## 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.

### 2018-2019 Speaker: Nalini Anantharaman (IRMA, University of Strasbourg, France)

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

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

**Abstract**:

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

**Abstract**:

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

**Abstract**:

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.

### 2016-2017 Speaker: Martin Hairer (University of Warwick)

#### 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.

### 2015-2016 Speaker: Fernando Codá Marques (Princeton University)

#### 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.