Dissipative Euler flows I
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Speaker: Camillo De Lellis (IAS)
Speaker: Camillo De Lellis (IAS)
Speaker: Camillo De Lellis (IAS)
Abstract: Recently P. Etingof, E. Frenkel, and D. Kazhdan, following earlier contributions by R. Langlands and J. Teschner, described an “analytic” approach to the geometric Langlands correspondence, in which the main ingredients are quantum states and operators acting on them rather than categories and functors. In this talk, I will review the gauge theory approach to the “categorical” version of geometric Langlands, and then, following the paper arXiv:2107.01732 with D. Gaiotto, I will explain how the same ingredients can be arranged differently to give a gauge theory interpretation of the “analytic” version of geometric Langlands.
Speaker: Edward Witten (IAS)
Abstract: In the past decade convex integration has been established as a powerful and versatile technique for the construction of weak solutions of various nonlinear systems of partial differential equations arising in fluid dynamics, including the Euler and Navier-Stokes equations. The existence theorems obtained in this way come at a high price: solutions are highly irregular, non-differentiable, and very much non-unique as there is usually infinitely many of them. Therefore this technique has often been thought of as a way to obtain mathematical counterexamples in the spirit of Weierstrass’ non-differentiable function, rather than advancing physical theory; “pathological”, “wild”, “paradoxical”, “counterintuitive” are some of the adjectives usually associated with solutions obtained via convex integration. In this lecture I would like to draw on some recent examples to show that there are many more sides to the story, and that, with proper usage and interpretation, the convex integration toolbox can indeed provide useful insights for problems in hydrodynamics.
Speaker: László Székelyhidi (IAS, Leipzig)
Congratulations to former Clay Research Fellow Ivan Corwin (2012-2016) who was selected to receive the 2021 Line and Michel Loève International Prize in Probability. The prize is awarded every two years to “recognize outstanding contributions by researchers in probability who are under 45 years old.”
Congratulations to CMI Fellow Amol Aggarwal who recently received the 2021 IAMP Early Career Award. Presented every three years at the International Congress on Mathematical Physics, the prize is awarded in recognition of a single achievement in mathematical physics by a scientist who is under 35 years of age.
Abstract: Measure rigidity theorems (especially Ratner’s theorem) have played a big role in dynamics, with many applications outside the field. Traditionally problems of this type have
been considered in the context of homogeneous dynamics and other systems with a lot of structure. In this talk I will discuss a circle of ideas which suggests that, under some conditions, some measure rigidity theorems may hold for general smooth actions of non- abelian groups.
Speaker: Alex Eskin (Chicago)
Congratulations to Aleksandr Logunov, Simion Filip, Jack Thorne, and Maryna Viazovska who were each awarded a European Mathematical Society Prize at the opening ceremony of the 8th European Congress of Mathematics this morning. Simion and Aleksandr are current Clay Research Fellows, Jack is a former Clay Research Fellow (2012-17), and Maryna won the Clay Research Award in 2017.
Ten European Mathematical Society Prizes are awarded at the European Congress of Mathematics every four years. These prizes recognise outstanding contributions to mathematics by European mathematicians under the age of 35. The list of all the Prize Winners can be found at https://8ecm.si/prizes
The presentation of the Prizes was the highlight of the the first day of the Congress, which was opened by the President of Slovenia, His Excellency Borut Pahor. The Prize Winners were introduced by the President of the Clay Mathematics Institute, Martin Bridson, chair of the Prize Committee for the Congress.
The citations explaining the awards to Simion, Aleksandr, Jack, and Maryna were as follows.
“A European Mathematical Society Prize is awarded to Simion Filipfor breakthroughs that have advanced the understanding of the geometry and dynamics of moduli spaces. Drawing ingeneously on ideas from Hodge theory and dynamical systems, he has resolved a string of major conjectures in the field. He proved that the orbit closures appearing in the celebrated rigidity theorem of Eskin-Mirzakhani-Mohammadi are quasi-projective algebraic varieties; he established a versionof Deligne semisimplicity for the Hodge bundle over every orbit closure; and with Eskin and Wright he extended the rigidity results of Eskin-Mirzakhani to the equivariant action on the natural flat bundle over the moduli space. He is also recognised for his work on the dynamics of the moduli space of complex surfaces, and for his results concerning Lyapunov exponents of higher-rank local systems.”
“A European Mathematical Society Prize is awardedto Aleksandr Logunovfor his fundamental contributions to the analysis of partial differential equations. In a remarkable series of papers (some joint with Eugenia Malinnikova), Logunov introduced new combinatorial-geometric techniques to study solutions of second order ellitpic PDEs. These techniques enabled him to solve longstanding problems of central importance in the field, including Nadirashvili’s conjecture that the zero set of a non-constant harmonic function in n-dimensional space has infinite (n-1)-dimensional volume, and a conjecture of S-T Yau concerning the optimal lower bound on the measure of the nodal sets for eigenfunctions of the Laplacian on compact manifolds.”
“A European Mathematical Society Prize is awardedto Jack Thornefor multiple breakthroughs in diverse areas of algebraic number theory. These include the proof, with Clozel, that the 5th, 6th, 7th and 8th symmetric power L-functions of an elliptic curve over the rational numbers, or of an elliptic modular form, are entire; the construction, with Harris, Lan, and Taylor, of systems of l-adic representations attached to all regular algebraic cuspidal automorphic forms on GL(n) over CM-fields; and the introduction of highly novel techniques and constructions in arithmetic invariant theory, which combine Lie theory with the study of singularities and their deformations, to obtain striking new results on the arithmetic of curves.”
“A European Mathematical Society Prize is awardedto Maryna Viazovskafor her groundbreaking work on discrete optimisation and in particular for her stunning and elegant resolution of the sphere packing problem in dimension 8, and her subsequent work with Cohn, Kumar, Miller and Radchenko resolving the sphere packing problem in dimension 24. She is also recognized for her work resolving significant cases of the Gross-Zagier Conjecture, and for her work with Bodarenko and Radchenko resolving a longstanding conjecture of Korevaar and Meyers on spherical designs. Her novel use of the theory of modular forms has had a dramatic impact in discrete geometry, analytic number theory, and harmonic analysis.”
Congratulations to Clay Research Fellow Ewain Gwynne who has won the inaugural David G Kendall Award for Young Researchers in recognition of his outstanding contributions to Liouville quantum gravity, Schramm-Loewner evolution and the Gaussian free field. The biennial award and lecture, given jointly by the Royal Statistical Society and the Bernoulli Society, recognises research in mathematical statistics and probability theory. It’s named after the first president of the Bernoulli Society and RSS Guy Medal in Gold recipient, David G Kendall.
The Clay Mathematics Institute is pleased to announce that Maggie Miller, Georgios Moschidis, Lisa Piccirillo, and Alexander Smith have been awarded Clay Research Fellowships.
Maggie Miller obtained her PhD in 2020 from Princeton University, where she was advised by David Gabai. She is currently an NSF Postdoctoral Fellow at the Massachusetts Institute of Technology.
Georgios Moschidis obtained his PhD in 2018 from Princeton University, where he was advised by Mihalis Dafermos. Since then he has been a Miller Research Fellow at the University of California, Berkeley.
Lisa Piccirillo obtained her PhD in 2018 from the University of Texas at Austin, where she was advised by John Luecke. She held an NSF Postdoctoral Fellowship at Brandeis University before moving to the Massachusetts Institute of Technology, as a Moore Instructor then Assistant Professor.
Alexander Smith obtained his PhD in 2020 from Harvard University, where he was advised by Noam Elkies and Mark Kisin. He currently holds a National Science Foundation Postdoctoral Fellowship at the Massachusetts Institute of Technology.
PROMYS Europe Connect is calling for applications for its 2021 online programme, to be held from 12 July to 6 August.
In view of continuing restrictions and uncertainty around Covid-19, PROMYS Europe Connect has been designed as a unique 4-week online programme that captures many of the key elements of the usual PROMYS Europe experience. PROMYS Europe is a challenging mathematics summer programme based at the University of Oxford, UK.
PROMYS Europe Connect is seeking
PROMYS Europe Connect is designed to encourage mathematically ambitious students who are at least 16 to explore the creative world of mathematics. Participants will tackle fundamental mathematical questions within a richly stimulating and supportive online community of fellow first-year students, returning students, undergraduate counsellors, research mentors, faculty, and visiting mathematicians.
First-year students will focus primarily on a series of very challenging problem sets, daily lectures, and exploration projects in Number Theory. There will also be a programme of talks, by guest mathematicians and the counsellors, on a wide range of mathematical subjects, as well as courses aimed primarily at students who are returning to PROMYS Europe for a second or third time.
PROMYS Europe is a partnership of Wadham College and the Mathematical Institute at the University of Oxford, the Clay Mathematics Institute, and PROMYS (Program in Mathematics for Young Scientists, founded in Boston in 1989).
The programme is dedicated to the principle that no one should be unable to attend for financial reasons. Most of the cost is covered by the partnership and by generous donations from supporters. In addition, full and partial financial aid is available for those who need it.
Applications for counsellors and students are available on the PROMYS Europe website. The closing date for counsellor applications is 7 February. The closing date for first-year student applications is 14 March, and students will need to allow enough time before the deadline to tackle the application problems. PROMYS Europe Connect will run online from 12 July to 6 August.