Acknowledged as the premier center for collaborative mathematical research, MSRI organizes and hosts semesterlength (or twosemesters duration) Programs that become the leading edge in that field of study. Mathematicians worldwide come to the Institute to engage in the research of classical fundamental mathematics, modern applied mathematics, statistics, computer science and other mathematical sciences. Upcoming MSRI Programs are listed as well as past Programs.
Current Programs

Hamiltonian systems, from topology to applications through analysis
Organizers: Rafael de la Llave (Georgia Institute of Technology), LEAD Albert Fathi (Georgia Institute of Technology; École Normale Supérieure de Lyon), Vadim Kaloshin (University of Maryland), Robert Littlejohn (University of California, Berkeley), Philip Morrison (University of Texas at Austin), Tere Seara (Universitat Politècnica de Catalunya), Sergei Tabachnikov (Pennsylvania State University), Amie Wilkinson (University of Chicago)The interdisciplinary nature of Hamiltonian systems is deeply ingrained in its history. Therefore the program will bring together the communities of mathematicians with the community of practitioners, mainly engineers, physicists, and theoretical chemists who use Hamiltonian systems daily. The program will cover not only the mathematical aspects of Hamiltonian systems but also their applications, mainly in space mechanics, physics and chemistry.
The mathematical aspects comprise celestial mechanics, variational methods, relations with PDE, Arnold diffusion and computation. The applications concern celestial mechanics, astrodynamics, motion of satellites, plasma physics, accelerator physics, theoretical chemistry, and atomic physics.
The goal of the program is to bring to the forefront both the theoretical aspects and the applications, by making available for applications the latest theoretical developments, and also by nurturing the theoretical mathematical aspects with new problems that come from concrete problems of applications.
Updated on Aug 20, 2018 08:16 AM PDT 
Complementary Program 201819
The Complementary Program has a limited number of memberships that are open to mathematicians whose interests are not closely related to the core programs; special consideration is given to mathematicians who are partners of an invited member of a core program.
Updated on Jan 02, 2018 10:45 AM PST
Upcoming Programs

Derived Algebraic Geometry
Organizers: Julie Bergner (University of Virginia), LEAD Bhargav Bhatt (University of Michigan), Dennis Gaitsgory (Harvard University), David Nadler (University of California, Berkeley), Nikita Rozenblyum (University of Chicago), Peter Scholze (Universität Bonn), Gabriele Vezzosi (Università di Firenze)Derived algebraic geometry is an extension of algebraic geometry that provides a convenient framework for directly treating nongeneric geometric situations (such as nontransverse intersections in intersection theory), in lieu of the more traditional perturbative approaches (such as the “moving” lemma). This direct approach, in addition to being conceptually satisfying, has the distinct advantage of preserving the symmetries of the situation, which makes it much more applicable. In particular, in recent years, such techniques have found applications in diverse areas of mathematics, ranging from arithmetic geometry, mathematical physics, geometric representation theory, and homotopy theory. This semester long program will be dedicated to exploring these directions further, and finding new connections.
Updated on Sep 12, 2018 09:26 AM PDT 
Birational Geometry and Moduli Spaces
Organizers: Antonella Grassi (University of Pennsylvania), LEAD Christopher Hacon (University of Utah), Sándor Kovács (University of Washington), Mircea Mustaţă (University of Michigan), Martin Olsson (University of California, Berkeley)Birational Geometry and Moduli Spaces are two important areas of Algebraic Geometry that have recently witnessed a flurry of activity and substantial progress on many fundamental open questions. In this program we aim to bring together key researchers in these and related areas to highlight the recent exciting progress and to explore future avenues of research.This program will focus on the following themes: Geometry and Derived Categories, Birational Algebraic Geometry, Moduli Spaces of Stable Varieties, Geometry in Characteristic p>0, and Applications of Algebraic Geometry: Elliptic Fibrations of CalabiYau Varieties in Geometry, Arithmetic and the Physics of String TheoryUpdated on Jan 31, 2017 07:46 PM PST 
Holomorphic Differentials in Mathematics and Physics
Organizers: LEAD Jayadev Athreya (University of Washington), Steven Bradlow (University of Illinois at UrbanaChampaign), Sergei Gukov (California Institute of Technology), Andrew Neitzke (University of Texas, Austin), Anna Wienhard (RuprechtKarlsUniversität Heidelberg), Anton Zorich (Institut de Mathematiques de Jussieu)Holomorphic differentials on Riemann surfaces have long held a distinguished place in low dimensional geometry, dynamics and representation theory. Recently it has become apparent that they constitute a common feature of several other highly active areas of current research in mathematics and also at the interface with physics. In some cases the areas themselves (such as stability conditions on Fukayatype categories, links to quantum integrable systems, or the physically derived construction of socalled spectral networks) are new, while in others the novelty lies more in the role of the holomorphic differentials (for example in the study of billiards in polygons, special  Hitchin or higher Teichmuller  components of representation varieties, asymptotic properties of Higgs bundle moduli spaces, or in new interactions with algebraic geometry).
It is remarkable how widely scattered are the motivating questions in these areas, and how diverse are the backgrounds of the researchers pursuing them. Bringing together experts in this wide variety of fields to explore common interests and discover unexpected connections is the main goal of our program. Our program will be of interest to those working in many different elds, including lowdimensional dynamical systems (via the connection to billiards); differential geometry (Higgs bundles and related moduli spaces); and different types of theoretical physics (electron transport and supersymmetric quantum field theory).
Updated on Apr 10, 2018 10:50 AM PDT 
Microlocal Analysis
Organizers: Pierre Albin (University of Illinois at UrbanaChampaign), Nalini Anantharaman (Université de Strasbourg), Kiril Datchev (Purdue University), Raluca Felea (Rochester Institute of Technology), Colin Guillarmou (Université de Paris XI (ParisSud)), LEAD Andras Vasy (Stanford University)Microlocal analysis provides tools for the precise analysis of problems arising in areas such as partial differential equations or integral geometry by working in the phase space, i.e. the cotangent bundle, of the underlying manifold. It has origins in areas such as quantum mechanics and hyperbolic equations, in addition to the development of a general PDE theory, and has expanded tremendously over the last 40 years to the analysis of singular spaces, integral geometry, nonlinear equations, scattering theory… This program will bring together researchers from various parts of the field to facilitate the transfer of ideas, and will also provide a comprehensive introduction to the field for postdocs and graduate students.
Updated on Apr 13, 2018 11:42 AM PDT 
Quantum Symmetries
Organizers: Vaughan Jones (Vanderbilt University), LEAD Scott Morrison (Australian National University), Victor Ostrik (University of Oregon), Emily Peters (Loyola University), Eric Rowell (Texas A & M University), LEAD Noah Snyder (Indiana University), Chelsea Walton (University of Illinois at UrbanaChampaign)Symmetry, as formalized by group theory, is ubiquitous across mathematics and science. Classical examples include point groups in crystallography, Noether's theorem relating differentiable symmetries and conserved quantities, and the classification of fundamental particles according to irreducible representations of the Poincaré group and the internal symmetry groups of the standard model. However, in some quantum settings, the notion of a group is no longer enough to capture all symmetries. Important motivating examples include Galoislike symmetries of von Neumann algebras, anyonic particles in condensed matter physics, and deformations of universal enveloping algebras. The language of tensor categories provides a unified framework to discuss these notions of quantum symmetry.Updated on Mar 22, 2018 11:21 AM PDT 
Higher Categories and Categorification
Organizers: David Ayala (Montana State University), Clark Barwick (Massachusetts Institute of Technology), David Nadler (University of California, Berkeley), LEAD Emily Riehl (Johns Hopkins University), Marcy Robertson (University of Melbourne), Peter Teichner (MaxPlanckInstitut für Mathematik), Dominic Verity (Macquarie University)Though many of the ideas in higher category theory find their origins in homotopy theory — for instance as expressed by Grothendieck’s “homotopy hypothesis” — the subject today interacts with a broad spectrum of areas of mathematical research. Unforeseen descent, or localtoglobal formulas, for familiar objects can be articulated in terms of higher invertible morphisms. Compatible associative deformations of a sequence of maps of spaces, or derived schemes, can putatively be represented by higher categories, as Koszul duality for E_nalgebras suggests. Higher categories offer unforeseen characterizing universal properties for familiar constructions such as Ktheory. Manifold theory is natively connected to higher category theory and adjunction data, a connection that is most famously articulated by the recently proven Cobordism Hypothesis.
In parallel, the idea of "categorification'' is playing an increasing role in algebraic geometry, representation theory, mathematical physics, and manifold theory, and higher categorical structures also appear in the very foundations of mathematics in the form of univalent foundations and homotopy type theory. A central mission of this semester will be to mitigate the exorbitantly high "cost of admission'' for mathematicians in other areas of research who aim to apply higher categorical technology and to create opportunities for potent collaborations between mathematicians from these different fields and experts from within higher category theory.Updated on Oct 05, 2018 12:21 PM PDT 
Random and Arithmetic Structures in Topology
Organizers: Nicolas Bergeron (Université de Paris VI (Pierre et Marie Curie)), Jeffrey Brock (Brown University), Alex Furman (University of Illinois at Chicago), Tsachik Gelander (Weizmann Institute of Science), Ursula Hamenstädt (Rheinische FriedrichWilhelmsUniversität Bonn), Fanny Kassel (Institut des Hautes Études Scientifiques (IHES)), LEAD Alan Reid (Rice University)The use of dynamical invariants has long been a staple of geometry and topology, from rigidity theorems, to classification theorems, to the general study of lattices and of the mapping class group. More recently, random structures in topology and notions of probabilistic geometric convergence have played a critical role in testing the robustness of conjectures in the arithmetic setting. The program will focus on invariants in topology, geometry, and the dynamics of group actions linked to random constructions.
Updated on Nov 16, 2017 02:50 PM PST 
Decidability, definability and computability in number theory
Organizers: Valentina Harizanov (George Washington University), Moshe Jarden (TelAviv University), Maryanthe Malliaris (University of Chicago), Barry Mazur (Harvard University), Russell Miller (Queens College, CUNY), Jonathan Pila (University of Oxford), LEAD Thomas Scanlon (University of California, Berkeley), Alexandra Shlapentokh (East Carolina University), Carlos Videla (Mount Royal University)This program is focused on the twoway interaction of logical ideas and techniques, such as definability from model theory and decidability from computability theory, with fundamental problems in number theory. These include analogues of Hilbert's tenth problem, isolating properties of fields of algebraic numbers which relate to undecidability, decision problems around linear recurrence and algebraic differential equations, the relation of transcendence results and conjectures to decidability and decision problems, and some problems in anabelian geometry and field arithmetic. We are interested in this specific interface across a range of problems and so intend to build a semester which is both more topically focused and more mathematically broad than a typical MSRI program.
Updated on Oct 05, 2018 09:07 AM PDT 
Mathematical problems in fluid dynamics
Organizers: Thomas Alazard (École Normale Supérieure; Centre National de la Recherche Scientifique (CNRS)), Hajer Bahouri (Université ParisEst Créteil ValdeMarne; Centre National de la Recherche Scientifique (CNRS)), 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 Jan 24, 2018 10:14 AM PST
Past Programs

Program Summer Research for Women in Mathematics
Organizers: Hélène Barcelo (MSRI  Mathematical Sciences Research Institute)See this LINK for the 2019 Summer Research for Women in Mathematics program.The purpose of the MSRI's program, Summer Research for Women in Mathematics, is to provide space and funds to groups of women mathematicians to work on a research project at MSRI. Research projects can arise from work initiated at a Women's Conference, or can be freestanding activities.Updated on Sep 11, 2018 01:32 PM PDT 
Program Complementary Program 201718
Updated on Nov 30, 2017 03:30 PM PST 
Program Group Representation Theory and Applications
Organizers: Robert Guralnick (University of Southern California), Alexander Kleshchev (University of Oregon), Gunter Malle (Universität Kaiserslautern), Gabriel Navarro (University of Valencia), Julia Pevtsova (University of Washington), Raphael Rouquier (University of California, Los Angeles), LEAD Pham Tiep (Rutgers University)Group Representation Theory is a central area of Algebra, with important and deep connections to areas as varied as topology, algebraic geometry, number theory, Lie theory, homological algebra, and mathematical physics. Born more than a century ago, the area still abounds with basic problems and fundamental conjectures, some of which have been open for over five decades. Very recent breakthroughs have led to the hope that some of these conjectures can finally be settled. In turn, recent results in group representation theory have helped achieve substantial progress in a vast number of applications.
The goal of the program is to investigate all these deep problems and the wealth of new results and directions, to obtain major progress in the area, and to explore further applications of group representation theory to other branches of mathematics.
Updated on Jan 12, 2018 04:00 PM PST 
Program Enumerative Geometry Beyond Numbers
Organizers: Mina Aganagic (University of California, Berkeley), Denis Auroux (University of California, Berkeley), Jim Bryan (University of British Columbia), LEAD Andrei Okounkov (Columbia University), Balazs Szendroi (University of Oxford)Traditional enumerative geometry asks certain questions to which the expected answer is a number: for instance, the number of lines incident with two points in the plane (1, Euclid), or the number of twisted cubic curves on a quintic threefold (317 206 375). It has however been recognized for some time that the numerics is often just the tip of the iceberg: a deeper exploration reveals interesting geometric, topological, representation, or knottheoretic structures. This semesterlong program will be devoted to these hidden structures behind enumerative invariants, concentrating on the core fields where these questions start: algebraic and symplectic geometry.
Updated on Jan 16, 2018 10:12 AM PST 
Program Geometric Functional Analysis and Applications
Organizers: Franck Barthe (Université de Toulouse III (Paul Sabatier)), Marianna Csornyei (University of Chicago), Boaz Klartag (Weizmann Institute of Science), Alexander Koldobsky (University of Missouri), Rafal Latala (University of Warsaw), LEAD Mark Rudelson (University of Michigan)Geometric functional analysis lies at the interface of convex geometry, functional analysis and probability. It has numerous applications ranging from geometry of numbers and random matrices in pure mathematics to geometric tomography and signal processing in engineering and numerical optimization and learning theory in computer science.
One of the directions of the program is classical convex geometry, with emphasis on connections with geometric tomography, the study of geometric properties of convex bodies based on information about their sections and projections. Methods of harmonic analysis play an important role here. A closely related direction is asymptotic geometric analysis studying geometric properties of high dimensional objects and normed spaces, especially asymptotics of their quantitative parameters as dimension tends to infinity. The main tools here are concentration of measure and related probabilistic results. Ideas developed in geometric functional analysis have led to progress in several areas of applied mathematics and computer science, including compressed sensing and random matrix methods. These applications as well as the problems coming from computer science will be also emphasised in our program.
Updated on Aug 23, 2017 03:38 PM PDT 
Program Geometric and Topological Combinatorics
Organizers: Jesus De Loera (University of California, Davis), Victor Reiner (University of Minnesota Twin Cities), LEAD Francisco Santos Leal (University of Cantabria), Francis Su (Harvey Mudd College), Rekha Thomas (University of Washington), Günter Ziegler (Freie Universität Berlin)Combinatorics is one of the fastest growing areas in contemporary Mathematics, and much of this growth is due to the connections and interactions with other areas of Mathematics. This program is devoted to the very vibrant and active area of interaction between Combinatorics with Geometry and Topology. That is, we focus on (1) the study of the combinatorial properties or structure of geometric and topological objects and (2) the development of geometric and topological techniques to answer combinatorial problems.
Key examples of geometric objects with intricate combinatorial structure are point configurations and matroids, hyperplane and subspace arrangements, polytopes and polyhedra, lattices, convex bodies, and sphere packings. Examples of topology in action answering combinatorial challenges are the by now classical Lovász’s solution of the Kneser conjecture, which yielded functorial approaches to graph coloring, and the more recent, extensive topological machinery leading to breakthroughs on Tverbergtype problems.Updated on Aug 28, 2017 11:26 AM PDT 
Program Summer Research 2017
Come spend time at MSRI in the summer! The Institute’s summer graduate schools and undergraduate program fill the lecture halls and some of the offices, but we have room for a modest number of visitors to come to do research singly or in small groups, while enjoying the excellent mathematical facilities, the great cultural opportunities of Berkeley, San Francisco and the Bay area, the gorgeous natural surroundings, and the cool weather.
We can provide offices, library facilities and bus passes—unfortunately not financial support. Though the auditoria are largely occupied, there are blackboards and ends of halls, so 26 people could comfortably collaborate with one another. We especially encourage such groups to apply together.
To make visits productive, we require at least a twoweek commitment. We strive for a wide mix of people, being sure to give special consideration to women, underrepresented groups, and researchers from nonresearch universities.
Updated on May 31, 2018 12:40 PM PDT 
Program Complementary Program (201617)
The Complementary Program has a limited number of memberships that are open to mathematicians whose interests are not closely related to the core programs; special consideration is given to mathematicians who are partners of an invited member of a core program.
Updated on Apr 14, 2017 10:04 AM PDT 
Program Analytic Number Theory
Organizers: Chantal David (Concordia University), Andrew Granville (Université de Montréal), Emmanuel Kowalski (ETH Zurich), Philippe Michel (École Polytechnique Fédérale de Lausanne (EPFL)), Kannan Soundararajan (Stanford University), LEAD Terence Tao (University of California, Los Angeles)Analytic number theory, and its applications and interactions, are currently experiencing intensive progress, in sometimes unexpected directions. In recent years, many important classical questions have seen spectacular advances based on new techniques; conversely, methods developed in analytic number theory have led to the solution of striking problems in other fields.
This program will not only give the leading researchers in the area further opportunities to work together, but more importantly give young people the occasion to learn about these topics, and to give them the tools to achieve the next breakthroughs.
Updated on Jul 10, 2015 03:54 PM PDT 
Program Harmonic Analysis
Organizers: LEAD Michael Christ (University of California, Berkeley), Allan Greenleaf (University of Rochester), Steven Hofmann (University of Missouri), LEAD Michael Lacey (Georgia Institute of Technology), Svitlana Mayboroda (University of Minnesota, Twin Cities), Betsy Stovall (University of WisconsinMadison), Brian Street (University of WisconsinMadison)The field of Harmonic Analysis dates back to the 19th century, and has its roots in the study of the decomposition of functions using Fourier series and the Fourier transform. In recent decades, the subject has undergone a rapid diversification and expansion, though the decomposition of functions and operators into simpler parts remains a central tool and theme.This program will bring together researchers representing the breadth of modern Harmonic Analysis and will seek to capitalize on and continue recent progress in four major directions:Restriction, Kakeya, and Geometric Incidence ProblemsAnalysis on Nonhomogeneous SpacesWeighted Norm InequalitiesQuantitative Rectifiability and Elliptic PDE.Many of these areas draw techniques from or have applications to other fields of mathematics, such as analytic number theory, partial differential equations, combinatorics, and geometric measure theory. In particular, we expect a lively interaction with the concurrent program.Updated on Aug 11, 2016 10:49 AM PDT 
Program Geometric Group Theory
Organizers: Ian Agol (University of California, Berkeley), Mladen Bestvina (University of Utah), Cornelia Drutu (University of Oxford), LEAD Mark Feighn (Rutgers University), Michah Sageev (TechnionIsrael Institute of Technology), Karen Vogtmann (University of Warwick)The field of geometric group theory emerged from Gromov’s insight that even mathematical objects such as groups, which are defined completely in algebraic terms, can be profitably viewed as geometric objects and studied with geometric techniques Contemporary geometric group theory has broadened its scope considerably, but retains this basic philosophy of reformulating in geometric terms problems from diverse areas of mathematics and then solving them with a variety of tools. The growing list of areas where this general approach has been successful includes lowdimensional topology, the theory of manifolds, algebraic topology, complex dynamics, combinatorial group theory, algebra, logic, the study of various classical families of groups, Riemannian geometry and representation theory.
The goals of this MSRI program are to bring together people from the various branches of the field in order to consolidate recent progress, chart new directions, and train the next generation of geometric group theorists.Updated on Aug 11, 2016 08:44 AM PDT 
Program Summer Research 2016
Come spend time at MSRI in the summer! The Institute’s summer graduate schools and undergraduate program fill the lecture halls and some of the offices, but we have room for a modest number of visitors to come to do research singly or in small groups, while enjoying the excellent mathematical facilities, the great cultural opportunities of Berkeley, San Francisco and the Bay area, the gorgeous natural surroundings, and the cool weather.
We can provide offices, library facilities and bus passes—unfortunately not financial support. Though the auditoria are largely occupied, there are blackboards and ends of halls, so 26 people could comfortably collaborate with one another. We especially encourage such groups to apply together.
To make visits productive, we require at least a twoweek commitment. We strive for a wide mix of people, being sure to give special consideration to women, underrepresented groups, and researchers from nonresearch universities.
Updated on Mar 22, 2016 11:58 AM PDT 
Program Complementary Program
Updated on Jul 13, 2016 09:06 AM PDT 
Program Differential Geometry
Organizers: Tobias Colding (Massachusetts Institute of Technology), Simon Donaldson (Imperial College, London), John Lott (University of California, Berkeley), Natasa Sesum (Rutgers University), Gang Tian (Princeton University), LEAD Jeff Viaclovsky (University of WisconsinMadison)Differential geometry is a subject with both deep roots and recent advances. Many old problems in the field have recently been solved, such as the Poincaré and geometrization conjectures by Perelman, the quarter pinching conjecture by BrendleSchoen, the Lawson Conjecture by Brendle, and the Willmore Conjecture by MarquesNeves. The solutions of these problems have introduced a wealth of new techniques into the field. This semesterlong program will focus on the following main themes:
(1) Einstein metrics and generalizations,
(2) Complex differential geometry,
(3) Spaces with curvature bounded from below,
(4) Geometric flows,
and particularly on the deep connections between these areas.Updated on Apr 21, 2015 03:40 PM PDT 
Program New Challenges in PDE: Deterministic Dynamics and Randomness in High and Infinite Dimensional Systems
Organizers: Kay Kirkpatrick (University of Illinois at UrbanaChampaign), Yvan Martel (École Polytechnique), Jonathan Mattingly (Duke University), Andrea Nahmod (University of Massachusetts, Amherst), Pierre Raphael (Université Nice SophiaAntipolis), Luc ReyBellet (University of Massachusetts, Amherst), LEAD Gigliola Staffilani (Massachusetts Institute of Technology), Daniel Tataru (University of California, Berkeley)The fundamental aim of this program is to bring together a core group of mathematicians from the general communities of nonlinear dispersive and stochastic partial differential equations whose research contains an underlying and unifying problem: quantitatively analyzing the dynamics of solutions arising from the flows generated by deterministic and nondeterministic evolution differential equations, or dynamical evolution of large physical systems, and in various regimes.
In recent years there has been spectacular progress within both communities in the understanding of this common problem. The main efforts exercised, so far mostly in parallel, have generated an incredible number of deep results, that are not just beautiful mathematically, but are also important to understand the complex natural phenomena around us. Yet, many open questions and challenges remain ahead of us. Hosting the proposed program at MSRI would be the most effective venue to explore the specific questions at the core of the unifying theme and to have a focused and open exchange of ideas, connections and mathematical tools leading to potential new paradigms. This special program will undoubtedly produce new and fundamental results in both areas, and possibly be the start of a new generation of researchers comfortable on both languages.
Updated on Sep 15, 2015 05:25 PM PDT 
Program Summer Research
Come spend time at MSRI in the summer! The Institute’s summer graduate schools and undergraduate program fill the lecture halls and some of the offices, but we have room for a modest number of visitors to come to do research singly or in small groups, while enjoying the excellent mathematical facilities, the great cultural opportunities of Berkeley, San Francisco and the Bay area, the gorgeous natural surroundings, and the cool weather.
We can provide offices, library facilities and bus passes—unfortunately not financial support. Though the auditoria are largely occupied, there are blackboards and ends of halls, so 26 people could comfortably collaborate with one another. We especially encourage such groups to apply together.
To make visits productive, we require at least a twoweek commitment. We strive for a wide mix of people, being sure to give special consideration to women, underrepresented groups, and researchers from nonresearch universities.
Updated on May 06, 2015 11:36 AM PDT 
Program Geometric and Arithmetic Aspects of Homogeneous Dynamics
Organizers: LEAD Dmitry Kleinbock (Brandeis University), Elon Lindenstrauss (The Hebrew University of Jerusalem), Hee Oh (Yale University), JeanFrançois Quint (Université de Bordeaux I), 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”.
Updated on Jan 12, 2015 10:58 AM PST 
Program Complementary Program (201415)
Updated on Feb 27, 2014 09:09 AM PST 
Program Dynamics on Moduli Spaces of Geometric Structures
Organizers: Richard Canary (University of Michigan), William Goldman (University of Maryland), François Labourie (Universite de Nice Sophia Antipolis), LEAD Howard Masur (University of Chicago), Anna Wienhard (RuprechtKarlsUniversität Heidelberg)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.
Updated on Apr 03, 2015 01:06 PM PDT 
Program Geometric Representation Theory
Organizers: LEAD David BenZvi (University of Texas, Austin), Ngô Bảo Châu (University of Chicago), Thomas Haines (University of Maryland), Florian Herzig (University of Toronto), Kevin McGerty (University of Oxford), David Nadler (University of California, Berkeley), Catharina Stroppel (Rheinische FriedrichWilhelmsUniversität Bonn), Eva Viehmann (TU München)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.
Updated on Aug 13, 2014 09:08 AM PDT 
Program New Geometric Methods in Number Theory and Automorphic Forms
Organizers: Pierre Colmez (Institut de Mathématiques de Jussieu), LEAD Wee Teck Gan (National University of Singapore), Michael Harris (Columbia University), Elena Mantovan (California Institute of Technology), Ariane Mézard (Institut de Mathématiques de Jussieu; École Normale Supérieure), 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 padic Langlands program, and periods of automorphic forms.
Updated on Oct 11, 2013 02:02 PM PDT 
Program Model Theory, Arithmetic Geometry and Number Theory
Organizers: Ehud Hrushovski (The Hebrew University of Jerusalem), François Loeser (Université de Paris VI (Pierre et Marie Curie)), David Marker (University of Illinois, Chicago), Thomas Scanlon (University of California, Berkeley), Sergei Starchenko (University of Notre Dame), LEAD 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 ominimality. Current areas of lively interaction include motivic integration, valued fields, diophantine geometry, and algebraic dynamics.
Updated on Feb 19, 2014 02:02 PM PST 
Program Algebraic Topology
Organizers: Vigleik Angeltveit (Australian National University), Andrew Blumberg (University of Texas, Austin), Gunnar Carlsson (Stanford University), Teena Gerhardt (Michigan State University), LEAD Michael Hill (University of California, Los Angeles), 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.
Updated on Jan 21, 2014 11:44 AM PST 
Program Mathematical General Relativity
Organizers: Yvonne ChoquetBruhat, Piotr Chrusciel (Universität Wien), Greg Galloway (University of Miami), Gerhard Huisken (Math. Forschungsinstitut Oberwolfach), LEAD James Isenberg (University of Oregon), Sergiu Klainerman (Princeton University), Igor Rodnianski (Massachusetts Institute of Technology), Richard Schoen (University of California, Irvine)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 semesterlong program aims to bring together researchers working in mathematical relativity, differential geometry, and PDE who wish to explore this rapidly growing area of mathematics.
Updated on Nov 05, 2013 04:41 PM PST 
Program Optimal Transport: Geometry and Dynamics
Organizers: Luigi Ambrosio (Scuola Normale Superiore), Yann Brenier (École Polytechnique), Panagiota Daskalopoulos (Columbia University), Lawrence Evans (University of California, Berkeley), Alessio Figalli (University of Texas, Austin), Wilfrid Gangbo (University of California, Los Angeles), LEAD Robert McCann (University of Toronto), Felix Otto (MaxPlanckInstitut für Mathematik in den Naturwissenschaften), Neil 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 longstanding 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.
Updated on Sep 29, 2013 11:41 PM PDT 
Program Noncommutative Algebraic Geometry and Representation Theory
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,CalabiYau algebras, deformation theory and Poisson structures, as well as the interplay of these subjects with the algebras appearing in representation theorylike enveloping algebras, symplectic reflection algebras and the many guises of Hecke algebras.
Updated on May 06, 2013 04:21 PM PDT 
Program Commutative Algebra
Organizers: David Eisenbud* (University of California, Berkeley), Srikanth Iyengar (University of Nebraska), Ezra Miller (Duke University), Anurag Singh (University of Utah), and Karen Smith (University of Michigan)Commutative algebra was born in the 19th century from algebraic geometry, invariant theory, and number theory. Today it is a mature field with activity on many fronts.
The yearlong program will highlight exciting recent developments in core areas such as free resolutions, homological and representation theoretic aspects, Rees algebras and integral closure, tight closure and singularities, and birational geometry. In addition, it will feature the important links to other areas such as algebraic topology, combinatorics, mathematical physics, noncommutative geometry, representation theory, singularity theory, and statistics. The program will reflect the wealth of interconnections suggested by these fields, and will introduce young researchers to these diverse areas.
New connections will be fostered through collaboration with the concurrent MSRI programs in Cluster Algebras (Fall 2012) and Noncommutative Algebraic Geometry and Representation Theory (Spring 2013).
For more detailed information about the program please see, http://www.math.utah.edu/ca/.
Updated on Aug 18, 2013 04:09 PM PDT 
Program Complementary Program 201213
Updated on May 21, 2013 12:44 PM PDT 
Program Cluster Algebras
Organizers: Sergey Fomin (University of Michigan), Bernhard Keller (Université Paris Diderot  Paris 7, France), Bernard Leclerc (Université de Caen BasseNormandie, France), Alexander Vainshtein* (University of Haifa, Israel), Lauren Williams (University of California, Berkeley)Cluster algebras were conceived in the Spring of 2000 as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. They are constructively defined commutative algebras with a distinguished set of generators (cluster variables) grouped into overlapping subsets (clusters) of fixed cardinality. Both the generators and the relations among them are not given from the outset, but are produced by an iterative process of successive mutations. Although this procedure appears counterintuitive at first, it turns out to encode a surprisingly widespread range of phenomena, which might explain the explosive development of the subject in recent years.
Cluster algebras provide a unifying algebraic/combinatorial framework for a wide variety of phenomena in settings as diverse as quiver representations, Teichmueller theory, invariant theory, tropical calculus, Poisson geometry, Lie theory, and polyhedral combinatorics.
Updated on May 06, 2013 04:25 PM PDT 
Program Random Spatial Processes
Organizers: Mireille BousquetMélou (Université de Bordeaux I, France), Richard Kenyon* (Brown University), Greg Lawler (University of Chicago), Andrei Okounkov (Columbia University), and Yuval Peres (Microsoft Research Laboratories)In recent years probability theory (and here we mean probability theory in the largest sense, comprising combinatorics, statistical mechanics, algorithms, simulation) has made immense progress in understanding the basic twodimensional models of statistical mechanics and random surfaces. Prior to the 1990s the major interests and achievements of probability theory were (with some exceptions for dimensions 4 or more) with respect to onedimensional objects: Brownian motion and stochastic processes, random trees, and the like. Inspired by work of physicists in the ’70s and ’80s on conformal invariance and field theories in two dimensions, a number of leading probabilists and combinatorialists began thinking about spatial process in two dimensions: percolation, polymers, dimer models, Ising models. Major breakthroughs by Kenyon, Schramm, Lawler, Werner, Smirnov, Sheffield, and others led to a rigorous underpinning of conformal invariance in twodimensional systems and paved the way for a new era of “twodimensional” probability theory.
Updated on Aug 10, 2015 02:30 PM PDT