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.

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.

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.

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.

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.

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.

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.

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”.

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 Brendle-Schoen, and the Willmore Conjecture by Marques-Neves. The solutions of these problems have introduced a wealth of new techniques into the field. This semester-long 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.

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
low-dimensional 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.

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 year-long 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/.

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.

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 counter-intuitive 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.

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 two-dimensional 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 one-dimensional 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 two-dimensional systems and paved the way for a new era of “two-dimensional” probability theory.

The fall 2011 program "Quantitative Geometry" is devoted to the investigation of geometric questions in which quantitative/asymptotic considerations are inherent and necessary for the formulation of the problems being studied. Such topics arise naturally in a wide range of mathematical disciplines, with significant relevance both to the internal development of the respective fields, as well as to applications in areas such as theoretical computer science. Examples of areas that will be covered by the program are: geometric group theory, the theory of Lipschitz functions (e.g., Lipschitz extension problems and structural aspects such as quantitative differentiation), large scale and coarse geometry, embeddings of metric spaces and their applications to algorithm design, geometric aspects of harmonic analysis and probability, quantitative aspects of linear and non-linear Banach space theory, quantitative aspects of geometric measure theory and isoperimetry, and metric invariants arising from embedding theory and Riemannian geometry. The MSRI program aims to crystallize the interactions between researchers in various relevant fields who might have a lack of common language, even though they are working on related questions.

This program aims at the study of various topics within the area of Free Boundaries Problems, from the viewpoints of theory and applications. Many problems in physics, industry, finance, biology, and other areas can be described by partial differential equations that exhibit apriori unknown sets, such as interfaces, moving boundaries, shocks, etc. The study of such sets, also known as free boundaries, often occupies a central position in such problems. The aim of this program is to gather experts in the field with knowledge of various applied and theoretical aspects of free boundary problems.

L -functions attached to modular forms and/or to algebraic varieties and algebraic number fields are prominent in quite a wide range of number theoretic issues, and our recent growth of understanding of the analytic properties of L-functions has already lead to profound applications regarding among other things the statistics related to arithmetic problems. This program will emphasize statistical aspects of L-functions, modular forms, and associated arithmetic and algebraic objects from several different perspectives — theoretical, algorithmic, and experimental.

The goal of this program is to showcase the many remarkable developments that have taken place in the past decade in Random Matrix Theory (RMT) and to spur on further developments on RMT and the related areas Interacting Particle Systems (IPS) and Integrable Systems (IS): IPS provides an arena in which RMT behavior is frequently observed, and IS provides tools which are often useful in analyzing RMT and IPS/RMT behavior.

Inverse Problems are problems where causes for a desired or an observed effect are to be determined. They lie at the heart of scientific inquiry and technological development. Applications include a number of medical as
well as other imaging techniques, location of oil and mineral deposits in the earth's substructure, creation of astrophysical images from telescope data, finding cracks and interfaces within materials, shape optimization,
model identification in growth processes and, more recently, modelling in the life sciences. During the last 10 years or so there has been significant developments both in the mathematical theory and applications of inverse problems. The purpose of the program would be to bring together people working on different aspects of the field, to appraise the current status of development and to encourage interaction between mathematicians and scientists and engineers working directly with the applications.

In the slightly more than two decades that have elapsed since the fields of Symplectic and Contact Topology were created, the field has grown enormously and unforeseen new connections within Mathematics and Physics have been found. The goals of the 2009-10 program at MSRI are to:
I. Promote the cross-pollination of ideas between different areas of symplectic and contact geometry;
II. Help assess and formulate the main outstanding fundamental problems and directions in the field;
III. Lead to new breakthroughs and solutions of some of the main problems in the area;
IV. Discover new applications of symplectic and contact geometry in mathematics and physics;
V. Educate a new generation of young mathematicians, giving them a broader view of the subject and the capability to employ techniques from different areas in their research.

The aims of this program will be to achieve the following goals:

1. Promote communication with related disciplines, including the symplectic geometry program in 2009-2010.
2. Lead to new breakthroughs in the subject and find new applications to low dimensional topology (knot theory, three-manifold topology, and smooth four manifold topology).
3. Educate a new generation of graduate students and PhD students in this exciting and rapidly-changing subject.

The program will focus on algebraic link homology and Heegaard Floer homology.

Tropical Geometry is the algebraic geometry over the min-plus algebra. It is a young subject that in recent years has both established itself as an area of its own right and unveiled its deep connections to numerous branches of pure and applied mathematics. From an algebraic geometric point of view, algebraic varieties over a field with non-archimedean valuation are replaced by polyhedral complexes, thereby retaining much of the information about the original varieties. From the point of view of complex geometry, the geometric combinatorial structure of tropical varieties is a maximal degeneration of a complex structure on a manifold.

The tropical transition from objects of algebraic geometry to the polyhedral realm is an extension of the classical theory of toric varieties. It opens problems on algebraic varieties to a completely new set of techniques, and has already led to remarkable results in Enumerative Algebraic Geometry, Dynamical Systems and Computational Algebra, among other fields, and to applications in Algebraic Statistics and Statistical Physics.

Current research centers on many open questions, i.e., representations over the integers or rings of positive characteristic, correspondence of characters and derived equivalences of blocks. Recently we have seen active interactions in group cohomology involving many areas of topology and algebra. The focus of this program will be on these areas with the goal of fostering emerging interdisciplinary connections among them.

Recent catalysts stimulating growth of this field in the last few decades have been the discovery of "crystals" and the development of the combinatorics of affine Lie groups.. Today the subject intersects several fields: combinatorics, representation theory, analysis, algebraic geometry, Lie theory, and mathematical physics. The goal of this program is to bring together experts in these areas together in one interdisciplinary setting.

In the 1980’s, attention to the geometric structures which cell complexes can carry shed light on earlier combinatorial and topological investigations into group theory, stimulating other provacative and innovative ideas over the past 20 years. As a consequence, geometric group theory has developed many different facets, including geometry, topology, analysis, logic.

These fields have each seen recent dramatic changes: new techniques developed, major conjectures solved, and new directions and connections forged. Yet progress has been made in parallel without the level of communication across these two fields that is warranted. This program will address the need to strengthen connections between these two fields, and reassess new directions for each.

The focus will be on geometric evolution equations, function theory and related elliptic and parabolic equations. Geometric flows have been applied to a variety of geometric, topological, analytical and physical problems. Linear and nonlinear elliptic and parabolic partial differential equations have been studied by continuous, discrete and computational methods. There are deep connections between the geometry and analysis of Riemannian and Kähler manifolds.

This program will take place at the interface of the theory and applications of dynamical systems. The goal will be to assess the current state-of-the-art and define directions for future research. Mathematicians who are developing a new generation of ideas in dynamical systems will be brought together with researchers who are using the techniques of dynamical systems in applied areas. A wide range of applications will be considered through four contextual settings around which the program will be organized. Some of the areas of concentration have greater emphasis on extending existing ideas in dynamical systems theory, rendering them more suitable for applications. Others are more directed toward seeking out potential areas of applications in which dynamical systems is likely to have a bigger role to play.
The four themes that will mold the semester are: (1) Extended dynamical systems, (2) Stochastic dynamical systems, (3) Control theory, and (4) Computation and modeling. The introductory workshop, which will be held in mid-January, will emphasize extended dynamical systems that occur as high-dimensional systems, such as on lattices or as partial differential equations. There will be a workshop on stochastic systems and control theory in March. The last theme will pervade the semester through seminar and working group activities.

The interplay between quantum field theory and mathematics during the past several decades has led to new concepts of mathematics, which will be explored and developed in this program. This includes: Stringy topology, branes and orbifolds, Generalized McKay correspondences and representation theory and Gromov-Witten theory.

Our focus will be rational and integral points on varieties of dimension > 1. Recently it has become clear that many branches of mathematics can be brought to bear on problems in the area: complex algebraic geometry, Galois and 4etale cohomology, transcendence theory and diophantine approximation, harmonic analysis, automorphic forms, and analytic number theory. Sometimes it is only by combining techniques that progress is made. We will bring together researchers from these various fields who have an interest in arithmetic applications, as well as specialists in arithmetic geometry itself.

The field of nonlinear dispersive equations has experienced a striking evolution over the last fifteen years. During that time many new ideas and techniques emerged, enabling one to work on problems which until quite recently seemed untouchable. The evolution process for this field has itsorigin in two ways of quantitatively measuring dispersion. One comes from harmonic analysis, which is used to establish certain dispersive (Lp) estimates for solutions to linear equations. The second has geometrical roots, namely in the analysis of vector fields generating the Lorentz groupassociated to the linear wave equation. Our semester program in nonlinear dispersive equations will bring together leading experts in both of these directions.

The research in nonlinear elliptic equations is one of the most developed in Mathematics, and of great importance because of its interaction with other areas within Mathematics and for its applications in broader scientific disciplines such as fluid dynamics, phase transitions, mathematical finance and image processing in computer science.

Graduate Students from MSRI Sponsoring Institutions may benominated to participate in this program.

Graduate Students from MSRI Sponsoring Institutions may benominated to participate in this program.

Graduate Students from MSRI Sponsoring Institutions may benominated to participate in this program.

The field of image analysis is one of the newest and most active sources of inspiration for applied mathematics. Present day mathematical challenges in image analysis span a wide range of mathematical territory.

The theory of complex hyperplane arrangements has undergone tremendous growth since its beginnings thirty years ago in the work of Arnol'd, Breiskorn, Deligne, and Hattori. Connections with generalized hypergeometric functions, conformal field theory, representations of braid groups, and other areas have stimulated fascinating research into topology of arrangement complements. Topological research leads in turn to many new combinatorial and algebraic questions about arrangements.

This MSRI Summer Graduate Program at the University of Oregon will provide an introduction to the material to be covered in the fall, 2004 MSRI program on Hyperplane Arrangements and Applications. See the program page for more information on the content.

Open only to graduate students nominated by MSRI's Academic Sponsors.

Please note, MSRI's Summer Graduate Programs are open only to students nominated by MSRI's Academic Sponsor universities.

As classical as the subject is, it is currently undergoing a very vigorous development, interacting strongly with theoretical physics, mechanics, topology, algebraic geometry, partial differential equations, the calculus of variations, integrable systems, and many other subjects. The five main topics on which we propose to concentrate the program are areas that have shown considerable growth in the last ten years: Complex geometry, calibrated geometries and special holonomy; Geometric analysis; Symplectic geometry and gauge theory; Geometry and physics; Riemannian and metric geometry.

The topological approach to real algebraic geometry is due to Hilbert who realized the advantages of considering topological properties of real algebraic plane curves. Much progress on Hilbert's work was achieved in the 1970's by the schools of Rokhlin and Arnold, including new objects and questions on complexification and complex algebraic geometry, relation to piecewise linear geometry and combinatorics, and enumerative geometry. This continues today with new topics such as amoebas, new connections such as that with symplectic geometry, and new challenges such as those posed by real polynomial systems.

Discrete and Computational Geometry deals with the structure and complexity of discrete geometric objects as well with the design of efficient computer algorithms for their manipulation. This area is by its nature interdisciplinary and has relations to many other vital mathematical fields, such as algebraic geometry, topology, combinatorics, and probability theory; at the same time it is on the cutting edge of modern applications such as geographic information systems, mathematical programming, coding theory, solid modeling, and computational structural biology.

Please note, MSRI's Summer Graduate Programs are open only to students nominated by MSRI's Academic Sponsor universities.

Please note, MSRI's Summer Graduate Programs are open only to students nominated by MSRI's Academic Sponsor universities.

Commutative algebra comes from several sources, the 19th century theory of equations, number theory, invariant theory and algebraic geometry. The field has experienced a striking evolution over the last fifteen years. During that period the outlook of the subject has been altered, new connections to other areas have been established, and powerful techniques have been developed.

The traditional mathematical study of semi-classical analysis has developed tremendously in the last thirty years following the introduction of microlocal analysis, that is local analysis in phase space, simultaneously in the space and Fourier transform variables. The purpose of this program is to bring together experts in traditional mathematical semi-classical analysis, in the new mathematics of "quantum chaos," and in physics and theoretical chemistry.

Quantum computation is an intellectually challenging and exciting area that touches on the foundations of both computer science and quantum physics.

Summer Graduate Program -- open only to students nominated by MSRI's Academic Sponsor universities.

Summer Graduate Program -- open only to students nominated by MSRI's Academic Sponsor universities, to be held in Vancouver, BC, Canada at the Pacific Institute of Mathematics facility of Simon Fraser University.

This program will discuss recent progress in the representation theory of infinite-dimensional algebras and superalgebras and their applications to other fields.

Algebraic stacks originally arose as solutions to moduli problems in which they were used to parametrize geometric objects in families.They have also arisen in studying homological properties of quotient singularities, non-abelian Hodge theory, string theory, etc. This program will focus on intersection theory on stacks, non-abelian Hodge theory and geometric n-stacks, perverse sheaves on stacks and the geometric Langlands program, D-brane charges in string theory, and moduli of gerbes and mirror symmetry.

This program will focus on recent advances in integral geometry,with a focus on theinterrelationships between integral geometry and the theory ofrepresentations (Penrosetransform in flag domains, horospherical transforms), complex geometry, symplectic geometry,algebraic analysis, and nonlinear differential equations.
There will be an Introductory Workshop in Inverse Problems and Integral Geometry August 13-24

In the last twenty years or so there have been substantial developments in the mathematical theory of inverse problems,and applications have arisen in many areas, ranging from geophysics to medical imaging to non-destructive evaluation of materials. The main topics of this program will be developments in inverse boundary value problems, and inverse scattering problems.
There will be an Introductory Workshop in Inverse Problems and Integral Geometry August 13-24

MSRI's second Graduate Summer Program for 2001.

MSRI's Summer Graduate Program I This summer graduate program, organized by Dan Rockmore and Dennis Healy, Jr., will introduce students to the world of signal processing. The course will cover standard tools of digital signal processing, but will also cover the exciting frontiers of the subject, including wavelets, ISP (integrated sensing and processing), image processing algorithms, etc. In addition, students will be briefly exposed to applications in various areas, such as biology, chemistry, medicine, music, and engineering.

The noncommutative mathematics of operator algebras has grown in many directions and has made unexpected connections with other parts of mathematics and physics. Since the 1984-85 MSRI program in Operator Algebras, developments have continued at a rapid pace, and interactions with other fields such as elementary particle physics and quantum groups continue to grow.

The past few decades have witnessed many new developments in the broad area of spectral theory of geometric operators, centered around the study of new spectral invariants and their application to problems in conformal geometry, classification of 4-manifolds, index theory, relationship with scattering theory and other topics. This program will bring together people working on different problems in these areas.

Number theorists have always made calculations, whether by hand, desk calculator, or computer. In recent years this predilection has extended in many directions, and has been reinforced by interest from other fields such as computer science, cryptography, and algebraic geometry. The Algorithmic Number Theory program at MSRI will cover these developments broadly, with an eye to making connections to some of these other areas.

SUMMER GRADUATE PROGRAMFor more information about this program, please see the original web page at:http://www.msri.org/calendar/sgp/sgp2/index.html

For more information about this program, please see the original web page at:http://www.msri.org/activities/programs/9900/noncomm/index.html

SUMMER GRADUATE PROGRAMFor more information about this program, please see the original web page at:http://www.msri.org/calendar/workshops/9900/Molecular_and_Cell_Biology/index.html

For more information about this program, please see the original web page at:http://www.msri.org/activities/programs/9900/fem/index.html

For more information about this event, please see the original web page at:http://www.msri.org/activities/programs/9900/galois/index.html

For more information, please see this programs original web page at http://www.msri.org/activities/events/9899/sgp99/bryant.html

For more information, please see the original program page at http://www.msri.org/activities/events/9899/sgp99/mahadevan.html

For more information about this program, please see the program's original web page at http://www.msri.org/activities/programs/9899/random/index.html

Please see the program's webpage at http://www.msri.org/activities/programs/9899/focm/index.html for more information.

Please see the program webpage at http://www.msri.org/activities/programs/9899/symbcomp/index.html for more information.

Please see the program webpage at http://www.msri.org/activities/programs/9798/sa/index.html for more information.

Please see the program webpage at http://www.msri.org/activities/programs/9798/mtf/index.html for more information about this program.

Please see the program webpage at http://www.msri.org/activities/programs/9798/ha/index.html for more information.

Please see the program webpage at http://www.msri.org/activities/programs/9697/ldt/index.html for more information.

Please see the program webpage at http://www.msri.org/activities/programs/9697/comb/index.html for more information.

Please see the program webpage at http://www.msri.org/activities/programs/9596/scv/ for more information.

Please see the program webpage at http://www.msri.org/activities/programs/9596/hs/ for more information.