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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.

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.