Viewing archives for 2017

Algebraic geometry, categories and trace forumulas

Abstract: In this talk I will present an approach to non-commutative algebraic geometry based on the following principle: a non-commutative variety simply is a category. This principle seems naive at first sight, but it has been very fruitful during the last 20 years, thanks to the works of many authors such as Kapranov, Bondal-Orlov, Rosenberg, Van den Bergh, Artin-Zhang, Konstevich-Soibelman, Keller, etc.

The purpose of this lecture is to explain how we can (or can not) “do geometry” with categories with a particular focus on cohomological and numerical aspects: Euler characteristic, Lefschetz’s type trace formula, etc. I will illustrate this by exploring its interactions with singularity theories, both in the classical complex analytic situation and in more arithmetic settings. In the last part of the lecture I will explain how this approach to non-commutative geometry can be used in order to make progress on the, still conjectural, Bloch’s conductor formula.

Details

Speaker: Bertrand Toën (Toulouse)

Introduction to decoupling

Abstract: In the last few years, Jean Bourgain and Ciprian Demeter have proven a variety of striking “decoupling” theorems in Fourier analysis. The method has applications both in PDE and in number theory. In particular, it resolved a question about diophantine equations raised by Vinogradov in the 1930s.

The proof of decoupling uses geometric ideas.  In particular, it gets a lot of mileage out of studying the problem at many different scales.

In this lecture, I will introduce decoupling and explain how it connects to number theory, and then I will sketch some of the ideas of the proof.

Details

Speaker: Larry Guth (MIT)

LMS Whitehead Prize

Former Clay Research Fellow Jack Thorne has been awarded a Whitehead Prize by the London Mathematical Society in recongnition of his contributions to number theory, and in particular to the Langlands program.

Landon T Clay 1926 – 2017

Landon Clay died peacefully at home in Peterborough, NH on July 29, surrounded by his wife and four sons.  His passing will be mourned by all who benefited from his vision and his generosity.

Obituary

Maryam Mirzakhani

It is with great sadness that we heard of the death earlier today of Maryam Mirzakhani.  Maryam was appointed to a Clay Research Fellowship in 2004 and was given a Clay Research Award in 2014 for her many and significant contributions to geometry and ergodic theory. The thoughts of the whole mathematical community will be with her family on this tragic day.

ICM 2018

Congratulations to Clay Research Fellows June Huh, James Maynard, John Pardon, Jack Thorne, and Miguel Walsh who have been invited to speak at the International Congress of Mathematicians in Rio de Janeiro in 2018.  Congratulations also to former Research Fellow Peter Scholze, CMI Scientific Advisory Board members Simon Donaldson and Andrei Okounkov, and Clay Research Awardees Rahul Pandharipande and Geordie Williamson who have been invited as Plenary Speakers.

Copley Medal

Andrew Wiles has been awarded the Copley Medal, the Royal Society’s oldest and most prestigious award. The medal is awarded annually for outstanding achievements in research in any branch of science and alternates between the physical and biological sciences.

Andrew Wiles is one of the world’s foremost mathematicians. His proof of Fermat’s Last Theorem the 1990s catapulted him to unexpected fame as both the mathematical and the wider world were gripped by the solving of a 300 year-old mystery. In 1637 Fermat had stated that there are no whole number solutions to the equation xn + yn = zn when n is greater than 2, unless xyz = 0. Fermat went on to claim that he had found a proof for the theorem, but said that the margin of the text he was making notes on was not wide enough to contain it.

After seven years of intense study in private at Princeton University, Andrew announced he had found a proof in 1993, combining three complex mathematical fields – modular forms, elliptic curves and Galois representations. He had not only solved the long-standing puzzle of the Theorem, but in doing so had created entirely new directions in mathematics. 

Andrew is a member of CMI’s Scientific Advisory Board and recipient of the 1999 Clay Research Award.

Alan T. Waterman Award

Clay Research Fellow John Pardon has been recognised by the National Science Foundation (NSF) with the  Alan T. Waterman Award.  Bestowed annually, the Waterman Award recognises outstanding researchers age 35 and under in NSF-supported fields of science and engineering.

2017 Clay Research Fellows

The Clay Mathematics Institute is pleased to announce that Peter Hintz and Akhil Mathew have been awarded Clay Research Fellowships.  

Peter Hintz studies hyperbolic partial differential equations arising in general relativity using methods from microlocal analysis, spectral and scattering theory, and dynamical systems. He obtained his PhD at Stanford in 2015, advised by András Vasy.

Akhil Mathew’s research focuses on homotopy theory, higher categories, and their applications. He is currently in his final year as a graduate student at Harvard University, working under the supervision of Jacob Lurie.

Clay Research Fellowships are awarded on the basis of the exceptional quality of candidates’ research and their promise to become mathematical leaders.