Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

Mathematical relativity is a very widely ranging area of mathematical study, spanning differential geometry, elliptic and hyperbolic PDE, and dynamical systems. We introduce in this workshop some of the leading areas of current interest associated with problems in cosmology, the theory of black holes, and the geometry and physics of the Cauchy problem (initial data constraints and evolution) for the Einstein equations.

The introductory workshop serves as an overview to the overlying programmatic theme. It aims to familiarize graduate students, postdocs, and non-experts to major and new topics of the current program. Though the audience is expected to have a general mathematical background, knowledge of technical terminology and recent findings is not assumed.

Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

Ever since the epic work of Yvonne Choquet-Bruhat on the well-posedness of Einstein's equations initiated the mathematical study of general relativity, women have played an important role in many areas of mathematical relativity. In this workshop, some of the leading women researchers in mathematical relativity present their work.

Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

The workshop is intended to give an overview of the research landscape surrounding optimal transportation, including its connections to geometry, design applications, and fully nonlinear partial differential equations.

As such, it will feature some survey lectures or minicourses by distinguished visitors and/or a few of the organizers of the theme semester, amounting to a kind of summer school. These will be complemented by a sampling of research lectures and short presentations from a spectrum of invited guests and other participants, including some who attended the previous week's {\em Connections for Women} workshop.

The introductory workshop aims to familiarize graduate students, postdocs, and non-experts to major and new topics of the current program. Though the audience is expected to have a general mathematical background, knowledge of technical terminology and recent findings is not assumed.

This two-day event aims to connect women graduate students and beginning researchers with more established female researchers who use optimal transportation in their work and can serve as professional contacts and potential role-models. As such, it will showcase a selection of lectures featuring female scientists, both established leaders and emerging researchers.

These lectures will be interspersed with networking and social events such as lunch or tea-time discussions led by successful researchers about (a) the particular opportunities and challenges facing women in science---including practical topics such as work-life balance and choosing a mentor, and (b) promising new directions in optimal transportation and related topics. Junior participants will be paired with more senior researchers in mentoring groups, and all participants will be encouraged to stay for the Introductory Workshop the following week, where they will have the opportunity to propose a short research communication.

In this two week program we will develop some of the mathematical foundations of seismic imaging that is a basic tool used in ``Imaging the Earth Interior". This is one of the components of the Mathematics of Planet Earth year in 2013.

The goal in seismic imaging is to determine the inner structure of the Earth from the crust to the inner core by using information provided by earthquakes in the case of the deep interior or by measuring the reflection of waves produced by acoustic or elastic sources on the surface of the Earth. The mathematics of seismic imaging involves solving inverse problems for the wave equation. No previous experience on inverse problems will be assumed.

The branches of number theory most directly related to automorphic forms have seen enormous progress over the past five years. Techniques introduced since 2008 have made it possible to prove many new arithmetic applications. The purpose of the current workshop is to drow the attention of young students or researchers to new questions that have arisen in the course of bringing several chapters in the Langlands program and related algebraic number theory to a close. We will focus especially on some precise questions of a geometric nature, or whose solutions seem to require new geometric insights. A graduate level in Number Theory is expected.

This two-week workshop will be devoted to the following subjects: Automorphy lifting theorems, p-adic local Langlands program, Characters of categorical representations and Hasse-Weil zeta function. During the first week, the lecturers present an open question and related mathematical objects. The first exercice sessions serve to direct the participants to an appropriate subject depending on their level. During the second week, the lecturers give some more advanced lectures on the field.

This Summer Graduate Workshop will be held in Park City, Utah.

The Graduate Summer School bridges the gap between a general graduate education in mathematics and the specific preparation necessary to do research on problems of current interest. In general, these students will have completed their first year, and in some cases, may already be working on a thesis. While a majority of the participants will be graduate students, some postdoctoral scholars and researchers may also be interested in attending.

We strongly recommend that graduate students have already had the equivalent of rigorous first year graduate-level courses in topology, algebra and analysis.

The main activity of the Graduate Summer School will be a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures will not duplicate standard courses available elsewhere. Each course will consist of lectures with problem sessions. Course assistants will be available for each lecture series. The participants of the Graduate Summer School meet three times each day for lectures, with one or two problem sessions scheduled each day as well.

The Second Pacific Rim Mathematical Association (PRIMA) Congress will be held at Shanghai Jiao Tong University in China, on June 24-28, 2013.

PRIMA is an association of mathematical sciences institutes, departments and societies from around the Pacific Rim, established in 2005 with the aim of promoting and facilitating the development of the mathematical sciences throughout the Pacific Rim region.

$1000 travel grants are available to representatives from MSRI Academic Sponsoring Institutions. These grants are available on a first-come, first-serve basis.

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Additional Travel Support Available from an NSF Grant

The NSF has awarded a substantial grant for travel by scientists at US universities to the PRIMA Congress in Shanghai. For further information and application details, please see https://www.mathprograms.org/db/programs/152

This Summer Graduate Workshop will be held in Montreal, Canada.

Homology theories of knots and links is a burgeoning field at the interface of mathematics with theoretical physics. The 2013 edition of the SMS will bring together leading researchers in mathematics and mathematical physics working in this area, with the aim to educate a new generation of scientists in this exciting subject. The school will provide a pedagogical review of the current state of the various constructions of knot homologies, and also encourage interactions between the communities in order to facilitate development of the unified picture.

Modern algebraic topology is a broad and vibrant field which has seen recent progress on classical problems as well as exciting new interactions with applied mathematics. This summer school will consist of a series of lecture by experts on major research directions, including several lectures on applied algebraic topology. Participants will also have the opportunity to have guided interaction with the seminal texts in the field, reading and speaking about the foundational papers.

The MSRI-UP summer program is designed for undergraduate students who have completed two years of university-level mathematics courses and would like to conduct research in the mathematical sciences. Due to funding restrictions, only U.S. citizens and permanent residents are eligible to apply and the program cannot accept foreign students regardless of funding. The academic portion of the 2013 program will be led by Dr. Rosa Orellana from Dartmouth College.

The Bay Area Differential Geometry Seminar meets 3 times each year and is a 1-day seminar on recent developments in differential geometry and geometric analysis, broadly interpreted. Typically, it runs from mid-morning until late afternoon, with 3-4 speakers. Lunch will be available and the final talk will be followed by dinner.

Location: Department of Mathematics, Stanford University

The workshop will examine the interplay between measures of singularities coming both from characteristic p methods of commutative algebra, and invariants of singularities coming from birational algebraic geometry. There is a long history of this interaction which arises via the "reduction to characteristic p" procedure. It is only in the last few years, however, that very concrete objects from both areas, namely generalized test ideals from commutative algebra and multiplier ideals from birational geometry, have been shown to be intimately connected. This workshop will explore this connection, as well as other topics used to study singularities such as jets schemes and valuations.

In recent years there have been increasing interactions between noncommutative algebra/representation theory on the one hand and algebraic geometry on the other. This workshop would aim to examine these interactions and, as importantly, to encourage the interactions between the three areas. The precise topics will become more precise nearer the time, but will certainly include:

Noncommutative algebraic geometry; Noncommutative resolutions of singularities and Calabi-Yau algebras; Symplectic reflection and related algebras; D-module theory; Deformation-quantization
 

This workshop will explore the fundamental problems of trying to assess students' mathematical proficiency, seeking to take a more comprehensive perspective on what it is to learn, know, and use mathematics. The advent of the Common Core State Standards both increases the demand and broadens the conception of what it is to be mathematically skillful, and opens new opportunities and challenges to improving our ability to assess what students understand and can do.

Recently there has been substantial progress in our understanding of the related questions of which hyperbolic groups are cubulated on the one hand, and which contain a surface subgroup on the other. The most spectacular combination of these two ideas has been in 3-manifold topology, which has seen the resolution of many long-standing conjectures. In turn, the resolution of these conjectures has led to a new point of view in geometric group theory, and the introduction of powerful new tools and structures. The goal of this conference will be to explore the further potential of these new tools and perspectives, and to encourage communication between researchers working in various related fields.

AWM launches a New Series of Biennial Research Symposia

AWM Research Symposium 2013 will be held at Santa Clara University March 16 -17, 2013. The symposium, the initial event in the series, will showcase the research of women in the mathematical professions. It will feature three plenary talks, special sessions on a broad range of research in pure and applied mathematics, poster sessions for graduate students, and a panel discussion of the "imposter syndrome." Join us next spring on the Santa Clara University campus.

The workshop will focus on recent breakthroughs in understanding and applications of free resolutions and on interactions of commutative algebra and representation theory, where algebraic geometry often appears as a third player. A specific goal is to stimulate further interaction between these fields.

This workshop will provide several short lecture series consisting two or three lectures each to introduce postdocs, graduate students and non-experts to some of the major themes of the conference. While the precise topics may change to reflect developments in the area, it is likely that we will run mini-series in the following subjects:

Noncommutative algebraic geometry; D-Module Theory; Derived Categories; Noncommutative Resolutions of Singularities; Deformation-Quantization; Symplectic Reflection Algebras; Growth Functions of Infinite Dimensional Algebras.

The Connections for Women workshop associated to the MSRI program in noncommutative algebraic geometry and representation theory is intended to bring together women who are working in these areas in all stages of their careers.

As the first event in the semester, this workshop will feature a "tapas menu" of current research and open questions: light but intriguing tastes, designed to encourage further exploration and interest. Talks will be aimed at a fairly general audience and will cover diverse topics within the theme of the program. In addition, there will be a poster session for graduate students and recent PhD recipients and a panel discussion on career issues, as well as free time for informal discussion.

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.

This workshop on Combinatorial Commutative Algebra aims to bring together researchers studying toric algebra and degenerations, simplicial objects such as monomial ideals and Stanley-Reisner rings, and their connections to tropical geometry, algebraic statistics, Hilbert schemes, D-modules, and hypergeometric functions.

Cluster algebras provide a unifying algebraic/combinatorial framework for a wide variety of phenomena in settings as diverse as quiver representations, Teichmuller theory, Poisson geometry, Lie theory, discrete integrable systems, and polyhedral combinatorics.

The workshop aims at presenting a broad view of the state-of-the-art understanding of the role of cluster algebras in all these areas, and their interactions with each other.

This workshop will take place at the opening of the MSRI special programs on Commutative Algebra and on Cluster Algebras. It will feature lecture series at different levels, to appeal to a wide variety of participants. There will be minicourses on the basics of cluster algebras, and others developing particular aspects of cluster algebras and commutative algebra.

This workshop will present basic notions from Commutative Algebra and Cluster Algebras, with a particular focus on providing background material. Additionally, the workshop aims to encourage and facilitate the exchange of ideas between researchers in Commutative Algebra and researchers in Cluster Algebras.

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/.

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.

The workshop will consist of two minicourses, together with a selection of topical lectures.

In the model theory course, o-minimality, and specifically the concrete example of the semi-algebraic sets of real numbers will provide the setting in which we introduce various fundamental results from model theory.
The algebraic dynamics course will allow the introduction of concepts and proof techniques from number theory and algebraic geometry in the context of applications involving model theory.

Toward the end of the workshop, the two minicourses will converge on the Pila-Wilkie theorem concerning points on analytic varieties, a result crucial in recent applications of o-minimality to diophantine geometry.

Mathematical general relativity is the study of mathematical problems related to Einstein's theory of gravitation. There are interesting connections between the physical theory and problems in differential geometry and partial differential equations.

The purpose of the workshop is to introduce graduate students to some fundamental aspects of mathematical general relativity, with particular emphasis on the geometry of the Einstein constraint equations and the Positive Mass Theorem. These topics will comprise a component of the upcoming semester program at MSRI in Fall 2013.

There will be mini-courses, as well as several research lectures. Students are expected to have had courses in graduate real analysis and Riemannian geometry, while a course in graduate-level partial differential equations is recommended.

This Summer Graduate Workshop will be held in Park City, Utah.

Some mobility between the Research in Mathematics and Graduate Summer School programs is expected and encouraged, but interested candidates should read the guidelines carefully and apply to the one program best suited to their field of study and experience. Postdoctoral scholars who are working in the field of Geometric Group Theory should apply to the Research Program in Mathematics, not to the Graduate Summer School.
Graduate students who are beyond their basic courses and recent PhDs in all fields of mathematics are encouraged to apply to the Graduate Summer School. Funding will go primarily to graduate students. Postdoctoral scholars not working in the field of Geometric Group Theory should also apply, but should be within four years of receipt of their PhD.
Deadline for submission of applications is January 31, 2012. Supplemental materials (such as Reference Letters) must be received in the PCMI office by February 4, 2012. Please plan accordingly. (Late applications may be accepted at the discretion of the organizers.) Response may be expected in early April. Financial support is available. Applicants are invited to request financial support by checking the appropriate boxes on the application form.

This Summer Graduate Workshop will be held in Montreal, Canada.

One of the cornerstones of the probabilistic approach to solving combinatorial problems is the following guiding principle: information about global structure can be obtained through local analysis. This principle is ubiquitous in probabilistic combinatorics. It arises in problems ranging from graph colouring, to Markov chain mixing times, to Szemerédi's regularity lemma and its applications, to the theory of influences. The 2012 Séminaire de Mathématiques Supérieures brings together experts in probabilistic combinatorics from around the world, to explain cutting edge research which in one way or another exhibits this principle.

This workshop will introduce some of the major themes of the MSRI program "Interactions between Noncommutative Algebra, Representation Theory, and Algebraic Geometry" to be held in the spring of 2013. There will be four mini-courses on the topics of noncommutative projective geometry, deformation theory, noncommutative resolutions of singularities, and symplectic reflection algebras. As well as providing theoretical background, the workshop will aim to equip participants with some intuition for the many open problems in this area through worked examples and experimental computer calculations.

The field of random media has been the object of intensive mathematical research over the last thirty years. It covers a variety of models, mainly from condensed matter physics, physical chemistry, and geology, where one is interested in materials which have defects or inhomogeneities. These features are taken into account by letting the medium be random. It has been found that this randomness can cause very unexpected effects in the large scale behavior of these models; on occasion these run contrary to the prevailing intuition. A feature of this area, which it has in common with other areas of statistical physics, is that what was initially thought to be just a simple toy model has turned out to be a major mathematical challenge.

Our understanding of the scaling limits of discrete statistical systems has shifted in recent years from the physicists' field-theoretical approaches to the more rigorous realm of probability theory and complex analysis. The aim of this workshop is to combine both discrete and continuous approaches, as well as the statistical physics/combinatorial and the probabilistic points of view. Topics include quantum gravity, planar maps, discrete conformal analysis, SLE, and other statistical models such as loop gases.

Over the last ten years there has been spectacular progress in the understanding of geometrical properties of random processes. Of particular importance in the study of these complex random systems is the aspect of their phase transition (in the wide sense of an abrupt change in macroscopic behavior caused by a small variation in some parameter) and critical phenomena, whose applications range from physics, to the performance of algorithms on networks, to the survival of a biological species.

Recent advances in the scope of rigorous scaling limits for discrete random systems, most notably for 2D systems such as percolation and the Ising model via SLE, have greatly contributed to the understanding of both the critical geometry of these systems and the behavior of dynamical stochastic processes modeling their evolution. While some of the techniques used in the analysis of these systems are model-specific, there is a remarkable interplay between them. The deep connection between percolation and interacting particle systems such as the Ising and Potts models has allowed one model to successfully draw tools and rigorous theory from the other.

The aim of this workshop is to share and attempt to push forward the state-of-the-art understanding of the geometry and dynamic evolution of these models, with a main focus on percolation, the random cluster model, Ising and other interacting particle systems on lattices.

The workshop will focus on recent developments concerning various quantitative aspects of "thin groups". These are discrete subgroups of semisimple Lie groups which are both « big » (i.e. Zariski dense) and « small » (i.e. of infinite co-volume). This dual nature leads to many intricate questions. Over the past few years, many new ideas and techniques, arising in particular from arithmetic combinatorics, have been involved in the study of such groups, leading for instance to far-reaching generalizations of the strong approximation theorem in which congruence quotients are shown to exhibit a spectral gap (super-strong approximation).

Simultaneously and sometimes surprisingly, the study of thin groups turns out to be of fundamental importance in a variety of subjects, including equidistribution of homogeneous flows and lattice points counting problems, dynamics on Teichmuller space, the Bourgain-Gamburd-Sarnak sieve in orbit, and arithmetic or geometric properties of certain types of monodromy groups and coverings. The workshop will gather a variety of experts from group theory, number theory, ergodic theory and harmonic analysis to present the accomplishments to date to a broad audience and discuss directions for further study.

The Bay Area Differential Geometry Seminar meets 3 times each year and is a 1-day seminar on recent developments in differential geometry and geometric analysis, broadly interpreted. Typically, it runs from mid-morning until late afternoon, with 3-4 speakers. Lunch will be available and the final talk will be followed by dinner.

Location: Stanford University

Research at the interface of lattice statistical mechanics and combinatorial problems of ``large sets" has been and exciting and fruitful field in the last decade or so. In this workshop we plan to develop a broad spectrum of methods and applications, spanning the spectrum from theoretical developments to the numerical end. This will cover the behaviour of lattice models at a macroscopic level (scaling limits at criticality and their connection with SLE) and also at a microscopic level (combinatorial and algebraic structures), as well as efficient enumeration techniques and Monte Carlo algorithms to generate these objects.

This 2-day workshop will bring together researchers from discrete mathematics, probability theory, theoretical computer science and statistical physics to explore topics at their interface. The focus will be on combinatorial structures, probabilistic algorithms and models that arise in the study of physical systems. This will include the study of phase transitions, probabilistic combinatorics, Markov chain Monte Carlo methods, and random structures and randomized algorithms.

Since discrete lattice models stand at the interface of these fields, the workshop will start with background talks in each of the following three areas: Statistical and mathematical physics; Combinatorics of lattice models; Sampling and computational issues. These talks will describe the general framework and recent developments in the field and will be followed with shorter talks highlighting recent research in the area.

The workshop will celebrate academic and gender diversity, bringing together women and men at junior and senior levels of their careers from mathematics, physics and computer science.

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.

Geometric problems which are inherently quantitative occur in various aspects of theoretical computer science, including
a) Algorithmic tasks for geometric questions such as clustering and proximity data structures.
b) Geometric methods in the design of approximation algorithms for combinatorial optimization problems, including the analysis of semidefinite programs and embedding methods.
c) Geometric questions arising from computational complexity, particularly in hardness of approximation. These include isoperimetric and Fourier analytic problems. These include isoperimetric and Fourier analytic problems.

This workshops aims to present recent progress in these directions.

The Mathematical Sciences Research Institute (MSRI), in conjunction with the Chern Institute of Mathematics (CIM) in Tianjin, China, celebrates the centennial of the birth of Shiing-Shen Chern, one of the greatest geometers of the 20th century and MSRI's co-founder. In commemoration of Chern's work, MSRI and CIM will hold a two-week international mathematics conference. During the first week, October 24 to 28, 2011, the conference will take place at CIM in Tianjin, China. During the second week, October 30 to November 5, 2011, the conference will be held at MSRI in Berkeley, California.

The auditorium at MSRI can seat about 140 participants. We advise early registration.

This workshop is devoted to various kinds of embeddings of metric spaces into Banach spaces, including biLipschitz embeddings, uniform embeddings, and coarse embeddings, as well as linear embeddings of finite dimensional spaces into low dimensional $\ell_p^n$ spaces. There will be an emphasis on the relevance to geometric group theory, and an exploration into the use of metric differentiation theory to effect embeddings.

"Probabilistic Reasoning in Quantitative Geometry" refers to the use of probabilistic techniques to prove geometric theorems that do not have any a priori probabilistic content. A classical instance of this approach is the probabilistic method to prove existence of geometric objects (examples include Dvoretzky's theorem, the Johnson-Lindenstrauss lemma, and the use of expanders and random graphs for geometric constructions). Other examples are the use of probabilistic geometric invariants in the local theory of Banach spaces (sums of independent random variables in the context of type and cotype, and martingale-based invariants), the more recent use of such invariants in metric geometry (e.g., Markov type in the context of embedding and extension problems), probabilistic tools in group theory, the use of probabilistic methods to prove geometric inequalities (e.g., maximal inequalities, singular integrals, Grothendieck inequalities), the use of probabilistic reasoning to prove metric embedding results such as Bourgain's embedding theorem (where the embedding is deterministic, but its analysis benefits from a probabilistic interpretation), probabilistic interpretations of curvature and their applications, and the use of probabilistic arguments in the context of isoperimetric problems (e.g., Gaussian, rearrangement, and transportation cost methods).

Differentiability of Lipschitz functions and tangents of sets

Quantitative Geometry deals with geometric questions in which quantitative or asymptotic considerations occur. The workshop will provide a mathematical introduction, a foretaste, of the many themes this exciting topic comprises: geometric group theory, theory of Lipschitz functions, large scale and coarse geometry, embeddings of metric spaces, quantitative aspects of Banach space theory, geometric measure theory and of isoperimetry, and more.

This workshop will provide an introduction to the program on Quantitative Geometry. There will be several short lecture series, given by speakers chosen for the accessibility of their lectures, designed to introduce non-specialists or students to some of the major themes of the program.

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.

Cluster algebras are a class of combinatorially defined rings that provide a unifying structure for phenomena in a variety of algebraic and geometric contexts. A partial list of related areas includes quiver representations, statistical physics, and Teichmuller theory. This summer workshop for graduate students will focus on the combinatorial aspects of cluster algebras, thereby providing a concrete introduction to this rapidly-growing field. Besides providing background on the fundamentals of cluster theory, the summer school will cover complementary topics such as total positivity, the polyhedral geometry of cluster complexes, cluster algebras from surfaces, and connections to statistical physics. No prior knowledge of cluster algebras will be assumed.

The workshop will consist of four mini-courses with accompanying tutorials. Students will also have opportunities for further exploration using computer packages in Java and Sage.

In cooperation with INdAM (Istituto Nazionale di Alta Matematica) and the SMI (Scuola Matematica Interuniversitaria), MSRI will sponsor a summer graduate workshop (SGW) on toric varieties in Cortona during summer of 2011; the workshop will reprise the very successful SGW on toric varieties held at MSRI in 2009.
Toric varieties are algebraic varieties defined by combinatorial data, and there is a wonderful interplay between algebra, combinatorics and geometry involved in their study. Many of the key concepts of abstract algebraic geometry (for example, constructing a variety by glueing affine pieces) have very concrete interpretations in the toric case, making toric varieties an ideal tool for introducing students to abstruse concepts.

Special restrictions apply, please see the workshop homepage.

Geometric Measure Theory (GMT) is a field of Mathematics that has contributed greatly to the development of the calculus of variations and geometric analysis. In recent years it has experienced a new boom with the development of GMT in the metric space setting which has lead to unexpected applications (for examples to questions arising from theoretical computer sciences). The goal of this summer graduate workshop is to introduce students to different aspects of this field. There will be 5 mini-courses and a couple of research lectures. We expect students to have a solid background in measure theory.

The study of moduli spaces of Riemann surface is a rich mixture of geometric topology, algebraic topology, complex analysis and algebraic geometry. Each community of researchers that studies these moduli spaces generates its own problems and its own techniques for solving them. However, it is not uncommon for researchers in one community to solve problems generated by another once they become aware of them. The goal of this summer school is to give graduate students a broad background in the various approaches to the study of moduli spaces of Riemann surfaces so that they will be aware of the problems and techniques of many of the communities that study these fascinating objects. Graduate student participants from the various communities will be encouraged to interact with their colleagues from the other communities of students in order to maximize cross fertilization.

Special restrictions apply, please see the workshop homepage.

In cooperation with the CRM (Centre de Recherches Mathematiques), the Fields Institute, and the PIMS (Pacific Insitute for Mathematical Sciences), MSRI will sponsor a summer graduate workshop on Metric measure spaces: geometric and analytic aspects in Montreal, Canada.
In recent decades, metric-measure spaces have emerged as a fruitful source of mathematical questions in their own right, and as indispensable tools for addressing classical problems in geometry, topology, dynamical systems and partial differential equations. The purpose of the 2011 summer school is to lead young scientists to the research frontier concerning the analysis and geometry of metric-measure spaces, by exposing them to a series of mini-courses featuring leading researchers who will present both the state-of-the-art and the exciting challenges which remain.

Special restrictions apply, please see the workshop homepage.

The MSRI-UP summer program is designed for undergraduate students who have completed two years of university-level mathematics courses and would like to conduct research in the mathematical sciences. Due to funding restrictions, only U.S. citizens and permanent residents are eligible to apply and the program cannot accept foreign students regardless of funding.

This workshop will focus on the classical Dirichlet space of holomorphic functions on the unit disk. This space is at the center of several active, interrelated areas of research that, viewed more broadly, focus on the interaction between function theoretic operator theory and potential theory. There are several goals of this Summer Graduate Workshop. First, mathematically, the workshop will demonstrate the basic properties of the Dirichlet space, then introduce the technique of Trees in Function Spaces. The workshop will show the interconnections between the areas of Complex Analysis, Function Theory, and Operator Theory and will also illustrate the real-variable analogues of the analytic result discussed.

SELECTION OF PARTICIPANTS IS NOW CLOSED.

The MSRI-UP summer program is designed for undergraduate students who have completed two years of university-level mathematics courses and would like to conduct research in the mathematical sciences. MSRI-UP includes summer research opportunities, mentoring, workshops on the graduate school application process, and follow-up support.

This workshop will involve a combination of theory and symbolic computation in commutative algebra. The lectures are intended to introduce three active areas of research: Boij-Söderberg theory, algebraic statistics, and integral closure. The lectures will be accompanied with tutorials on the computer algebra system Macaulay 2.

Refreshments after lecture at La Val's Pizza.

Refreshments after lecture at La Val's Pizza

Our workshop will highlight some work relevant to or carried out during our program at the MSRI, including statistical results about ranks for elliptic curves, zeros of L-functions, curves over finite fields, as well as algorithms for L-functions, point counting, and automorphic forms.