Zygmund-Calderón Lectures in Analysis


2011-2012 Speaker: Demetrios Christodoulou (ETH, Zurich)



Lecture 1: Hyperbolic p.d.e. and Lorentzian geometry


Monday May 7, 4-5 PM in Ryerson 251

In the first part of the lecture I shall begin with a discussion of how Euler-Lagrange systems of partial differential equations of hyperbolic type arise in classical continuum physics. I shall proceed to discuss the causal structure of spacetime, that is, of the manifold of independent variables, defined by a solution of the system of equations, and how Lorentzian geometry comes into play. In the second part of the lecture I shall focus on what are perhaps the two most important Euler-Lagrange systems of partial differential equations of hyperbolic type, namely the Euler equations of fluid mechanics, and the Einstein equations of general relativity. The Euler equations govern the motion of a perfect fluid and were first formulated in 1756, but despite the lapse of two and a half centuries, we are still far from adequately understanding the observed phenomena which are supposed to lie within their domain of validity. Among these phenomena are the formation and evolution of shocks in compressible fluids. This shall be the topic of my second and third lecture. The Einstein equations of general relativity, formulated in 1915, govern the geometry of physical spacetime itself, gravitation being the manifestation of spacetime curvature. I shall briefly discuss the global stability of Minkowski spacetime, the trivial solution of the Einstein equations, proved in my joint work with Klainerman (published in 1993), and contrast this with what happens in the case of the Euler equations, where there is a family of trivial solutions, the constant states. A basic notion in general relativity is the concept of a trapped surface. The existence of such a surface in a spacetime implies according to a 1965 theorem of Penrose that the spacetime must come to an end. It also implies that there is a region of the spacetime, called a black hole, which is inaccessible to observation from infinity. The formation of trapped surfaces shall be the topic of my fourth lecture.

Lecture 2: The analysis of shock formation in 3-dimensional fluids, Part A


Tuesday May 8, 4:30-5:30 PM in Eckhart 206

See combined abstract with Lecture 3 below.

Lecture 3: The analysis of shock formation in 3-dimensional fluids, Part B


Wednesday May 9, 4-5 PM in Ryerson 251

In 2007 I published a monograph which treated the relativistic Euler equations in three space dimensions for a perfect fluid with an arbitrary equation of state. In this monograph I considered initial data which outside a sphere coincide with the data corresponding to a constant state. Under a suitable restriction on the size of the initial departure from the constant state, I established theorems which gave a complete description of the maximal classical development. In particular, I showed that the boundary of the domain of the maximal classical development has a singular part where the inverse density of the wave fronts vanishes, signalling shock formation. In fact, the theorems which I established gave a complete picture of shock formation in three-dimensional fluids. In my lectures I shall give a simplified presentation of these results, assuming from the outset that the initial conditions are irrotational. The basic geometric concept on which the analysis is based is that of the acoustical spacetime manifold. The analysis features the interplay of the original system of equations with another system, the acoustical structure equations, which governs the causal structure of the acoustical manifold. The acoustical geometry degenerates as shocks form, nevertheless things remain smooth relative to a different differential structure, which is what permits a complete analysis of the singular boundary.

Lecture 4: The short pulse method and the formation of trapped surfaces in general relativity


Thursday May 10, 3-4 PM in Ryerson 251

In 1965 Penrose introduced the fundamental concept of a trapped surface, on the basis of which he proved a theorem which asserts that a spacetime containing such a surface must be incomplete. The presence of a trapped surface implies, moreover, that there is a region of spacetime, the black hole, which is inaccessible to observation from infinity. A major challenge since that time had been to find out how trapped surfaces actually form, by analyzing the dynamics of gravitational collapse. In a monograph published in 2009 I achieved this aim by establishing the formation of trapped surfaces in pure general relativity through the focusing of incoming gravitational waves. The theorems proved therein constitute the first foray into the long-time dynamics of general relativity in the large, that is, when the initial data are no longer confined to a suitable neighborhood of trivial data. The main new method which this work introduces, the short pulse method, applies to general systems of Euler-Lagrange equations of hyperbolic type, and provides means to tackle problems which have hitherto been inaccessible. The method capitalizes on the assumption that the initial data, although smooth, change abruptly as we cross a certain surface, so there is a small parameter which corresponds to the distance within which the change is effected. A calculus is built in which this small parameter everywhere enters. This calculus is used to demonstrate that when the parameter in question is suitably small we have long time existence independently of the size of the initial data. And in the case of the Einstein equations, when this size is suitably large then trapped surfaces eventually form.

Previous years

2010-2011: Assaf Naor (Courant Institute)



Nonlinear Dvoretzky theory

April 14 at 4:30pm in Ry 251

The classical Dvoretzky theorem asserts that for every integer k>1 and every target distortion D>1 there exists an integer n=n(k,D) such that any n-dimensional normed space contains a subspace of dimension k that embeds into Hilbert space with distortion D. Variants of this phenomenon for general metric spaces have been studied for 25 years, with a variety of applications. In this talk we will discuss the solution of the nonlinear Dvoretzky problem of Bourgain-Figiel-Milman, as well as more recent work on Tao's Dvoretzky problem for Hausdorff dimension. A sample result along these lines (obtained jointly with Manor Mendel) is that for every epsilon > 0, any n-point metric space has a subset of size n^{1-epsilon} which embeds into Hilbert space with distortion O(1/epsilon); a result that is optimal up to constant factors. We will also describe subtle connections between nonlinear Dvoretzky theory and theoretical computer science, as well as the appearance of a variety of probabilistic tools in the study of such problems, including random walks on metric spaces and randomized Calderon-Zygmund decompositions.


Towards a calculus for non-linear spectral gaps

April 15 at 4pm in E 206

The spectral gap of a symmetric stochastic matrix is the reciprocal of the best constant in its associated Poincare inequality. This inequality can be formulated in purely metric terms, where the metric is Hilbertian. This immediately allows one to define the spectral gap of a matrix with respect to other, non- Euclidean, geometries; a standard procedure that is used a lot in embedding theory, most strikingly as a method to prove non-embeddability in the coarse category. Motivated by a combinatorial approach to the construction of bounded degree graph families which do not admit a coarse embedding into any uniformly convex normed space (such spaces were first constructed by V. Lafforgue), we will naturally arrive at questions related to the behavior of non-linear spectral gaps under graph operations such as powering and zig-zag products. We will also discuss the issue of constructing base graphs for these iterative constructions, which leads to new analytic and geometric challenges.

Joint work with Manor Mendel


The Ribe program

April 18 at 4pm in E 202

A 1976 theorem of Ribe asserts that finite dimensional linear properties of normed spaces are preserved under uniformly continuous homemorphisms. This important rigidity result inspired a research program, known today as the Ribe program, which was formulated by Bourgain in 1986. Its goal is to find explicit manifestations of the phenomenon discovered by Ribe, namely, to reformulate key linear properties of Banach spaces using only the notion of distance, i.e., without referring to their linear structure in any way. Once this is done, one can attempt to study when metric spaces which have nothing to do with linear spaces have these properties. Over the past 25 years, this point of view led to various interesting results in metric geometry. In this talk we will explain some of the milestones of the Ribe program, describe some recent progress, and discuss some challenging problems that remain open.

2009-2010 Speaker: Daniel Tataru (Berkeley)



Energy critical dispersive equations

March 5 at 4pm in RY 251

Nonlinear wave and Schroedinger equations model arise in the modeling of many physical phenomena. In the study of both local and global behavior of solutions to such equations, the game being played is between the linear, dispersive effects, and the nonlinear, ode type dynamics introduced by the nonlinearities. In many instances one or the other of these factors has the upper hand. However, for energy critical equations these two driving forces are in a delicate balance. The aim of this talk is to provide an overview of the ideas, results and open problems in this very active area of current research.


Global solutions for defocusing problems

March 8 at 4pm in E 202

In this talk we consider a subclass of the family of energy critical dispersive equations, namely that of defocusing problems. These are equations for which the nonlinear dynamics act in conjunction with the linear flow to contribute to the dispersion of the solutions. Several problems will be discussed, beginning with the nonlinear wave equation and NLS, and continuing with a more in depth presentation of wave maps and Schroedinger maps.


Near soliton dynamics for focusing problems: from stability to blow-up

March 9 at 4:30pm in E 206

In brief, focusing problems are those for which the nonlinear effects can contribute to the concentration of the solutions. A common feature of focusing energy critical dispersive equations is that they admit nontrivial steady states (solitons); these are non-unique due to scaling. This talk will be devoted to the analysis of the dynamics of solutions which stay close to the soliton family. For different problems these dynamics exhibit widely different properties, from stability to blow-up; these properties are closely related to the spectral resolution for the linearized evolution. Problems to be discussed include the nonlinear wave equation, wave maps, Yang-Mills and Schroedinger maps.

2008-2009 Speaker: Maciej Zworski

What is Microlocal Analysis?

Monday April 20th 4:00pm-5:00pm in E 202

One possible answer is that microlocal analysis explores the classical/quantum correspondence in the theory of partial differential equations. Its historical origins in mathematics were rather different, and the Calderon-Zygmund theory of singular integrals played a central role in the development of the subject. But over the years the parallels with the classical/quantum correspondence have become a unified theme of the subject in its many guises.

I will attempt to present this unifying theme by discussing the following questions (with the answers provided by the listed authors):
- Can random matrices enhance classical/quantum correspondence? (Bordeaux, Hager, Sjostrand)
- Do smooth (non-analytic) functions appear in nature? (Melrose, Taylor, Lebeau)
- Does considering primes as semiclassical parameters, p = hbar, make sense? (Belov-Kanel, Kontsevitch)


What are Quantum Resonances?

Tuesday April 21st 4:30pm-5:30pm in E 206

Quantum resonances describe metastable states and replace eigenvalues in more realistic situations in which decay or escape are possible. Under many names (scattering poles, quasinormal modes, zeros of zeta functions) they appear in many branches of mathematics and physics: typically, the real part is interpreted as rest energy or frequency and the imaginary part as the rate of decay. I will give clear mathematical definitions in some simple settings illustrated by "live" numerical experiments. I will also present some applications of results on resonances to linear and non-linear PDEs.


Microlocal Methods in the Study of Resonances.

Wednesday April 22nd 4:15pm-5:15pm in E 203

Metastable states are both intuitively and mathematically associated with trapping: that is already seen in calling eigenstates "bound states". Trapping is a geometric or classical condition and the study of the relation between trapping and the distribution of resonances has been successfully explored by microlocal methods. I will discuss recent results on chaotic scattering: the fractal Weyl laws for the density of resonances and estimating quantum decay rates using topological pressure of the dynamical system. These results have been obtained in joint work with Sjostrand, and with Nonnenmacher. They will also be illustrated using numerical and experimental results.

2007-2008 Speaker: Gunther Uhlman

Visibility and Invisibility

Monday April 14, 4:00 pm, Room 202 Eckhart Hall, 1118 E 58th Street

We give a general introduction to visibility and invisibility in relation to Calderón's inverse problem. Visibility refers to finding the electrical conductivity of a medium by making voltages and current measurements at the boundary. This problem arose in geophysical prospection and it has been proposed as a diagnostic tool in medical imaging, particular early breast cancer detection. Invisibility refers to finding non- homogeneous conductivities that give rise to the same boundary measurements as an homogeneous conductivity. The conductivity equation has transformation laws that allow for design of conductivities that steer the currents around a hidden region, returning it to its original path on the far side.


Travel Time Tomography

Tuesday April 15, 4:30 pm, Room 206 Eckhart Hall, 1118 E 58th Street

In this lecture we will describe a surprising connection between Calderón's inverse problem and travel time tomography. This latter problem consists in determining the index of refraction (sound speed) of a medium by measuring the travel times of sound waves going though the medium. This problem arises in determining the inner structure of the Earth by measuring the travel times of seismic waves as well as in ultrasound imaging.


Invisibility

Wednesday April 16, 4:00pm, Room 202 Eckhart Hall, 1118 E 58th Street

We describe recent theoretical and experimental progress on making objects invisible to detection by electromagnetic waves, acoustic waves and quantum waves. Similar to the conductivity equation, Maxwell's equations and Helmholtz equations have transformation laws that allow for design of electromagnetic material parameters that steer light around a hidden region, returning it to its original path on the far side. Not only would observers be unaware of the contents of the hidden region, they would not even be aware that something was being hidden. The object, which would have no shadow, is said to be cloaked. We recount the recent history of the subject and discuss some of the mathematical and physical issues involved.