of the Dirichlet energy and it culminates into two regularity results.

In a recent joint work with Emanuele Spadaro we revisit the theory of $Q$-valued functions, providing shorter versions of Almgren's proofs. In particular we show how most of them can be understood by combining clean ideas from the theory of elliptic PDEs with elementary combinatorial arguments. At the same time we propose a second (intrinsic) approach to the theory, which, at the expense of introducing some arguments of more analytic flavour, reduces further the combinatorics. Finally, using ideas of Chang and White, we improve upon the regularity of solutions of minimizers on $2$-dimensional domains, achieving the optimal statement.

#### Fengbo Hang

Courant Institute, New York University

##### Hopf Degree and Generalized Faddeev Model

The Hopf degree for a map from S^{4n-1} to S^{2n} has a classical integral formula. It is interesting to know whether such integral remains integer for suitable weakly differentiable maps. We will discuss some answers to this question and the minimization problem for Faddeev model in dimension 4n-1 (joint work with F. H. Lin and Y. S. Yang).

#### Igor Rodnianski

Princeton University

##### Evolution Problem in General Relativity: From Rough Space-Times to Black Holes.

I will start by explaining basic principles of General Relativity, including connections between causal geometry and various physical phenomena. I will review the all familiar Einstein-vacuum space-times:

Minkowski and Schwarzschild, the latter giving the simplest example of a black hole space-time, describe basic monotonicity property of the Einstein equations and its use in the Penrose incompleteness theorem.

I will then discuss the evolution problem in General Relativity, explain how to construct rough space-times, give a new breakdown criteria and a uniqueness result. I will also explore several stability problems, including that of black holes, where, in particular, I will illuminate the role of the celebrated red-shift effect.

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2007

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April 20-22, 2007

#### Carlos Kenig

University of Chicago

##### The Energy Critical, Focusing, Non-Linear Schrödinger and Wave Equations

We will discuss a point of view on obtaining optimal global well-posedness and scattering results for critical dispersive and wave equation problems. The program applies to both focusing and defocusing problems. The specific applications of the method to be discussed are to the focusing, energy critical non linear Schrodinger and wave equations. The method is a blend of ideas from elliptic and parabolic problems, with oscillatory integral techniques. This is joint work with Frank Merle.

#### Juan Luis Vazquez

Universidad Autónoma de Madrid

##### The Theory of Fast Diffusion Equations. Main features and recent news.

We will review the main mathematical features of the nonlinear heat °ow called the Fast Diffusion Equation δtu = Δum, m < 1. Much is known nowadays about this flow, posed inRn, in a bounded open subset or on a manifold. Surprising phenomena appear, like loss of regularity (solutions with Lp initial data may not be bounded), extinction in finite time, even lack of existence or lack of uniqueness for classes of small and smooth initial data.

As novelties, we will present geometrical results related to the 2-d Ricci °ow and the description of asymptotics using weighted functional inequali- ties of Hardy-Sobolev type.

*Background Reference:* J. L. Vázquez. "Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type", Oxford Lecture Notes in Maths. and its Applications 33, Oxford University Press, 2006.

Abstract PDF: Juan Luis Vazquez

#### Pierre Raphael

Princeton University

##### Blow up for some nonlinear dispersive PDE's

The focusing nonlinear Schrodinger equation iut +Δu+ u|u|p-1 = 0 inRN is a universal model for the self trapping of waves in a nonlinear medium. The cases p = 3 and N = 2,3 are of particular physical relevance but are quite different from the mathematical point of view. The phenomenon we aim at describing is the concentration of the nonlinear wave which mathematical counterpart is the finite time blow up of the solution to the Cauchy problem. A general understanding of the singularity formation for NLS is still widely open.

I will start by reviewing the series of results obtained in collaboration with Frank Merle for the L2 critical NLS iut + Δu + u|u|4/N = 0 in RN and which prove the existence and stability of "log-log" type blow up scenario. I will then explan how these results may be extended to various situations and in particular provide tools to construct new blow up solutions in the L2 super critical setting. I will also explain how the intuition based on the use of Liouville type classification theorem has recently allowed us to prove the blow up of the critical norm in super critical settings.

I will conclude by presenting a recent work joint with Mohammed Lemou anf Florian Mehats which concerns the quite unexpected adaptation of these techniques to nonlinear kinetic Vlasov-Poisson type problems emerging from astrophysiscs.

#### Ovidiu Savin

Columbia University

##### Symmetry of global solutions to certain fully nonlinear elliptic equations

Abstract 1: We consider bounded global solutions to fully nonlinear equations of the type $F(D^2u)=f(u)$. The main assumption on $F$ and $f$ is that there exists a one dimensional solutions $g$ that solves the equation in all directions. We show that solutions with Lipschitz level sets depend on one variable, that is the level sets are in fact hyperplanes. In the particular case when $F=\triangle$ and $f(u)=u3-u$ the same result was obtained by Barlow, Bass and Gui by probabilistic methods.

#### Sylvia Serfaty

Courant Institute

##### Vortices in the 2D Ginzburg-Landau model with magnetic field.

We review some results obtained jointly with Etienne Sandier on the Ginzburg-Landau model of superconductivity in 2 dimensions, and described in our recent book. In a superconducting sample, according to the intensity of the applied field, phase transitions happen and vortices appear in certain regimes. By studying energy minimizers in a suitable parameter regime, we extract limiting problems that describe the optimal repartition of vortices according to the applied field. We will also describe some of our more recent results on the subject.

*Background Reference:* Etienne Sandier, Sylvia Serfaty, Vortices in the Magnetic Ginzburg-Landau Model, Progress in Nonlinear Differential Equations 70, Birkhauser, 2007.

#### Alexander Volberg

Michigan State University

##### Equation of Monge-Ampère and Bellman Solutions for Certain Harmonic Analysis Problems (After Slavin-Stokolos)

Bellman function method in Harmonic Analysis was introduced by Donald Burkholder for finding the norm in L^p of martingale transform. Later it became clear that scope of the method is relatively wide, in particular Nazarov-Treil-Volberg obtained by this method a necessary and sufficient condition for the two weight martingale transform to be bounded (which was used then in Bellman estimates of Ahlfors-Beurling transform by Petermichl and myself). Recently Slavin and Stokolos made an important observation how Monge-Ampère equation may help to find the exact Bellman function of certain Harmonic Analysis problems. They illustrated their idea by finding the elegant and short way to find the Bellman function of dyadic maximal operator (A. Melas' recent result). We will illustrate this approach by this and couple of other examples, where Monge-Ampère allows us to find exact Bellman function of a Harmonic Analysis problem. In particular, to find best constants and extremal functions (or extremal sequences).

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2006

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April 7-9, 2006

#### Wilhelm Schlag

University of Chicago

##### Spectral theory and applications to nonlinear PDE

We will discuss some recent work, most of it joint with Joachim Krieger at Harvard, concerning nonlinear PDE that allow for a family of nonlinear bound-states (e.g., standing waves). These families can be either stable or unstable under small perturbations. We will describe some new results on the unstable case and show that stability can be achieved provided the perturbations are chosen on suitable finite co-dimensional manifolds. Some conjectures and possible further work will be discussed.

#### Charles Fefferman

Princeton University

##### Fitting a Smooth Function to Data

Fix positive integers *m*,*n*, and suppose we are given *N* points in *Rn+1*. We compute a function *F* in *Cm(Rn)*, whose graph passes through (or close to) all (or nearly all) of the given points, and whose *Cm* norm has the smallest possible order of magnitude. Joint work with Bo'az Klartag.

#### Isabelle Gallagher

Universite Paris 7

##### Mathematical analysis of equatorial waves

In this talk we will consider a model of rotating fluids, describing the motion of the ocean in the equatorial zone. This model is known as the Saint-Venant, or shallow-water type system, to which a rotation term is added whose amplitude is linear with respect to the latitude; in particular it vanishes at the equator. After a quick physical introduction to the model, we describe the various waves involved and the resonances associated with those waves. We then exhibit the limit system (as the rotation becomes large), obtained as usual by filtering out the waves, and study its wellposedness. Finally we present three types of convergence results: a weak convergence result towards a linear, geostrophic equation, a "hybrid" strong convergence result of the filtered solutions towards a weak solution to the limit system, and finally a strong convergence result of the filtered solutions towards the unique strong solution to the limit system for smooth enough initial data. In particular we show that there are no confined equatorial waves in the mean motion as the rotation becomes large.

#### Alexandru Ionescu

University of Wisconsin

##### Low-regularity solutions of nonlinear equations

I will discuss some recent work with C. Kenig on local and global well-posedness of several nonlinear dispersive equations. The main models I will consider are the Benjamin-Ono equation (BO), the Kadomtsev-Petviashvili I equation (KP-I), and Schrödinger maps.

#### Diego Maldonado

University of Maryland

##### On the Monge-Ampere equation and its linearization

During the 90's, Luis Caffarelli pioneered a geometric approach to the study of convex solutions *u* to the Monge-Ampere equation det*D*2*u* = μ. One ground-breaking consequence of that approach is the *C*1,α-regularity result for *u* when μ verifies a doubling property. In this talk, we will go over a new proof for this theorem that can be extended to the context of the Heisenberg group (and Carnot groups in general). Then, we will show the connections between the mentioned geometric approach and two topics of independent interest: the real analysis on spaces of homogeneous type and the theory of quasi-conformal mappings. Finally, we will prove a weak reverse-Holder inequality for non-negative solutions to the linearized Monge-Ampere equation that involves techniques developed by E. Fabes and D. Stroock in the 80's. The material of this talk is based on several collaborations with Luca Capogna, Liliana Forzani, and Leonid Kovalev.

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2005

####
April 8-10, 2005

#### Luca Capogna

University of Arkansas

##### Mean curvature flow and the isoperimetric problem in the Heisenberg group

I will describe recent results (joint with Mario Bonk, U. Michigan) concerning a notion of mean curvature flow in the Heisenberg group. We derive an equation for the flow, characterize self-similar solution and prove basic existence and comparison results. We also study the induced flow on the Legendrian foliation of the manifolds and indicate how it relates to the isoperimetric problem in the Heisenberg group.

#### Svatlana jitomirskaya

University of California, Irvine

##### The ten martini problem

In these talks I will describe recent advances to the theory of homogenization of first- and second-order partial differential equations set in general stationary ergodic environments.

#### Vladimir Maz'ya

Ohio State University

##### Unsolved mysteries of solutions to PDEs near the boundary

Throughout its long history, specialists in the theory of partial differential equations gained a deep insight into the boundary behaviour of solutions.Yet despite the apparent progress in this area achieved during the last century, there are fundamental unsolved problems and surprising paradoxes related to solvability, spectral, and asymptotic properties of boundary value problems in domains with irregular boundaries. I shall formulate some challenging questions arising naturally when one deals with unrestricted, polyhedral, Lipschitz graph, fractal and convex domains.

#### Natasa Pavlovic

Princeton University

##### Dyadic models for the equations of fluid motion

In this talk we shall introduce a scalar dyadic model for the Euler and the Navier-Stokes equations in three dimensions and will discuss some of the results that were obtained for these models. For the dyadic Euler equations we prove finite time blow-up, while in the context of the dyadic Navier-Stokes equations with hyper-dissipation we prove finite time blow-up in case when the degree of dissipation is sufficiently small (joint work with Nets Katz). These results can be generalized to analogous results for a vector dyadic model (joint work with Susan Friedlander). Also time permitting, we shall present some results for the actual Navier-Stokes equations that are inspired by observing similar phenomena present in dyadic models.

#### Steve Wainger

University of Wisconsin

##### Some discrete operators arising in Harmonic Analysis

A great deal of attention has been given to integral operators acting on functions defined on R(d) where the integration is over a submanifold of R(d) of positive co-dimension. In these talks we will discuss operators acting on functios defined on Z(d)-points in R(d) with integral coordinates. Integration is replaced by summation and the sum is not over all of Z(d), but rather over certain arithmetic subsets of Z(d).

#### Yu Yuan

University of Washington

##### Global solutions to special Lagrangian equations

We survey some recent results on global solutions to special Lagrangian equations. These elliptic equations arise in calibrated geometry and have applications to string theory. The global results have close relation to the regularity theory of the special Lagrangian equations. In the minimal surface equation case, one has the classical Bernstein theorem.

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2004

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April 23-25, 2004

#### H. Brezis

Rutger University

##### New estimates for the Laplacian, the div-curl, and related elliptic systems.

I will present a recent joint work with J. Bourgain concerning new estimates for integrals on loops, for the Laplacian , for the div-curl system, and more general first order elliptic systems in L^1 .

#### Alex Kiselev

University of Wisconsin - Madison

##### Spectrum and dynamics of Schrëodinger operators with decaying potentials.

We review recent progress in spectral and scattering theory of Schrëodinger operators. In particular, we will discuss sharp results on the rate of decay of potential needed for asymptotic completeness of (modified) wave operators in dimension one. The counterexample which shows sharpness of the result involves the construction of potentials which lead to imbeeded singular continuous spectrum. The inspiration for this contruction goes back to the classical Wigner von Neumann example of positive imbedded eigenvalue for a Schrëodinger operator with potential decaying at a Coulomb rate.

#### Alexander Nagel

University of Wisconsin - Madison

##### Regularity of the Kohn-Laplacian in decoupled domains

We obtain optimal estimates for solutions of the Kohn-Laplacian on decoupled domains, where the eigenvalues of the Levi form can degenerate at different rates. In domains with comparable eigenvalues, it is known that the relevant singular integral operators are variants of the standard classical Calderon-Zygmund operators. In contrast, for decoupled domains one is led to the study of operators which are more related to product theory and flag kernels.

#### S.R.S Varadhan

Courant Institute

##### Homogenization of Random Hamilton-Jacobi-Bellman equations and applications to large Deviations in a quenched Random Environment.

The problem of establishing a quenched large deviation principle for a diffusion in a random environment is a special case of the following larger class of problems. Under suitable scaling, the Hamilton-Jacobi-Bellman type equation that describes the optimal value of a controlled diffusion, in a random environment, i.e with a random cost function, is to be replaced by a first order Hamilton-Jacobi equation. We will review the literature and discuss some new results.

#### Sijue Wu

University of Michigan

##### Recent Progress in Mathematical Analysis of Vortex Sheets

The vortex sheet problem serves as a prototype for the evolution of the vorticity in fluid flows. One can think for example of the wake of an airfoil as a typical problem of this type. This problem can be described by the incompressible Euler equation, where the initial vorticity is ideally a finite Radon measure supported on a curve. The issue is to determine the specific nature of the evolution of this curve--the vortex sheet, after the singularity formation time. We answer this question through results on the regularity of the vortex sheet, and the existence and nonexistence of solutions to the initial value problem.

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2003

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April 25-27, 2003

#### Michael Christ

University of California, Berkeley

##### Illposedness of the nonlinear Schrodinger equation

Wellposedness of the nonlinear Schrodinger equation in Sobolev spaces has been extensively studied, and it has been shown that wellposedness holds provided the Sobolev exponent exceeds a certain threshold, depending on the spatial dimension and the degree of nonlinearity.

##### The d-bar Neumann problem, magnetic Schrodinger operators, and the Aharonov-Bohm phenomenon

The d-bar Neumann problem is a non-coercive boundary value problem for Laplace's equation on domains in C^n. The regularity properties of solutions depend on the complex geometry of the boundary, in a manner which has been extensively studied but is still only partially understood. One fundamental problem is to characterize compactness of the Neumann operator. A sufficient condition, known as (P), has been introduced and studied by Catlin and Sibony. We investigate the necessity of condition (P) for the special class of domains possessing a one-dimensional symmetry group. For these domains, both (P) and compactness are equivalent to certain properties of Schrodinger operators with both electric and magnetic fields. The relation between (P) and compactness is then closely linked with diagmagnetic and paramagnetic inequalities. Our main result is an example showing how another effect, quite different from (P), can create compactness. The construction is based on an extreme form of the Aharonov-Bohm effect. However, our example is not at all smooth, and we present strong evidence that for smoothly bounded domains with symmetry, property (P) is equivalent to compactness. This is joint work with S. Fu.

#### Ronald Coifman

Yale University

##### Challenges in Analysis: High Dimensional Geometry and Approximation

The so called curse of dimensionality is well known in statistics and other fields involving dependence on a large number of parameters. In these lectures we make the point that these are issues involving Harmonic Analysis. The talks will discuss effective functional and geometric approximation in high dimensions. In particular we will discuss various issues involved in approximating empirical functions of a large number of parameters including geometric analysis of data sets embedded in high dimensions. Such analysis can be achieved through Harmonic Analysis and operator theory on the data. We will also discuss effective low dimensional functional approximation (around 10-20 dimensions ). These mathematical issues will be illustrated on a variety of examples from biology, chemistry, mutimedia ..

#### Alex Iosevich

University of Missouri-Columbia

##### Analysis and combinatronics of distances set

Let E be a subset of the unit cube in dimensions two or greater. Let D_K(E) denote the set of distances between pairs of elements of E with respect to the distance induced by a convex body K symmetric with respect to the origin.The Falconer Distance Problem (FDP) asks whether one can conclude that D_K(E) has positive Lebesgue measure if the Hausdorff dimension of E is sufficiently large. Let S be a finite discrete subset of Euclidean space in dimensions two or greater. The Falconer Distance Problem can be viewed as a natural continuous analog of the Erdos Distance Problem (EDP) which asks for the smallest possible size of D_K(S) in terms of the size of S. We shall discuss the FDP and EDP, and their dependence on the geometric properties of K. We shall also discuss applications to problems in analysis and geometric combinatorics.

#### Gerd Mockenhaupt

Georgia Institute of Technology

##### On the Hardy-Littlewood majorant property

Hardy and Littlewood observed that Lp-spaces on the torus have the majorant property if p is a positive even integer. For other values of p it is known that the majorant property falls to hold. We will discuss a linearized variant of the majorant problem which relates it to restriction problems for Fourier series to frequency sets E contained in a finite interval [0,N]. One is then asking for bounds on the quantity Bp(E). While for a random selection of a frequency set E contained in an interval of length N the constants Bp(E) are at most of logarithmic growth in N there are sets Ep in [0,N] for whihc one has power growth in N (provided p is not an even integer). This is joint work with Wilhelm Schlag.

#### Camil Muscalu

University of California, Los Angeles

##### Multilinear singular integrals, Part 2

Three years ago, Christoph Thiele gavea talk at the Riviere-Fabes symposium, where he presented a theorem he obtained in collaboration with Terry Tao and myselft, which generalized the results on the bilinear Hilbert transorm. The purpose of my talk is to describe what happened afterwords with this theory og multilinear singular integrals and their Carleson type maximal analogs/ Most of the results are joint work with Terry Tao and Christoph Thiele.

#### Mikhail Safonov

University of Minnesota, Minneapolis

##### Mean value theorem for harmonic functions: some unusual applications

We discuss two applications of the classical mean value theorem (MVT) for harmonic (or subharmonic) functions. The first one is based only on a simple consequence of the MVT, so-called growth theorem, which is also true for more general second order elliptic equations Lu=0. We use it in order to control the boundary behavior of solutions to the equation Lu=f in a bounded domain, where f may blow near the boundary. As another application, we show that the well-known interior Schauder type estimates for solutions to the Poisson equation can be derived on the grounds of the MVT solely. These two facts are the core of the theory of intermediate Schauder estimates, which was developed by D. Gilbarg, L. Hörmander, and J. H. Mikhael. Their methods use a barrier technique, which works only for Lipschitz domains. By our approach, this theory can be extended to more general domains.

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2002

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April 5-7, 2002

#### Principal Speakers

**Professors Daniel Stroock** (MIT) and **Robert Fefferman** (Chicago) gave two lectures each: "A differentiable structure on the space of probability measures" and "Hodge theory on certain non-compact Riemannian manifolds" (Professor Stroock), and "Some issues in harmonic analysis related to the work of Fabes and Rivière on non-isotropic dilations: I) Maximal functions, and II) Singular integrals" (Professor Fefferman).

#### Invited Speakers

**Professor P. Daskalopoulos** (UC Irvine), "Gauss curvature flow with flat sides: geometry and regularity of the interface";

**Professor S. Hoffman** (University of Missouri, Columbia), "The solution of the square root problem of Kato"

**Professor M. Mitrea** (University of Missouri, Columbia), "Elliptic boundary value problems on Sobolev-Besov spaces"

**Professor Terence Tao** (UCLA), "Global regularity of wave maps."