
Mathematical problems in fluid dynamics
Organizers: Thomas Alazard (Ecole Normale Supérieure ParisSaclay; Centre National de la Recherche Scientifique (CNRS)), Hajer Bahouri (Centre National de la Recherche Scientifique (CNRS); Institut de Mathématiques de Jussieu), Mihaela Ifrim (University of WisconsinMadison), Igor Kukavica (University of Southern California), David Lannes (Université de Bordeaux I; Centre National de la Recherche Scientifique (CNRS)), LEAD Daniel Tataru (University of California, Berkeley)Fluid dynamics is one of the classical areas of partial differential equations, and has been the subject of extensive research over hundreds of years. It is perhaps one of the most challenging and exciting fields of scientific pursuit simply because of the complexity of the subject and the endless breadth of applications.
The focus of the program is on incompressible fluids, where water is a primary example. The fundamental equations in this area are the wellknown Euler equations for inviscid fluids, and the NavierStokes equations for the viscous fluids. Relating the two is the problem of the zero viscosity limit, and its connection to the phenomena of turbulence. Water waves, or more generally interface problems in fluids, represent another target area for the program. Both theoretical and numerical aspects will be considered.
Updated on Apr 25, 2019 02:32 PM PDT 
Universality and Integrability in Random Matrix Theory and Interacting Particle Systems
Organizers: LEAD Ivan Corwin (Columbia University), Percy Deift (New York University, Courant Institute), Ioana Dumitriu (University of California, San Diego), Alice Guionnet (École Normale Supérieure de Lyon), Alexander Its (Indiana UniversityPurdue University Indianapolis), Herbert Spohn (Technische Universität München), HorngTzer Yau (Harvard University)The past decade has seen tremendous progress in understanding the behavior of large random matrices and interacting particle systems. Complementary methods have emerged to prove universality of these behaviors, as well as to probe their precise nature using integrable, or exactly solvable models. This program seeks to reinforce and expand the fruitful interaction at the interface of these areas, as well as to showcase some of the important developments and applications of the past decade.
Updated on Apr 20, 2020 11:12 AM PDT 
The Analysis and Geometry of Random Spaces
Organizers: LEAD Mario Bonk (University of California, Los Angeles), Joan Lind (University of Tennessee), Steffen Rohde (University of Washington), Eero Saksman (University of Helsinki), Fredrik Viklund (Royal Institute of Technology), JangMei Wu (University of Illinois at UrbanaChampaign)This program is devoted to the investigation of universal analytic and geometric objects that arise from natural probabilistic constructions, often motivated by models in mathematical physics. Prominent examples for recent developments are the SchrammLoewner evolution, the continuum random tree, Bernoulli percolation on the integers, random surfaces produced by Liouville Quantum Gravity, and Jordan curves and dendrites obtained from random conformal weldings and laminations. The lack of regularity of these random structures often results in a failure of classical methods of analysis. One goal of this program is to enrich the analytic toolbox to better handle these rough structures.
Updated on Nov 20, 2019 02:12 PM PST 
Complex Dynamics: from special families to natural generalizations in one and several variables
Organizers: LEAD Sarah Koch (University of Michigan), Jasmin Raissy (Institut de Mathématiques de Toulouse), Dierk Schleicher (Université d'AixMarseille (AMU)), Mitsuhiro Shishikura (Kyoto University), Dylan Thurston (Indiana University)Holomorphic dynamics is a vibrant field of mathematics that has seen profound progress over the past 40 years. It has numerous interconnections to other fields of mathematics and beyond.
Our semester will focus on three selected classes of dynamical systems: rational maps (postcritically finite and beyond); transcendental maps; and maps in several complex variables. We will put particular emphasis on the interactions between each these, and on connections with adjacent areas of mathematics.
Updated on Nov 20, 2019 02:12 PM PST 
Floer Homotopy Theory
Organizers: Mohammed Abouzaid (Columbia University), Andrew Blumberg (University of Texas, Austin), Kristen Hendricks (Rutgers University), Robert Lipshitz (University of Oregon), LEAD Ciprian Manolescu (Stanford University), Nathalie Wahl (University of Copenhagen)The development of Floer theory in its early years can be seen as a parallel to the emergence of algebraic topology in the first half of the 20th century, going from counting invariants to homology groups, and beyond that to the construction of algebraic structures on these homology groups and their underlying chain complexes. In continuing work that started in the latter part of the 20th century, algebraic topologists and homotopy theorists have developed deep methods for refining these constructions, motivated in large part by the application of understanding the classification of manifolds. The goal of this program is to relate these developments to Floer theory with the dual aims of (i) making progress in understanding symplectic and lowdimensional topology, and (ii) providing a new set of geometrically motivated questions in homotopy theory.
Updated on Nov 25, 2019 01:27 PM PST 
Analytic and Geometric Aspects of Gauge Theory
Organizers: Laura Fredrickson (Stanford University), Rafe Mazzeo (Stanford University), Tomasz Mrowka (Massachusetts Institute of Technology), Laura Schaposnik (University of Illinois at Chicago), LEAD Thomas Walpuski (HumboldtUniversität)The mathematics and physics around gauge theory have, since their first interaction in the mid 1970’s, prompted tremendous developments in both mathematics and physics. Deep and fundamental tools in partial differential equations have been developed to provide rigorous foundations for the mathematical study of gauge theories. This led to ongoing revolutions in the understanding of manifolds of dimensions 3 and 4 and presaged the development of symplectic topology. Ideas from quantum field theory have provided deep insights into new directions and conjectures on the structure of gauge theories and suggested many potential applications. The focus of this program will be those parts of gauge theory which hold promise for new applications to geometry and topology and require development of new analytic tools for their study.
Updated on Feb 05, 2020 10:22 AM PST 
Algebraic Cycles, LValues, and Euler Systems
Organizers: Henri Darmon (McGill University), Ellen Eischen (University of Oregon), LEAD Benjamin Howard (Boston College), David Loeffler (University of Warwick), Christopher Skinner (Princeton University), Sarah Zerbes (University College London), Wei Zhang (Massachusetts Institute of Technology)The fundamental conjecture of Birch and SwinnertonDyer relating the Mordell–Weil ranks of elliptic curves to their Lfunctions is one of the most important and motivating problems in number theory. It resides at the heart of a collection of important conjectures (due especially to Deligne, Beilinson, Bloch and Kato) that connect values of Lfunctions and their leading terms to cycles and Galois cohomology groups.
The study of special algebraic cycles on Shimura varieties has led to progress in our understanding of these conjectures. The arithmetic intersection numbers and the padic regulators of special cycles are directly related to the values and derivatives of Lfunctions, as shown in the pioneering theorem of GrossZagier and its padic avatars for Heegner points on modular curves. The cohomology classes of special cycles (and related constructions such as Eisenstein classes) form the foundation of the theory of Euler systems, providing one of the most powerful methods known to prove vanishing or finiteness results for Selmer groups of Galois representations.
The goal of this semester is to bring together researchers working on different aspects of this young but fastdeveloping subject, and to make progress on understanding the mysterious relations between Lfunctions, Euler systems, and algebraic cycles.
Updated on Feb 25, 2020 11:41 AM PST

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