January 14, 2013
to May 24, 2013
Organizers: Mike Artin (Massachusetts Institute of Technology), Viktor Ginzburg (University of Chicago), Catharina Stroppel (Universität Bonn , Germany), Toby Stafford* (University of Manchester, United Kingdom), Michel Van den Bergh (Universiteit Hasselt, Belgium), Efim Zelmanov (University of California, San Diego)
Over the last few decades noncommutative algebraic geometry (in its many forms) has become increasingly important, both within noncommutative algebra/representation theory, as well as having significant applications to algebraic geometry and other neighbouring areas. The goal of this program is to explore and expand upon these subjects and their interactions. Topics of particular interest include noncommutative projective algebraic geometry, noncommutative resolutions of (commutative or noncommutative) singularities,Calabi-Yau algebras, deformation theory and Poisson structures, as well as the interplay of these subjects with the algebras appearing in representation theory--like enveloping algebras, symplectic reflection algebras and the many guises of Hecke algebras.
August 19, 2013
to December 20, 2013
Organizers: Yvonne Choquet-Bruhat (University of Paris), Piotr T. Chruściel (University of Vienna), Greg Galloway (University of Miami), Gerhard Huisken (Albert Einstein Institute), James Isenberg* (University of Oregon), Sergiu Klainerman (Princeton University),Igor Rodniansky (Princeton University), Richard Schoen (Stanford University)
The study of Einstein's general relativistic gravitational field equation, which has for many years played a crucial role in the modeling of physical cosmology and astrophysical phenomena, is increasingly a source for interesting and challenging problems in geometric analysis and PDE. In nonlinear hyperbolic PDE theory, the problem of determining if the Kerr black hole is stable has sparked a flurry of activity, leading to outstanding progress in the study of scattering and asymptotic behavior of solutions of wave equations on black hole backgrounds. The spectacular recent results of Christodoulou on trapped surface formation have likewise stimulated important advances in hyperbolic PDE. At the same time, the study of initial data for Einstein's equation has generated a wide variety of challenging problems in Riemannian geometry and elliptic PDE theory. These include issues, such as the Penrose inequality, related to the asymptotically defined mass of an astrophysical systems, as well as questions concerning the construction of non constant mean curvature solutions of the Einstein constraint equations. This semester-long program aims to bring together researchers working in mathematical relativity, differential geometry, and PDE who wish to explore this rapidly growing area of mathematics.
August 19, 2013
to December 20, 2013
Organizers: Luigi Ambrosio (Scuola Normale Superiore di Pisa), Yann Brenier (CNRS, Universit´e de Nice), Panagiota Daskolopoulos (Columbia University), Lawrence C Evans (University of California at Berkeley), Alessio Figalli (UT Austin), Wilfrid Gangbo (Georgia Institute of Technology), Robert J McCann* (University of Toronto), Felix Otto (Universit¨at Bonn), and Neil S Trudinger (Australian National University)
In the past two decades, the theory of optimal transportation has emerged as a fertile field of inquiry, and a diverse tool for exploring applications within and beyond mathematics. This transformation occurred partly because long-standing issues could finally be resolved, but also because unexpected connections emerged which linked these questions to classical problems in geometry, partial differential equations, nonlinear dynamics, natural sciences, design problems and economics. The aim of this program will be to gather experts in optimal transport and areas of potential application to catalyze new investigations, disseminate progress, and invigorate ongoing exploration.
January 20, 2014
to May 23, 2014
Organizers: Ehud Hrushovski (Hebrew University, Israel) , François Loeser (Université Pierre et Marie Curie, France), David Marker (University of Illinois, Chicago), Thomas Scanlon ( University of California, Berkeley), Sergei Starchenko (University of Notre Dame), Carol Wood* ( Wesleyan University)
The program aims to further the flourishing interaction between model theory and other parts of mathematics, especially number theory and arithmetic geometry. At present the model theoretical tools in use arise primarily from geometric stability theory and o-minimality. Current areas of lively interaction include motivic integration, valued fields, diophantine geometry, and algebraic dynamics.
January 20, 2014
to May 23, 2014
Organizers: Vigleik Angeltveit (Australian National University), Andrew J. Blumberg (University of Texas-Austin), Gunnar Carlsson (Stanford University), Teena Gerhardt (Michigan State University), Michael A. Hill* (University of Virginia), and Jacob Lurie (Harvard University)
Algebraic topology touches almost every branch of modern mathematics. Algebra, geometry, topology, analysis, algebraic geometry, and number theory all influence and in turn are influenced by the methods of algebraic topology. The goals of this 2014 program at MSRI are:
Bring together algebraic topology researchers from all subdisciplines, reconnecting the pieces of the field
Identify the fundamental problems and goals in the field, uncovering the broader themes and connections
Connect young researchers with the field, broadening their perspective and introducing them to the myriad approaches and techniques.
August 11, 2014
to December 12, 2014
Organizers: Pierre Colmez (Institut de Mathématiques de Jussieu), Wee Teck Gan* (University of California, San Diego), Michael Harris (Institut de Mathématiques de Jussieu), Elena Mantovan (California Institute of Technology), Ariane Mézard (Université de Versailles Saint-Quentin), Akshay Venkatesh (Stanford University)
The branches of number theory most directly related to the arithmetic of automorphic forms have seen much recent progress, with the resolution of many longstanding conjectures. These breakthroughs have largely been achieved by the discovery of new geometric techniques and insights. The goal of this program is to highlight new geometric structures and new questions of a geometric nature which seem most crucial for further development. In particular, the program will emphasize geometric questions arising in the study of Shimura varieties, the p-adic Langlands program, and periods of automorphic forms.
August 18, 2014
to December 19, 2014
Organizers: David Ben-Zvi* (University of Texas), Thomas Haines (University of Maryland), Florian Herzig (University of Toronto), Kevin McGerty (University of Oxford), David Nadler (University of California, Berkeley), Ngo Bao Chau (University of Chicago), Catharina Stroppel (University of Bonn), and Eva Viehmann (University of Bonn),
The fundamental aims of geometric representation theory are to uncover the deeper geometric and categorical structures underlying the familiar objects of representation theory and harmonic analysis, and to apply the resulting insights to the resolution of classical problems. One of the main sources of inspiration for the field is the Langlands philosophy, a vast nonabelian generalization of the Fourier transform of classical harmonic analysis, which serves as a visionary roadmap for the subject and places it at the heart of number theory. A primary goal of the proposed MSRI program is to explore the
potential impact of geometric methods and ideas in the Langlands program by bringing together researchers working in the diverse areas impacted by the Langlands philosophy, with a particular emphasis on representation theory over local fields.
Another focus comes from theoretical physics, where new perspectives on the central objects of geometric representation theory arise in the study supersymmetric gauge theory, integrable systems and topological string theory. The impact of these ideas is only beginning to be absorbed and the program will provide a forum for their dissemination and development.
January 12, 2015
to May 22, 2015
Organizers: Richard D. Canary (University of Michigan), William Goldman (University of Maryland), Francois Labourie (Université Paris-Sud) , Howard Masur* (University of Chicago), and Anna Wienhard (Princeton University)
The program will focus on the deformation theory of geometric structures on manifolds, and the resulting geometry and dynamics. This subject is formally a subfield of differential geometry and topology, with a heavy infusion of Lie theory. Its richness stems from close relations to dynamical systems, algebraic geometry, representation theory, Lie theory, partial differential equations, number theory, and complex analysis.
January 19, 2015
to May 29, 2015
Organizers: Dmitry Kleinbock* (Brandeis University), Elon Lindenstrauss (Hebrew University of Jerusalem), Hee Oh (Brown University), Jean-Francois Quint (Université Paris 13), Alireza Salehi Golsefidy (University of California, San Diego)
Homogeneous dynamics is the study of asymptotic properties of the action of subgroups of Lie groups on their homogeneous spaces. This includes many classical examples of dynamical systems, such as linear Anosov diffeomorphisms of tori and geodesic flows on negatively curved manifolds. This topic is related to many branches of mathematics, in particular, number theory and geometry. Some directions to be explored in this program include: measure rigidity of multidimensional diagonal groups; effectivization, sparse equidistribution and sieving; random walks, stationary measures and stiff actions; ergodic theory of thin groups; measure classification in positive characteristic. It is a companion program to “Dynamics on moduli spaces of geometric structures”.
