Zygmund-Calderón Lectures in Analysis

The Zygmund-Calderón Lectures are named after Antoni Zygmund (1900-1992) and Alberto Calderón (1920-1998). Zygmund was on the faculty of the University of Chicago from 1947 until his retirement in 1980. He received the National Medal of Science in 1986. Calderón was a graduate student in mathematics at the University of Buenos Aires when he met Zygmund in 1948. He became a student of Zygmund's at Chicago, graduating in 1950. He returned as a faculty member from 1959 to 1972 and again from 1975 until his retirement in 1985. The lectures were known as the Zygmund Lectures until Calderón died, at which time they were renamed the Zygmund-Calderón Lectures.

2023-2024 Speaker: Thomas Ducyckaerts (Sorbonne Paris Nord University)

Lecture I. Generalized soliton resolution for nonlinear dispersive equations

According to the soliton resolution conjecture, solutions of nonlinear dispersive equations should behave asymptotically, for infinitely large times, as the sum of decoupled solitary waves and a radiation term. In this lecture, we will first give the history of this conjecture. We will next report on progresses made on focusing wave equations in the last 10 years, including the proof of the soliton resolution for radially symmetric solutions of the energy-critical equation. We will also give examples of solutions that do not fit into a strict soliton resolution scenario, leading to a weaker conjecture on the expected asymptotic behaviors of solutions of dispersive equations.

This lecture can be followed by non-specialist. It is based on joint works with Charles Collot, Carlos E. Kenig, Yvan Martel, Frank Merle, Giuseppe Negro.

Lecture II. On classification of non-radiative solutions for various energy-critical wave equations

In the modern theory of dispersive nonlinear equations, the study of the asymptotic dynamics of solutions is often reduced to the proof of a rigidity theorem, classifying solutions with a specific nondispersive behavior. In this lecture, we will focus on the classification of non-radiative solutions of wave equations, that are solutions such that their energy in the exterior of a wave cone vanishes asymptotically in both time directions. We will see that this classification depends crucially on the dimension, and that it is a powerful tool to solve the soliton resolution conjecture for the energy-critical nonlinear equations in the radial case.

This lecture is based on joint works with Charles Collot, Carlos E. Kenig, Yvan Martel, Frank Merle.

Lecture III. Classification of radial nonlinear waves outside a ball.

This lecture concerns a simple model of nonlinear dispersive equation where one can give a complete description of the dynamics, in the spirit of the soliton resolution conjecture presented in Lecture I. We will consider radial solutions of the focusing non-linear wave equation outside a ball, in dimension 3 of space, with Dirichlet conditions at the boundary. We will show that any global radial solution of the equation is written asymptotically as the sum of a stationary solution and a solution of the linear wave equation, and that the set of initial data leading to a given stationary solution is a submanifold of finite explicit codimension.

This lecture is based on joint work with Jianwei Yang.

2022-2023 Speaker: Tamar Ziegler (Henry and Manya Noskwith Chair in Mathematics at the Einstein Institute of Mathematics of the Hebrew University, Jerusalem.)

Lecture 1: Sign Patterns of the Mobius Function

Monday April 24, E-202, 4pm

The first talk will focus on major advances in dynamics, additive combinatorics, and analytic number theory leading to progress in our understanding of sign patters of the Mobius function. 
The Mobius function is one of the most important arithmetic functions. There is a vague yet well known principle regarding its randomness properties called the “Mobius randomness law". It basically states that the Mobius function should be orthogonal to any "structured" sequence. P. Sarnak suggested a far reaching conjecture as a possible formalization of this principle. He conjectured that "structured sequences" should correspond to sequences arising from deterministic dynamical systems. Sarnak’s conjecture follows from Chowla’s conjecture - which is the mobius version of the prime tuple conjecture. I will describe progress in recent years towards these conjectures.

Lecture 2: Approximate Cohomology

Tuesday, April 25, E-206, 3:30pm

The second talk will describe application of ideas from dynamics and additive combinatorics to a question about approximate homomorphisms. Say that a function f from an abelian group G to Hom(V,W) is an approximate homomorphism if f(x+y)-f(x)-f(y) is of bounded rank uniformly for all x,y in G. Is there a homomorphism h such that (f-h)(x) is of bounded rank for all x in G ? 

Lecture 3: High Strength Varieties

Thursday, April 27, E-206, 3:30pm

In the third talk I will describe work with D. Kazhdan where we obtain results in algebraic geometry, inspired by questions in additive combinatorics, via analysis over finite fields. Specifically we are interested in quantitative properties of polynomial rings that are independent of the number of variables. A sample problem is the following : Let V be a complex vector space, P a polynomial of degree d, and X the null set of P,  X={v|P(v)=0}. Consider a function f:X —> C which is a polynomial of degree d on lines in X. When is f the restriction of a degree d polynomial on V? A key tool is a universality property satisfied by high strength varieties.

Lecture 1: Singularity Formation in the Super Critical Range


The 6th Clay problem has a very simple formulation: may incompressible fluids form singularities? In dimension two, the problem is critical and Leray’s answer is 100 years old: no. In dimension three, the problem is open and belongs to the class of super critical problems. In this first lecture, I will illustrate the concept of criticality on canonical non linear models, including fluid and dispersive type equations, and describe classical open problems in the field.

Lecture 2: Describing Singularities


In the last forty years, an immense amount of energy has been put into the search for singularities of non linear waves and their description using very different approaches: formal computations, numerical evidence, rigorous derivations, ... In the last ten years, a unified vision has emerged which I will present to describe all known mechanisms of singularity formation. This will motivate the search for new unexpected scenarii.

Lecture 3: Defocusing Singularities and Compressible Fluids 


The Non Linear Schrodinger equation with defocusing non linearity is a classical model for which singularity formation was explicitely conjectured not to occur in the energy super critical range. We will claim the opposite: in some regimes, blow up can occur. The structure of this new class of singularities is deeply related to compressible fluid mechanics and the description of large compressible shock waves, in connection with a classical problem: how may viscosity influence the formation of fluid singularities? I will present a series of joint works with Merle (IHES), Rodnianski (Princeton) and Szeftel (Paris) which open a breach for the study of large fluid singularities.

Lecture 1: Area-Mininimizing Surfaces


The famous Plateau problem concerns surfaces which minimize the area among those spanning a given contour. It is known since long that in many situations we cannot expect that the minimizer is completely regular. In particular the problem takes different aspects depending on what meaning we give to the word ``surfaces'', ``area'', and ``spanning''. In this lecture I will examine one of the most popular approaches to the Plateau problem, that of currents, and in particular the progress that has been made thus far in the analysis of the singular set the minimizers, at the boundary (i.e. on the assigned contour) and in the interior (i.e. away from it). I will in particular emphasize the difference between two type of singularities: those of conical type, which are much easier to detect, and those of branching type, which are instead much harder and were first tackled in the late seventies in the pioneering work of Almgren.

Lecture 2: Harmonic Multivalued Functions


Harmonic multivalued functions have been introduced in the late seventies by Almgren to analyze branching singularities of area-minimizing oriented surfaces (technically integral currents) in codimension higher than 1. Almgren's theory has been recently expanded in several directions, in particular different notions of multivalued functions are needed to analyze singularities in different situations. I will review several recent progress, their applications, and some of the most challenging remaining open questions.

Lecture 3: Center Manifolds


A pivotal role in the analysis of branching singularities is played by the ``center manifolds''. Roughly speaking a center manifold is a surface which is equidistant from the various branches of a minimal surface in a neighborhood of a branching singularity. I will give an intuition on how such an object can be constructed in different situations, which challenges one must overcome, and why it plays such in important role in different contexts.

Lecture 1: From spectral to spatial delocalization: quantum ergodicity on graphs

Monday, November 12, 2018, 2:00pm-2:50pm, Eckhart 207


A hundred years ago, Einstein wondered about quantization conditions for classically ergodic systems. His 1917 paper may be considered to be the starting point of the field of ``quantum chaos''. Although a mathematical description of the spectrum of Schrödinger operators associated to ergodic classical dynamics is still missing, a lot of progress has been made on the delocalization of the associated eigenfunctions.

In Lecture I, we will look at notions of delocalization for eigenfunctions of discrete Schrödinger operators on graphs. We will mostly discuss the notion of ``quantum ergodicity'', but will find connections with other notions used in the theory of Anderson (de)localization.

Lecture 2: Topics in quantum chaos

Monday, November 12, 2018, 4:00pm-5:00pm, Eckhart 202


In Lecture II, we will survey the main conjectures of the domain and describe recent advances on the delocalization of eigenfunctions of the laplacian of negatively curved manifolds.

Lecture 3: Delocalization on random regular graphs

Tuesday, November 13, 2018, 1:00- 1:50pm, Eckhart 207


In Lecture I, we only discuss deterministic graphs. In Lecture III, we will discuss recent advances for random graphs, mostly by Bauerschmidt-Huang-Knowles-Yau and by Backhausz-Szegedy.

Lecture 1: Taming infinities

Monday, October 25, 2016, 4:30pm–5:30pm, Ryerson 251

Abstract: Some physical and mathematical theories have the unfortunate feature that if one takes them at face value, many quantities of interest appear to be infinite! Various techniques, usually going under the common name of “renormalisation” have been developed over the years to address this, allowing mathematicians and physicists to tame these infinities. We will tip our toes into some of the mathematical aspects of these techniques and we will see how they have recently been used to make precise analytical statements about the solutions of some equations whose meaning was not even clear until now.

Lectures 2 and 3: The BPHZ theorem for stochastic PDEs

Tuesday, October 26, 2016, 4:30pm–5:30pm, Eckhart 202

Wednesday, October 27, 2016, 4pm–5pm, Eckhart 202

Abstract: The Bogoliubov-Parasiuk-Hepp-Zimmermann theorem is a cornerstone of perturbative quantum field theory: it provides a consistent way of "renormalising" the diverging integrals appearing there to turn them into bona fide distributions. Although the original article by Bogoliubov and Parasiuk goes back to the late 50s, it took about four decades for it to be fully understood. In the first lecture, we will formulate the BPHZ theorem as a purely analytic question and show how its solution arises very naturally from purely algebraic considerations. In the second lecture, we will show how a very similar structure arises in the context of singular stochastic PDEs and we will present some very recent progress on its understanding, both from the algebraic and the analytical point of view.

Lecture 1: Min-max theory for the area functional - a panorama

Tuesday, January 19, 2016, 4:30pm–5:30pm, Eckhart 133

Abstract: In this talk we will give a current panorama of the min-max theory for the area functional, initially devised by Almgren in the 1960s and improved by Pitts (1981). This is a deep high-dimensional version of the variational theory of closed geodesics. The setting is very general, being that of Geometric Measure Theory, and the main application until very recently was the construction of minimal varieties of any dimension in a compact Riemannian manifold. In the past few years we have discovered new applications of this old theory, including a proof of the Willmore conjecture, of the Freedman-He-Wang conjecture, and of Yau's conjecture (about the existence of infinitely many minimal hypersurfaces) in the positive Ricci curvature setting. We will give an overview of these results and describe open problems and future directions. Most of the material covered in these lectures is based on joint work with Andre Neves.

Lecture 2: Multiparameter sweepouts and a proof of the Willmore conjecture

Wednesday, January 20, 2016, 4pm–5pm, Eckhart 202

Abstract: In 1965, T. J. Willmore conjectured what should be the optimal shape of a torus immersed in three-dimensional Euclidean space. He predicted that the Clifford torus, more precisely a stereographic projection of it, is the minimizer of the Willmore energy - the total integral of the square of the mean curvature. This is a conformally invariant problem. In this lecture we will explain a proof of this conjecture that exploits a connection with the min-max theory of minimal surfaces. A crucial ingredient is the discovery of new five-parameter sweepouts of surfaces in the three-sphere that turn out to be homotopically nontrivial. The solution is based on the study of the geometric and topological properties of such families. We will also describe how to prove the Freedman-He-Wang conjecture (joint with I. Agol and A. Neves) about the Möbius energy of links using similar ideas.

Lecture 3: The case of fluctuations around a global equilibrium.

Thursday, January 21, 2016, 4:30pm–5:30pm, Eckhart 202

Abstract: The space of cycles in a compact Riemannian manifold has very rich topological structure. The space of hypercycles, for instance, taken with coefficients modulo two, is weakly homotopically equivalent to the infinite dimensional real projective space. We will explain how to use this structure, together with Lusternik-Schnirelman theory and work of Gromov and Guth, to prove that every compact Riemannian manifold of positive Ricci curvature contains infinitely many embedded minimal hypersurfaces. Then we will discuss more recent work in which we prove the first Morse index bounds of the theory. The main difficulty comes from the problem of multiplicity, which we are able to settle in the classical one-parameter case.

Past Zygmund Lecturers include: Charles Fefferman, Jean-Pierre Kahane, Elias Stein, Yves Meyer, Donald L. Burkholder, Lennart Carleson, Luis Caffarelli, Louis Nirenberg, Jean Bourgain, Nicolai Krylov, Rick Schoen, Thomas Wolff, Ronald Coifman, Eugene Fabes, and Guido Weiss. The Zygmund-Calderón lecturers have been: Charles Fefferman and Elias Stein (who shared the lectures in the first year of their new existence), W. Timothy Gowers, Peter Jones, Terry Tao, Michael Christ, Stephen Wainger, Ben Green, Frank Merle, Alice Chang, Gunther Uhlmann, Maciej Zworski, Assaf Naor, Demetrios Christodoulou, Vladimir Sverak, and Laure Saint-Raymond.