While their original aim was to explain the strange behavior of certain magnetic alloys, the study of spin glass models has led to a far-reaching and beautiful physical theory whose techniques have been applied to a myriad of problems in theoretical computer science, statistics, optimization and biology. As many of the physical predictions can be formulated as purely mathematical questions, often extremely hard, about large random systems in high dimensions, in recent decades a new area of research has emerged in probability theory around these problems.

Mathematically, a mean-field spin glass model is a Gaussian process (random function) on the discrete hypercube or the sphere in high dimensions. A fundamental challenge in their analysis is, roughly speaking, to understand the size and structure of their super-level sets as the dimension tends to infinity, which are often studied through smooth objects like the free energy and Gibbs measure whose origin is in statistical physics. The aim of the summer school is to introduce students to landmark results on the latter while emphasizing the techniques and ideas that were developed to obtain them, as well as exposing the students to some recent research topics.

This two week summer school, jointly organized by SLMath with OIST, will offer the following two mini-courses:

1. Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form
   This course will present some recent developments in the theory of divergence-measure fields via measure-theoretic analysis and its applications to the analysis of nonlinear PDEs of conservative form – nonlinear conservation laws.
2. Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs
   This course will present some recent developments precisely characterizing the regularity of the point at ∞ for second order elliptic and parabolic PDEs and broadly extending the role of the Wiener test in classical analysis.

Product of projective lines embedded in projective 3-space

This summer graduate school focuses on modern homological techniques in commutative algebra, specifically those involving multigraded and differential graded structures. These topics have a long and rich history, but neither is generally covered in graduate courses. Moreover, recent developments have exhibited exciting interplay between the two subjects. 

The purpose of the school is to introduce the participants to modern themes on these topics, including Koszul duality for toric varieties and differential graded algebra structures on resolutions. The school will consist of two lectures each day and carefully planned problem sessions designed to reinforce the foundational material, with an emphasis on using computational tools such as the symbolic algebra program Macaulay2. 

ALCF Visualization and Data Analytics Team; Adam Burrows and the Princeton Supernova Theory Group, Princeton University

This summer school will give an accessible introduction to the mathematical study of general relativity, a field which in the past has had barriers to entry due to its interdisciplinary nature, and whose study has been concentrated at specific institutions, to a wider audience of students studying at institutions throughout the U.S., Europe and Greece. Another goal of the summer school will be to demonstrate the common underlying mathematical themes in many problems which traditionally have been studied by separate research communities.

In the last few years, there have been extraordinary developments in many aspects of curve theory. Beginning with many examples in low genus, this summer school will introduce the participants to the background behind these developments in the following areas:

1. moduli spaces of stable curves
2. Brill–Noether theory
3. the extrinsic geometry of the curves in projective space

We will also include an introduction to some open problems at the forefront of these active areas.

This summer school will familiarize students with the basic problems of the mathematical theory of Euclidean quantum fields. The lectures will introduce some of its prominent models and analyze them via the so called “stochastic quantization” methods, involving recently developed stochastic and PDE techniques. This is an area which is highly interdisciplinary combining ideas ranging from the theory of partial differential equations, to stochastic analysis, to mathematical physics. Our goal is to bring together students who are perhaps familiar with some but not all of these subjects and teach them how to integrate these different tools to solve cutting-edge problems of Euclidean quantum field theory.

This summer school provides the mathematical background to recognize Koszul duality in representation theory. The school is especially oriented toward applications in the local Langlands program, with an emphasis on real groups. As Koszul duality patterns have been initially observed in the context of Hecke algebras, our school will also introduce the students to Hecke algebras and their categorifications.

The image of a large sphere isometrically embedded into a small space through a C^1 embedding. (Attributions: E. Bartzos, V. Borrelli, R. Denis, F. Lazarus, D. Rohmer, B. Thibert)

This two week summer school, jointly organized by SLMath with RIKEN, will introduce graduate students to the theory of h-principles.  After building up the theory from basic smooth topology, we will focus on more recent developments of the theory, particularly applications to symplectic and contact geometry, fluid dynamics, and foliation theory.

h-principles in smooth topology (Emmy Murphy)
Riemannian geometry and applications to fluid dynamics (Dominik Inauen)
Contact and symplectic flexibility (Emmy Murphy)
Foliation theory and diffeomorphism groups (Takashi Tsuboi)

This two week summer school, jointly organized by SLMath with IBM Zurich, will introduce students to the mathematics and algorithms used in the design and analysis of quantum-safe cryptosystems. Each week will be dedicated to two of the four families of quantum-safe schemes.

a Calabi-Yau manifold

This summer school will serve as an introduction to the SLMath program "Special geometric structures and analysis". There will be two mini-courses: one in Geometric Measure theory and the other in Microlocal Analysis. The aim is to give the basic notions of two subjects also treated during the program.

This summer school will focus on the introductory notions related to the passage of Newtonian and quantum many-body dynamics to kinetic collisional models of Boltzmann flow models arising in statistical sciences in connection to model reductions when continuum macro dynamics arises; and their numerical schemes associated to transport of kinetic processes in classical and data driven mean field dynamics incorporating recent tools from computational kinetics and data science tools. There will be two sets of lectures: “From Newton to Boltzmann to Fluid dynamics”, and “Kinetic collisional theory in mean field regimes: analysis, discrete approximations, and applications”. Each lecture series will be accompanied by a collaboration session, led by the lecturer and teaching assistants. The purpose of the collaboration sessions is to encourage and strengthen higher-level thinking of the materials taught in the lectures and to direct further reading for interested students. Interactive learning activities will be conducted. For example, students will be given problem sets associated with the lectures and will work in small groups to discuss concepts and/or find solutions to assigned problems. The students will also be encouraged to give oral or poster presentations on their solutions or other materials relevant to the course.

The MSRI-UP summer program is designed to serve a diverse group of undergraduate students who would like to conduct research in the mathematical sciences.

In 2024, MSRI-UP will focus on Mathematical Endocrinology. The research program will be led by Dr. Erica J. Graham, Associate Professor in the Department of Mathematics at Bryn Mawr College.

This school will present various developments in Riemannian and Kähler geometry around the notion of curvature seen as a tool to describe and understand the geometry of the objects. The school will give graduate students the opportunity to learn key ideas and techniques of the field, with an emphasis on solidifying foundations in view of potential future research. The first week will be centered around the question of the existence of Kähler metrics with special curvature properties and the famous Yau-Tian-Donaldson conjecture. The second week will focus on geometric flows in Riemannian and complex geometry. 

"May 12 Initiative" Annual Workshop

The Simons Laufer Mathematical Sciences Institute (SLMath) celebrates the "May 12 Initiative" with a panel discussion and social event open to all on the topic "Being a Woman in Mathematics". This is a hybrid event taking place on Zoom and in person at SLMath. This event is free and open to worldwide participation.

If you plan to participate online, please connect using this LINK.  

The undergraduate mathematics education system remains a huge barrier to college completion and to equity in higher education. The problem in entry level mathematics courses is not a lack of innovation. Numerous projects and institutions have created, piloted, and occasionally replicated effective reform efforts that overcame particular challenges, like the need to improve pedagogical practices or attend to gender equity. The biggest barrier to systemic reform – implementing many of these research-backed innovations at scale – is a structural one, particularly at large research-focused institutions. This workshop will bring together a group of stakeholders to explore a new avenue for change, the rise of teaching-focused faculty at research-intensive institutions who increasingly influence introductory coursework. By creating a network that connects these faculty across institutions, change at scale across 50, 100, or even more institutions becomes possible – on issues ranging from pedagogy to equity to curricular innovation. Creating such structures would also allow for bringing future innovations to scale much more quickly than is currently possible.

Drawing by Tina Zhang.

This workshop is about the recent MIP*=RE result from quantum computational complexity, and the resulting resolution of the Connes embedding problem from the theory of von Neumann algebras. MIP*=RE connects the disparate areas of computational complexity theory, quantum information, operator algebras, and approximate representation theory. The aim of this workshop is to bridge this divide, by giving an in-depth exposition of the techniques used in the proof of MIP*=RE, and highlighting perspectives on the MIP*=RE result from operator algebras and approximate representation theory. In particular, this workshop will highlight connections with group stability, something that has not been covered in previous workshops. In addition to increasing understanding of the MIP*=RE proof, we hope that this will open up further applications of the ideas behind MIP*=RE in operator algebras.

Several executions of a 3-dimensional sumcheck protocol with a random order of directions (thanks to Dev Ojha for creating the diagram)

Proofs are at the foundations of mathematics. Viewed through the lens of theoretical computer science, verifying the correctness of a mathematical proof is a fundamental computational task. Indeed, the P versus NP problem, which deals precisely with the complexity of proof verification, is one of the most important open problems in all of mathematics.

The complexity-theoretic study of proof verification has led to exciting reenvisionings of mathematical proofs. For example, probabilistically checkable proofs (PCPs) admit local-to-global structure that allows verifying a proof by reading only a minuscule portion of it. As another example, interactive proofs allow for verification via a conversation between a prover and a verifier, instead of the traditional static sequence of logical statements. The study of such proof systems has drawn upon deep mathematical tools to derive numerous applications to the theory of computation and beyond.

In recent years, such probabilistic proofs received much attention due to a new motivation, delegation of computation, which is the emphasis of this summer school. This paradigm admits ultra-fast protocols that allow one party to check the correctness of the computation performed by another, untrusted, party. These protocols have even been realized within recently-deployed technology, for example, as part of cryptographic constructions known as succinct non-interactive arguments of knowledge (SNARKs).

This summer school will provide an introduction to the field of probabilistic proofs and the beautiful mathematics behind it, as well as prepare students for conducting cutting-edge research in this area.

This summer school will introduce graduate students to sketching-based approaches to computational linear and multi-linear algebra. Sketching here refers to a set of techniques for compressing a matrix, to one with fewer rows, or columns, or entries, usually via various kinds of random linear maps. We will discuss matrix computations, tensor algebras, and such sketching techniques, together with their applications and analysis.

The goal of the summer school is for the students to first become familiar with the concept of concentration of measure in different settings (Euclidean, Riemannian and discrete), and the main open problems surrounding it. The students will later become familiar with the proof techniques that involve the different types of localization and obtain expertise on the ways to apply the localization techniques. After attending the graduate school, the students are expected to have the necessary background that would give them a chance to both conduct research around open problems in concentration of measure, find new applications to existing localization techniques and perhaps also develop new localization techniques.

Schur quartic x 4−xy3 = z 4−zu3 and several of the 64 lines that it contains

Derived algebraic geometry is an ‘update’ of algebraic geometry using ‘derived’ (roughly speaking, homological) techniques. This requires recasting the very foundations of the field: rings have to be replaced by differential graded algebras (or other forms of derived rings), categories by higher categories, and so on. The result is a powerful set of new tools, useful both within algebraic geometry and in related areas. The school serves as an introduction to these techniques, focusing on their applications.

The school is built around two related courses on geometric (‘derived spaces’) and categorical (‘derived categories’) aspects of the theory. Our goal is to explain the key ideas and concepts, while trying to keep technicalities to a minimum.

The overarching goal of this summer school is to expose the students both to modern forms of unsupervised learning — in the form of geometrical and topological data analysis — and to supervised learning — in the form of (deep) neural networks applied to regression/classification problems. The organizers have opted for a lighter exposure to a broader range of topics. Using the metaphor of a meal, we are offering 2 + 2 samplers — geometry and topology for data analysis + theoretical and practical deep learning — rather than 1 + 1 main dishes. The main goal, thus, is to inspire the students to learn more about one or several of the topics covered in the school.

The expected learning outcomes for students attending the school are the following:

1. An introduction to how concepts and tools from geometry and topology can be leveraged to perform data analysis in situations where the data are not labeled.

2. An introduction to recent and ongoing theoretical and methodological/practical developments in the use of neural networks for data analysis (deep learning).

Soap bubble: equilibrium solution of the rescaled mean curvature flow and constant curvature surface.

This graduate summer school will introduce students to two important and inter-related fields of differential geometry: geometric flows and minimal surfaces.

Geometric flows have had far reaching influences on numerous branches of mathematics and other scientific disciplines. An outstanding example is the completion of Hamilton’s Ricci flow program by Perelman, leading to the resolution of the Poincare conjecture and Thurston’s geometrization conjecture for 3-manifolds. In this part of the summer school, students will be guided through basic topics and ideas in the study of geometric flows.

Since Penrose used variations of volume to formulate and study black holes in general relativity (in his Nobel prize-winning work), the intriguing connections between minimal surfaces and general relativity have been a strong driving force for the modern developments of both research areas. This part of the summer school will introduce students to the basic theory of minimal submanifolds and its applications in Riemannian geometry and general relativity.

The curriculum of this program will be accessible and will have a broad appeal to graduate students from a variety of mathematical areas, introducing some of the latest developments in each area and the remaining open problems therein, while simultaneously emphasizing their synergy.

This two week school will focus on spectral theory of periodic, almost-periodic, and random operators.  The main aim of this school is to teach the students who work in one of these areas, methods used in parallel problems, explain the similarities between all these areas and show them the `big picture'.

This school is associated with an upcoming research program at MSRI under the same title. The goal of the school is to equip students unfamiliar with these topics with the mathematical and theoretical computer science toolbox that forms the foundation of market and mechanism design.

A basic enzymatic mechanism

The aim of the course is to learn how tools from algebraic geometry (in particular, from computational and real algebraic geometry) can be used to analyze standard models in molecular biology. Particularly, these models are key ingredients in the development of Systems and Synthetic biology, two active research areas focusing on understanding, modifying, and implementing the design principles of living systems.

We will focus on the mathematical aspects of the methods, and exemplify and apply the theory to real networks, thereby introducing the participants to relevant problems and mechanisms in molecular biology. As a counterpart, however, the participants will also see how this field has in the past challenged current methods, mainly in the realm of real algebraic geometry, and has given rise to new general and purely theoretical results on polynomial equations. We will end our lectures with an overview of open questions in both fields.

The MSRI-UP summer program is designed to serve a diverse group of undergraduate students who would like to conduct research in the mathematical sciences.

In 2023, MSRI-UP will focus on Topological Data Analysis. The research program will be led by Dr. Jose Perea, Associate Professor in the Department of Mathematics and the Khoury College of Computer Sciences at Northeastern University.

Some basic concepts in mathlib and the dependencies between them

Computational proof assistants now make it possible to develop global, digital mathematical libraries with theorems that are fully checked by computer. This summer school will introduce students to the new technology and the ideas behind it, and will encourage them to think about the goals and benefits of formalized mathematics. Students will learn to use the Lean interactive proof assistant, and by the end of the session they will be in a position to formalize mathematics on their own, join the Lean community, and contribute to its mathematical library.

Commutative Algebra has seen an extraordinary development in the last few years. Long standing conjectures have been proven and new connections to different areas of mathematics have been built.This summer graduate school will consist of three mini-courses (5 lectures each) on fundamental topics in commutative algebra that are not covered in the standard courses. Each course will be accompanied by problem sessions focused on research. Five general colloquium-style lectures will be given by invited scholars who will also attend the school and help with afternoon research activities. 

2023 Pilot Program: The goal of MAY-UP is to provide students with a glimpse into Linear Algebra; and the ways in which this topic may arise both theoretically and computationally in their future studies. Material will include an introduction to matrices as well as basic matrix operations. We will also provide students with introductory programming skills in Python, including development environment setup and matrix manipulations via code.

The Simons Laufer Mathematical Sciences Institute (SLMath), formerly MSRI, celebrates May 12 with a panel discussion and social event open to all on the topic "Pathways in Mathematics". This is a hybrid event taking place on Zoom and in person at SLMath and satellite institutions.

In 2022-23, SLMath (formerly MSRI) celebrates 40 years of serving the mathematical sciences community through our topic-focused programs and workshops, and the general public via our national and global outreach initiatives. Director Tatiana Toro and Deputy Director Hélène Barcelo invite the community to join us for a symposium to reflect upon these four decades of extraordinary activity.  This celebration will feature special guest speakers, panel discussions and an evening reception.

The workshop Critical Issues in Mathematics Education: Mentoring for Equity aims to reach a broad audience of faculty and students in postsecondary mathematical sciences. Participants will learn about the evidence base for effective mentoring, with a focus on culturally responsive mentoring that supports all students and faculty along their mathematical paths. The workshop includes a combination of discussion of research evidence, review and adaptation of practical tools, and explicit training in effective mentoring, including how to bring these tools back to participants’ home institutions. The workshop intertwines objectives of increasing participants’ knowledge of the scholarship on effective mentoring, and engages participants in interactive activities to develop tangible skills as mentors and as mentor-trainers. Participants should come with a growth mindset, prepared to reflect on their experiences as mentors and mentees, and actively contribute to activities that build skills for implementing best mentoring practices.  This workshop will cultivate local and national mentoring communities that bring effective tools and strategies to mentoring, so that mentees can persist and thrive in research, teaching, education, and throughout their education and careers. One focus will be on addressing the individual mentoring needs of all faculty and students, including those who have been historically-marginalized in mathematics education and careers.

ALL FUNDING FOR THIS WORKSHOP HAS BEEN ALLOCATED

As part of the Mathematical Sciences Collaborative Diversity Initiatives, the six NSF-funded U.S. mathematics institutes will host their annual SACNAS pre-conference event, the 2022 Modern Math Workshop (MMW). The Modern Math Workshop encourages undergraduates from underrepresented groups to pursue careers in the mathematical sciences, and builds research and networking opportunities among undergraduates, graduate students and recent PhDs.

Graph of the Motzkin polynomial, which is nonnegative but not a sum of squares.

The study of nonnegative polynomials and sums of squares is a classical area of real algebraic geometry dating back to Hilbert’s 17th problem. It also has rich connections to real analysis via duality and moment problems. In the last 15 years, sums of squares relaxations have found a wide array of applications from very applied areas (e.g., robotics, computer vision, and machine learning) to theoretical applications (e.g., extremal combinatorics, theoretical computer science). Also, an intimate connection between sums of squares and classical algebraic geometry has been found. Work in this area requires a blend of ideas and techniques from algebraic geometry, convex geometry and representation theory. After an introduction to nonnegative polynomials, sums of squares and semidefinite optimization, we will focus on the following three topics:

• Sums of squares on real varieties (sets defined by real polynomial equations) and connections with classical algebraic geometry.
• Sums of squares method for proving graph density inequalities in extremal combinatorics. Here addition and multiplication take place in the gluing algebra of partially labelled graphs.
• Sums of squares relaxations for convex hulls of real varieties and theta-bodies with applications in optimization.

The summer school will give a self-contained introduction aimed at beginning graduate students, and introduce participants to the latest developments. In addition to attending the lectures, students will meet in intensive problem and discussion sessions that will explore and extend the topics developed in the lectures.

A tropical stable map and the corresponding floor diagram

Enumerative geometry and the theory of moduli spaces of curves are two cornerstones of modern algebraic geometry; the two subjects have had a significant influence on each other. In the last 15 years, discrete and combinatorial methods, systematized within tropical geometry, have begun to provide new avenues of access into these two subjects. The goal of this summer school is to give students crash courses in tropical and logarithmic geometry, with a particular focus on the applications in enumerative geometry and moduli theory.  The school will consist of three courses of seven lectures each:

1. Enumeration of tropical curves/ by Hannah Markwig
2. Curve counting in tropical and algebraic geometry by Renzo Cavalieri
3. Logarithmic geometry and stable map/s by Dhruv Ranganathan

Popular visualization of the MNIST dataset

This school is offered in partnership with Istituto Nazionale di Alta Matematica (INdAM) and the Courant Institute of Mathematical Sciences.

Learning theory is a rich field at the intersection of statistics, probability, computer science, and optimization. Over the last decades the statistical learning approach has been successfully applied to many problems of great interest, such as bioinformatics, computer vision, speech processing, robotics, and information retrieval. These impressive successes relied crucially on the mathematical foundation of statistical learning.

Recently, deep neural networks have demonstrated stunning empirical results across many applications like vision, natural language processing, and reinforcement learning. The field is now booming with new mathematical problems, and in particular, the challenge of providing theoretical foundations for deep learning techniques is still largely open. On the other hand, learning theory already has a rich history, with many beautiful connections to various areas of mathematics (e.g., probability theory, high dimensional geometry, game theory). The purpose of the summer school is to introduce graduate students (and advanced undergraduates) to these foundational results, as well as to expose them to the new and exciting modern challenges that arise in deep learning and reinforcement learning.

Recently, progress in the field of topological methods in discrete mathematics has been rapid and has generated a lot of activity with the resolution of major open problems, the emergence of new lines of inquiry, and the development of new tools. These exciting new developments have not been digested into a textbook treatment. The two main goals of this school are to:

1. Provide graduate students with a thorough introduction to novel topological techniques and to a handful of their applications in the fields of combinatorics and discrete geometry with short glimpses into mathematical mechanics and algorithm complexity.
2. Expose students to current research, and guide them in research on open problems in discrete mathematics using modern topological tools.

The summer school will lead participants from appealing, simple-to-state problems at confluence of combinatorics, geometry, and topology to sophisticated topological methods that are required for their resolution. In recent years topological methods have found numerous novel applications in mathematics and beyond, such as in data science, machine learning, economics, the social sciences, and biology.  The problems we will discuss are particularly well-suited to rapidly put students in a position to approach related research questions.

Fluid-flow stream function color-coded by vorticity in 3D flat torus calculated by K. Nakai (The University of Tokyo)

This school is offered in partnership with the National Center for Theoretical Sciences.

The purpose of the workshop is to introduce graduate students to fundamental results on the Navier-Stokes and the Euler equations, with special emphasis on the solvability of its initial value problem with rough initial data as well as the large time behavior of a solution. These topics have long research history. However, recent studies clarify the problems from a broad point of view, not only from analysis but also from detailed studies of orbit of the flow.

The PCMI graduate summer school program in 2022 will consist of a sequence of 11 minicourses. The lecturers and topics for these minicourses are listed below. Each minicourse is accompanied by a problem session. The topics are arranged so that there is good material and opportunities for learning both for less experienced students as well as more advanced students. Beyond their attendance in these minicourse sessions, all graduate participants will be able to take part in the substantial other benefits of a PCMI session. This includes the opportunity to interact with the researchers in residence and take part in the research seminar component of PCMI. Many graduate students also interact in significant ways with the undergraduate cohort,,the undergraduate faculty cohort, and may also participate in the many pedagogically focused activities which form part of the K-12 Teacher Leadership Program and the Workshop for Equity in Mathematics Education. PCMI includes numerous cross-program activities to help members from all these groups interact with one another.

Several geometric ideas in the context of a surface: hyperbolic metric, CAT(0) inequality, Gromov hyperbolicity/coarse median geometry, nonpositively-curved square tiling, Besikovitch inequality. (Picture by M. Hagen and A. Sisto.)

The purpose of the summer school is to introduce graduate students to key mainstream directions in the recent development of geometry, which sprang from Riemannian Geometry in an attempt to use its methods in various contexts of non-smooth geometry. This concerns recent developments in metric generalizations of the theory of nonpositively curved spaces and discretizations of methods in geometry, geometric measure theory and global analysis. The metric geometry perspective gave rise to new results and problems in Riemannian Geometry as well.

All these themes are intertwined and have developed either together or greatly influencing one another. The summer school will introduce some of the latest developments and the remaining open problems in these very modern areas, and will emphasize their synergy.

Image by Prof. Robert Lipshitz

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg--Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.
 
The goal of the summer school is to provide participants the tools in symplectic geometry and stable homotopy theory required to work on Floer homotopy theory. Students will come away with a basic understanding of some of the key techniques, questions, and challenges in both of these fields. The summer school may be particularly valuable for participants with a solid understanding of one of the two fields who want to learn more about the other and the connections between them.

by DeDelphin Sénizergues

The topic of random graphs is at the forefront of applied probability, and it is one of the central topics in multidisciplinary science where mathematical ideas are used to model and understand the real world. At the same time, random graphs pose challenging mathematical problems that have attracted the attention from probabilists and combinatorialists since the 1960, with the pioneering work of Erdös and Rényi. Around the turn of the millennium, very large data sets started to become available, and several applied disciplines started to realize that many real-world networks, even though they are from various different origins, share many fascinating features. In particular, many of such networks are small worlds, meaning that graph distances in them are typically quite small, and they are scalefree, in the sense that there are enormous differences in the number of connections that their elements make. In particular, such networks are quite different from the classical random graph models, such as proposed by Erdös and Rényi.

Algebraic Theory Of Differential And Difference Equations, Model Theory And Their Applications

The purpose of the summer school will be to introduce graduate students to effective methods in algebraic theories of differential and difference equations with emphasis on their model-theoretic foundations and to demonstrate recent applications of these techniques to studying dynamic models arising in sciences. While these topics comprise a coherent and rich subject, they appear in graduate coursework in at best a piecemeal way, and then only as components of classes for other aims. With this Summer Graduate School, students will learn both the theoretical basis of differential and difference algebra and how to use these methods to solve practical problems. Beyond the lectures, the graduate students will meet daily in problem sessions and will participate in one-on-one mentoring sessions with the lecturers and organizers.

This school is offered in partnership with the Australian Mathematical Sciences Institute and the University of Hawaii, Hilo.

Representation Theory has undergone a revolution in recent years, with the development of what is now known as higher representation theory. In particular, the notion of categorification has led to the resolution of many problems previously considered to be intractable.

The school will begin by providing students with a brief but thorough introduction to what could be termed the “bread and butter of modern representation theory”, i.e., compact Lie groups and their representation theory; character theory; structure theory of algebraic groups.

We will then continue on to a number of more specialized topics. The final mix will depend on discussions with the prospective lecturers, but we envisage such topics as:

• modular representation theory of finite groups (blocks, defect groups, Broué’s conjecture);

• perverse sheaves and the geometric Satake correspondence;

• the representation theory of real Lie groups.

photo courtesy of Panagiota Daskalopoulos

[The image on this vase from Minoan Crete, dated on 1500-2000 BC, resembles an ancient solution to the Curve shortening flow - one of the most basic geometric flows. The vase is at Heraklion Archaeological Museum]

This summer graduate school is a collaboration between MSRI and the FORTH-IACM Institute in Crete. The purpose of the school is to introduce graduate students to some of the most important geometric evolution equations. Information about the location of the summer school can be found here.

This is an area of geometric analysis that lies at the interface of differential geometry and partial differential equations. The lectures will begin with an introduction to nonlinear diffusion equations and continue with classical results on the Ricci Flow, the  Mean curvature flow and other fully non-linear extrinsic flows such as the Gauss curvature flow. The lectures will also include geometric applications such as isoperimetric inequalities, topological applications such as the Poincaré onjecture, as well as recent important developments related to the study of singularities and ancient solutions.

The MSRI-UP summer program is designed to serve a diverse group of undergraduate students who would like to conduct research in the mathematical sciences.

In 2022, MSRI-UP will focus on Algebraic Methods in Mathematical Biology. The research program will be led by Dr. Anne Shiu, Associate Professor of Mathematics at Texas A&M University.

L 2 Spectra of K for apertures π 15 , · · · 14π 15 , π

The field of Integral Equations has a long and distinguished history, being the driving force behind many fundamental developments in various areas of mathematics including Harmonic Analysis, Partial Differential Equations, Potential Theory, Scattering Theory, Functional Analysis, Complex Analysis, Operator Theory, Mathematical Physics and Numerical Analysis.

This school will:

1. introduce graduate students to the systematic study of integral equations;
2. present some of the latest theoretical advancements in the field and open problems; and
3. involve participants in a hands-on discovery lab focused on deriving results about integral operators in two dimensions relevant for both the theoretical and numerical treatment of Integral Equations in two dimensions. The curriculum of this program will be accessible and will have a broad appeal to graduate students from a variety of mathematical areas (both theoretical and applied).

MSRI's 2022 Celebration of Women in Math event will be for graduate students, with a focus on "How to build a Career in Math".  It will be a hybrid workshop, with online and in-person activities at satellite institutions.

The event will include a panel discussion, social activities, and breakout sessions on the following topics:

• Finding (having) mentors
• How to build a network and collaborations
• How to become an independent researcher
• How to balance teaching/research/admin/life

Registration is open. 

The world is changing, along with perceptions. Many call for the improvement of mathematics teaching and learning, for both citizenry and STEM preparation. To achieve sustainable change, though, the focus needs to extend from individuals to systems. It is not enough to change one classroom or one course. Transformation requires change at all levels: in teaching, programmatic practices, and institutions. This workshop will bring together teachers and researchers from universities, community colleges, and K-12 schools to explore the reasons for and processes by which change in university mathematics departments is initiated, promoted, and sustained and lessons learned from change efforts in K-12. It will review what we know about change at all levels and reflect on stories of failure and success.

MSRI and the Mathematical Science Institutes Diversity Initiative (MSIDI) are pleased to announce that the 2021 Blackwell-Tapia Conference (rescheduled from Fall 2020), will be held simultaneously at four locations nationwide.  The conference will celebrate the 2020 Blackwell-Tapia prize winner, Tatiana Toro (University of Washington), who has recently been announced as the next Director of MSRI, effective August 2022.

ONLY REGISTRATIONS FOR VIRTUAL PARTICIPATION ARE BEING ACCEPTED AS OF NOVEMBER 8.

Choose from four host sites nationwide:

Mathematical Sciences Research Institute (MSRI): Berkeley, California
Institute for Pure and Applied Mathematics (IPAM): Los Angeles, California
Institute for Mathematical and Statistical Innovation (IMSI): Chicago, Illinois
Institute for Advanced Study (IAS): Princeton, New Jersey

Several executions of a 3-dimensional sumcheck protocol with a random order of directions (thanks to Dev Ojha for creating the diagram)

Proofs are at the foundations of mathematics. Viewed through the lens of theoretical computer science, verifying the correctness of a mathematical proof is a fundamental computational task. Indeed, the P versus NP problem, which deals precisely with the complexity of proof verification, is one of the most important open problems in all of mathematics.

The complexity-theoretic study of proof verification has led to exciting reenvisionings of mathematical proofs. For example, probabilistically checkable proofs (PCPs) admit local-to-global structure that allows verifying a proof by reading only a minuscule portion of it. As another example, interactive proofs allow for verification via a conversation between a prover and a verifier, instead of the traditional static sequence of logical statements. The study of such proof systems has drawn upon deep mathematical tools to derive numerous applications to the theory of computation and beyond.

In recent years, such probabilistic proofs received much attention due to a new motivation, delegation of computation, which is the emphasis of this summer school. This paradigm admits ultra-fast protocols that allow one party to check the correctness of the computation performed by another, untrusted, party. These protocols have even been realized within recently-deployed technology, for example, as part of cryptographic constructions known as succinct non-interactive arguments of knowledge (SNARKs).

This summer school will provide an introduction to the field of probabilistic proofs and the beautiful mathematics behind it, as well as prepare students for conducting cutting-edge research in this area.

a random quasiconformal map obtained from Beltrami equation by randomly assigning the values of +-1/2 for the Beltrami coefficient on small squares subdividing the unit square

This Summer Graduate School will cover basic tools that are instrumental in Random Conformal Geometry (the investigation of analytic and geometric objects that arise from natural probabilistic constructions, often motivated by models in mathematical physics) and are at the foundation of the subsequent semester-long program  "The Analysis and Geometry of Random Spaces".  Specific topics are Conformal Field Theory, Brownian Loops and related processes, Quasiconformal Maps, as well as Loewner Energy and Teichmüller Theory.

Image by Nick Schmitt

Figure 1. A rotationally symmetric solution to the self-duality equations on an open and dense subset of the torus. Singularities appear where the surface intersects the ideal boundary at infinity of the hyperbolic 3-space visualized by the wireframe.

Gauge theory is a geometric language used to formulate many fundamental physical phenomena, which has also had profound impact on our understanding of topology. The main idea is to study the space of solutions to partial differential equations admitting a very large group of local symmetries. Starting in the late 1970s, mathematicians began to unravel surprising connections between gauge theory and many aspects of geometric analysis, algebraic geometry and low-dimensional topology. This influence of gauge theory in geometry and topology is pervasive nowadays, and new developments continue to emerge.

The goal of the summer school is to introduce students to the foundational aspects of gauge theory, and explore their relations to geometric analysis and low-dimensional topology. By the end of the two-week program, the students will understand the relevant analytic and geometric aspects of several partial differential equations of current interest (including the Yang-Mills ASD equations, the Seiberg-Witten equations, and the Hitchin equations) and some of their most impactful applications to problems in geometry and topology.

This summer school will introduce graduate students to sketching-based approaches to computational linear and multi-linear algebra. Sketching here refers to a set of techniques for compressing a matrix, to one with fewer rows, or columns, or entries, usually via various kinds of random linear maps. We will discuss matrix computations, tensor algebras, and such sketching techniques, together with their applications and analysis.

The MSRI-UP summer program is designed to serve a diverse group of undergraduate students who would like to conduct research in the mathematical sciences.

In 2021, MSRI-UP will focus on Parking Functions: Choose your own adventure. The research program will be led by Dr. Pamela E. Harris, Associate Professor of Mathematics at Williams College.

Mathematics and Racial Justice Workshop

The overarching goal of the Workshop on Mathematics and Racial Justice is to explore the role that mathematics plays in today’s movement for racial justice. For the purposes of this workshop, racial justice is the result of intentional, active and sustained anti-racist practices that identify and dismantle racist structures and policies that operate to oppress, disenfranchise, harm, and devalue Black people. This workshop will bring together mathematicians, statisticians, computer scientists, and STEM educators as well as members of the general public interested in using the tools of these disciplines to critically examine and eradicate racial disparities in society. Researchers with expertise or interest in problems at the intersection of mathematics, statistics and racial justice are encouraged to participate. This workshop will take place over two weeks and will include sessions on Bias in Algorithms and Technology; Fair Division, Allocation, and Representation; Public Health Disparities; and Racial Inequities in Mathematics Education.

The Corvaja-Zannier proof of Siegel's theorem using subspaces. Illustrated by Sofía Pastén Vásquez.

The theory of Diophantine equations is understood today as the study of algebraic points in algebraic varieties, and it is often the case that algebraic points of arithmetic relevance are expected to be sparse.

This summer school will introduce the participants to two of the main techniques in the subject: (i) the filtration method to prove algebraic degeneracy of integral points by means of the subspace theorem, leading to special cases of conjectures by Bombieri, Lang, and Vojta, and (ii) unlikely intersections through o-minimality and bi-algebraic geometry, leading to results in the context of the Manin-Mumford conjecture, the André-Oort conjecture, and generalizations. This SGS should provide an entry point to a very active research area in modern number theory.

The Sherrington-Kirkpatrick model, aka the randomly-weighted complete graph. Edge weights are indicated using grayscale. Six distinguished vertices have been randomly chosen; edges between those vertices are shaded black to form a "hidden signal".

The courses in this summer school focus on mathematical models of group dynamics, how to describe their dynamics and their scaling limits, and the connection to discrete and continuous optimization problems.

The phrase "group dynamics" is used loosely here -- it may refer to species migration, the spread of a virus, or the propagation of electrons through an inhomogeneous medium, to name a few examples. Very commonly, such systems can be described via stochastic processes which approximately behave like the solution of an appropriate partial differential equation in the large-population limit.

Microlocal analysis originated in the study of linear partial differential equations (PDEs) in the high-frequency regime, through a combination of ideas from Fourier analysis and classical Hamiltonian mechanics. In parallel, similar ideas and methods had been developed since the early times of quantum mechanics, the smallness of Planck’s constant allowing to use semiclassical methods. The junction between these two points of view (microlocal and semiclassical) only emerged in 1970s, and has taken its full place in the PDE community in the last 20 years. This methodology resulted in major advances in the understanding of linear and nonlinear PDEs in the last 50 years. Moreover, microlocal methods continue to find new applications in diverse areas of mathematical analysis, such as the spectral theory of nonselfadjoint operators, scattering theory, and inverse problems.

NOTE: The introductory sessions for this workshop will be held online the morning of April 29th.  Additional sessions will be held when it is once again possible to meet in person.  Times listed on schedule is in Pacfic Standard Time.

The world is changing, along with perceptions. Many call for the improvement of mathematics teaching and learning, for both citizenry and STEM preparation. To achieve sustainable change, though, the focus needs to extend from individuals to systems. It is not enough to change one classroom or one course. Transformation requires change at all levels: in teaching, programmatic practices, and institutions. This workshop will bring together teachers and researchers from universities, community colleges, and K-12 schools to explore the reasons for and processes by which change in university mathematics departments is initiated, promoted, and sustained and lessons learned from change efforts in K-12. It will review what we know about change at all levels and reflect on stories of failure and success.

Speaker Abstracts

The advent of reliable computing machines, computer algebra systems, and multiple precision computational packages diminished the need for tables of reference values for computing function values by interpolation, but today's numerical analysts, scientific researchers, and software developers still need a way to confirm the accuracy of numerical algorithms that compute mathematical function values. The field of validated computation of mathematical functions explores the development of multiple precision codes that compute certifiably accurate function values that can be used to test the accuracy of function data from personal, commercial, or publicly available codes. We discuss the analysis used to obtain reliable error bounds for floating point approximations and describe the implementation of the work in a publicly available beta site. 

In the context of several key states in the U.S.A, we will review the basics of COVID-19 and consider the post-lockdown dynamics.  In particular we will discuss the main drivers of the disease and the drawbacks to a natural herd immunity strategy. This talk represents joint work with Kamal Barley, Keisha Cook and Abba Gumel.

The largest multidisciplinary and multicultural STEM diversity event in the country, the SACNAS conference serves to equip, empower, and energize participants for their academic and professional paths in STEM.

For more information, click HERE.

Overturning wave, artistic drawing by E. Ifrim

Due to the COVID-19 pandemic, this summer school will be held online.

The purpose of this two weeks school is to introduce graduate students to the state of the art methods and results in the study of incompressible Euler’s equations in general, and water waves in particular. This is a research area which is highly relevant to many real life problems, and in which substantial progress has been made in the last decade.

The goal is to present the main current research directions in water waves. We will begin with the physical derivation of the equations, and present some of the analytic tools needed in study. The final goal will be two-fold, namely (i) to understand the local solvability of the Cauchy problem for water waves, as well as (ii) to describe the long time behavior of solutions.

Through the lectures and associated problem sessions, students will learn about a number of new analysis tools which are not routinely taught in a graduate school curriculum. The goal is to help students acquire the knowledge needed in order to start research in water waves and Euler equations.

Due to the COVID-19 pandemic, this summer school will be held online.

Probability theory, statistics as well as mathematical physics have increasingly been used in computer science. The goal of this school is to provide a unique opportunity for graduate students and young researchers to developed multi-disciplinary skills in a rapidly evolving area of mathematics.

The topics would include spin glasses, constraint satisfiability, randomized algorithms, Monte-Carlo Markov chains and high-dimensional statistics, sparse and random graphs, computational complexity, estimation and approximation algorithms. Those topics will fall into two main categories, on the one hand problems related to spin glasses and on the other hand random algorithms.

The part of the summer school dedicated to spin glasses will be split into three parts: an introductory course about traditional spin glasses followed by two more advanced courses where spin glasses meet computer science in addition to a talk on dynamics of spin glasses. The part of the summer school on random algorithms will consist of an introductory course on phase transitions in large random structures, followed by advanced courses on theoretical bounds for computational complexity in reconstruction and inference, and on understanding rare events in random graphs and models of statistical mechanics.

The two introductory courses on spin glasses and on random algorithms will be accompanied by three exercises sessions of one hour. A one hour exercises session will follow each of the three sessions of a course for both the introductory course on spin glasses and the introductory course on random algorithms. Exercises sessions will be led by an assistant, but will primarily focus on participation of the students.

The MSRI-UP summer program is designed to serve a diverse group of undergraduate students who would like to conduct research in the mathematical sciences.

In 2020, MSRI-Up will focus on Branched Covers of Curves. The research program will be led by Dr. Edray Goins, Professor of Mathematics at Pomona College.

Due to the COVID-19 virus outbreak, the Critical Issues in Mathematics Education 2020 workshop was held online. The full workshop description and list of talks can be found HERE.

On May 22 portions of the Critical Issues in Mathematics Education 2020: Today’s Mathematics, Social Justice, and Implications for Schools workshop will be streamed online via Zoom.

Friday 5/22: 12pm PST (3pm eastern time)

12:00 - 1:00
Rico Gutstein, Preparing Students Today for Whatever Tomorrow Brings

Due to the COVID-19 virus outbreak, the Critical Issues in Mathematics Education 2020 workshop was held online. The full workshop description and list of talks can be found HERE.

On May 15 portions of the Critical Issues in Mathematics Education 2020: Today’s Mathematics, Social Justice, and Implications for Schools workshop will be streamed online via Zoom.

Friday 5/15: 12pm PST (3pm eastern time)

12:00 - 1:00
Dan Reinholz, Preparing teachers to notice, name, and disrupt racial and gender inequity

Due to the COVID-19 virus outbreak, the Critical Issues in Mathematics Education 2020 workshop was held online. The full workshop description and list of talks can be found HERE.

Friday 5/8: 12pm PST (3pm eastern time)

12:00 - 1:00
Nathan Alexander, Mathematical Models in the Sociological Imagination
Lincoln Chandler, Pursuing Racial Equity within Schools 

Due to the COVID-19 virus outbreak, the Critical Issues in Mathematics Education 2020 workshop was held online. The full workshop description and list of talks can be found HERE.

Friday 5/01: 12pm PST (3pm eastern time)

12:00 - 1:00 Hyman Bass, 'Mathematics and Social Justice': An undergraduate course. What could this be?

Due to the COVID-19 virus outbreak, the Critical Issues in Mathematics Education 2020 workshop was held online. The full workshop description and list of talks can be found HERE.

Friday 4/24: 12pm PST (3pm eastern time)

12:00 - 1:00 Padmanabhan Seshaiyer, K-12 to Post-Secondary Viewpoint Critical Issues in Mathematics Education

Due to the COVID-19 virus outbreak, the Critical Issues in Mathematics Education 2020 workshop was held online. The full workshop description and list of talks can be found HERE.

Friday 4/17: 12pm PST (3pm eastern time)

12:00 - 1:00 Some unintended consequences of active learning

Sage Forbes-Gray, Sunset Park High School,  Brooklyn, NY, Mfa Master Teacher 
Sharon Collins - New Heights Academy Charter School, NYC, MfA Master Teacher; 
Kate Belin - Fannie Lou High School, NYC, MfA Master Teacher; 

Moderator: Courtney Ginsberg, MfA
Host: Katherine Stevenson, CSUN

Due to the COVID-19 virus outbreak, the Critical Issues in Mathematics Education 2020 workshop was held online. The full workshop description and list of talks can be found HERE.

Friday 4/10: 12pm PST (3pm eastern time)

12:00 - 1:00 Estrella Johnson, Some unintended consequences of active learning

Due to the COVID-19 virus outbreak, the Critical Issues in Mathematics Education 2020 workshop was held online. The full workshop description and list of talks can be found HERE.

Friday 3/27: Starting at 12pm PST (3pm eastern time)

12:00p - 1:00p
Nicol Turner Lee, Brookings Inst., Center for Tech Innov. -  Unconscious Bias
Saber Khan, Processing Foundation, leader of #EthicalCS -  Identity & Ethics

Due to the COVID-19 virus outbreak, the Critical Issues in Mathematics Education 2020 workshop was held online. The full workshop description and list of talks can be found HERE.

Friday 3/20: Starting at 12pm PST (3pm eastern time)

12:00p - 12:45p Lisa Goldberg, Hot Hands: What Data Science Can (and Can't) Tell Us About Basketball Trends
12:45p - 1:00p Discussion with Lisa and Kate on:  What Bayes tells us about our ability to reason about randomness

Due to the COVID-19 virus outbreak, the Critical Issues in Mathematics Education 2020 workshop was held online. The full workshop description and list of talks can be found HERE.

On March 12 and March 13, portions of the Critical Issues in Mathematics Education 2020: Today’s Mathematics, Social Justice, and Implications for Schools workshop will be streamed online via Zoom. Only the talks below will are scheduled at this time.  Further talks may be scheduled at a later date, and you will be notified when we know more.

Please see the schedule below, as well as links to the two sessions.
 

Thursday 3/12: Starting at 9am PST (noon eastern time)
9:00 - 9:10 Welcoming remarks
9:10 - 9:15 Introduction to CIME 2020 plan and speaker David Daley
9:15 - 9:55 David Daley, Why Your Vote Doesn't Count
9:55 - 10:00 Kate Stevenson, introduction of activity
10:00-10:30 Mathical Book Prize Announcement

Friday 3/13: Starting at 9am PST (noon eastern time)
9:00 - 9:05 Introduction of speaker Wesley Pegden
9:05 - 9:45 Wesley Pegden, Bringing Mathematics to the Courtroom
9:45 - 10:00 Q&A
 

Download Official Notice of Change

A formal Notice of Change letter is available here, which can be shared with your institution, funding agency, and others.

MSRI will host a Symposium on the occasion of Julia Robinson’s 100th birthday on Monday, December 9, 2019 at MSRI. Julia Robinson (1919-1985) was an internationally renowned logician of the twentieth century. She was a trailblazer in mathematics as well as in many other ways: she was the first woman president of the American Mathematical Society, and the first woman mathematician elected to membership in the National Academy of Sciences.

Participating speakers in this day-long celebration of her work and of current mathematics insprired by her research include: Martin Davis, Kirsten Eisentrager, Yuri Matiyasevich, and  Lou van den Dries. Following the symposium, Lenore Blum will give a public lecture at UC Berkeley.

As part of the Mathematical Sciences Collaborative Diversity Initiatives, six mathematics institutes are pleased to host their annual SACNAS pre-conference event, the 2019 Modern Math Workshop (MMW). The Modern Math Workshop is intended to encourage minority undergraduates to pursue careers in the mathematical sciences and to assist undergraduates, graduate students and recent PhDs in building their research networks.

This simplicial fan in 3-dimensional space

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 gluing affine pieces) have very concrete interpretations in the toric case, making toric varieties an ideal tool for introducing students to abstruse concepts.

The image of a large sphere isometrically embedded into a small space through a C^1 embedding. (Attributions: E. Bartzos, V. Borrelli, R. Denis, F. Lazarus, D. Rohmer, B. Thibert)

This two week summer school will introduce graduate students to the theory of h-principles.  After building up the theory from basic smooth topology, we will focus on more recent developments of the theory, particularly applications to symplectic and contact geometry, fluid dynamics, and foliation theory.

Popular visualization of the MNIST dataset

Learning theory is a rich field at the intersection of statistics, probability, computer science, and optimization. Over the last decades the statistical learning approach has been successfully applied to many problems of great interest, such as bioinformatics, computer vision, speech processing, robotics, and information retrieval. These impressive successes relied crucially on the mathematical foundation of statistical learning.

Recently, deep neural networks have demonstrated stunning empirical results across many applications like vision, natural language processing, and reinforcement learning. The field is now booming with new mathematical problems, and in particular, the challenge of providing theoretical foundations for deep learning techniques is still largely open. On the other hand, learning theory already has a rich history, with many beautiful connections to various areas of mathematics (e.g., probability theory, high dimensional geometry, game theory). The purpose of the summer school is to introduce graduate students (and advanced undergraduates) to these foundational results, as well as to expose them to the new and exciting modern challenges that arise in deep learning and reinforcement learning.

Fluid-flow stream function color-coded by vorticity in 3D flat torus calculated by K. Nakai (The University of Tokyo)

The purpose of the workshop is to introduce graduate students to fundamental results on the Navier-Stokes and the Euler equations, with special emphasis on the solvability of its initial value problem with rough initial data as well as the large time behavior of a solution. These topics have long research history. However, recent studies clarify the problems from a broad point of view, not only from analysis but also from detailed studies of orbit of the flow.

from distinct distances in the plane to line incidences in R^3

In the past eight years, a number of longstanding open problems in combinatorics were resolved using a new set of algebraic techniques. In this summer school, we will discuss these new techniques as well as some exciting recent developments.

A Holomorphic Curve

Symplectic topology is a fast developing branch of geometry that has seen phenomenal growth in the last twenty years. This two weeks long summer school, organized in the setting of the Séminaire de Mathématiques Supérieures, intends to survey some of the key directions of development in the subject today thus covering: advances in homological mirror symmetry; applications to hamiltonian dynamics; persistent homology phenomena; implications of flexibility and the dichotomy flexibility/rigidity; legendrian contact homology; embedded contact homology and four-dimensional holomorphic techniques and others. With the collaboration of many of the top researchers in the field today, the school intends to serve as an introduction and guideline to students and young researchers who are interested in accessing this diverse subject. 

Rips's δ-thin triangle condition for Gromov hyperbolicity of metric spaces (Stomatapoll)

Geometric group theory studies discrete groups by understanding the connections between algebraic properties of these groups and topological and geometric properties of the spaces on which they act. The aim of this summer school is to  introduce graduate students to specific central topics and recent developments in geometric group theory. The school will also include students presentations to give the participants an opportunity to practice their speaking skills in mathematics.  Finally, we hope that this meeting will help connect Latin American students with their American and Canadian counterparts in an environment that encourages discussion and collaboration. 

An illustration of an adaptation of Quillen's classical homological stability spectral sequence argument

This summer school will give an introduction to representation stability, the study of algebraic structural properties and stability phenomena exhibited by sequences of representations of finite or classical groups -- including sequences arising in connection to hyperplane arrangements, configuration spaces, mapping class groups, arithmetic groups, classical representation theory, Deligne categories, and twisted commutative algebras.  Representation stability incorporates tools from commutative algebra, category theory, representation theory, algebraic combinatorics, algebraic geometry, and algebraic topology. This workshop will assume minimal prerequisites, and students in varied disciplines are encouraged to apply. 

The MSRI-UP summer program is designed to serve a diverse group of undergraduate students who would like to conduct research in the mathematical sciences.

In 2019, MSRI-Up will focus on the application of combinatorial arguments and techniques to enumerate, examine, and investigate the existence of discrete mathematical structures with certain properties. The areas of interest for these applications encompass a wide range of mathematical fields and will include algebra, number theory, and graph theory, through weight multiplicity computations, the study of vector partition functions, and graph domination problems, respectively. The research program will be led by Dr. Pamela E. Harris, Assistant Professor of Mathematics at Williams College.

Is your department interested in helping graduate students learn to teach? Perhaps your department is considering starting a teaching-focused professional development program. Or maybe your department has a program but is interested in updating and enhancing it.

Many departments now offer pre-semester orientations, semester-long seminars, and other opportunities for graduate students who are new to teaching so they will be well-equipped to provide high-quality instruction to undergraduates. The purpose of this workshop is to support faculty from departments that are considering starting a teaching-focused professional development program or, for departments that have a program, to learn ways to improve it.

The figure represents a blow-up. The so called blow-up algebras or Rees rings are the algebraic realizations of such blow-ups.

Linkage is a method for classifying ideals in local rings. Residual intersections is a generalization of linkage to the case where the two `linked' ideals  need not have the same codimension. Residual intersections are ubiquitous: they play an important role in the study of blowups, branch and multiple point loci, secant varieties, and Gauss images; they appear naturally in intersection theory; and they have close connections with integral closures of ideals. 

Commutative algebraists have long used the Frobenius or p-th power map to study commutative rings containing a finite field. The theory of tight closure and test ideals has widespread applications to the study of symbolic powers and to Briancon-Skoda type theorems for equi-characteristic rings.

Numerical conditions for the integral dependence of ideals and modules have a wealth of applications, not the least of which is in equisingularity theory. There is a long history of generalized criteria for integral dependence of ideals and modules based on variants of the Hilbert-Samuel and the Buchsbaum-Rim multiplicity that still require some remnants of finite length assumptions.

The Rees ring and the special fiber ring of an ideal arise in the process of blowing up a variety along a subvariety. Rees rings and special fiber rings also describe, respectively, the graphs and the images of rational maps between projective spaces. A difficult open problem in commutative algebra, algebraic geometry, elimination theory, and geometric modeling is to determine explicitly the equations defining graphs and images of rational maps.

The school will consist of the following four courses with exercise sessions plus a Macaulay2 workshop

• Linkage and residual intersections
• Characteristic p methods and applications
• Blowup algebras
• Multiplicity theory

Mathematical Modeling (MM) now has increased visibility in the education system and in the public domain. It appears as a content standard for high school mathematics and a mathematical practice standard across the K-12 curriculum (Common Core Standards; and other states’ standards in mathematics education).  Job opportunities are increasing in business, industry and government for those trained in the mathematical sciences. Quantitative reasoning is foundational for civic engagement and decision-making for addressing complex social, economic, and technological issues. Therefore, we must take action to support and sustain a significant increase in the teaching and learning of mathematical modeling from Kindergarten through Graduate School.

Mathematical modeling is an iterative process by which mathematical concepts and structures are used to analyze or gain qualitative and quantitative understanding of real world situations. Through modeling students can make genuine mathematical choices and decisions that take into consideration relevant contexts and experiences.

Mathematical modeling can be a vehicle to accomplish multiple pedagogical and mathematical goals. Modeling can be used to introduce new material, solidify student understanding of previously learned concepts, connectthe world to the classroom, make concrete the usefulness (maybe even the advantages) of being mathematically proficient, and provide a rich context to promote awareness of issues of equity, socio-political injustices, and cultural relevance in mathematics.

A critical issue in math education is that although mathematical modeling is part of the K-12 curriculum, the great majority of teachers have little experience with mathematical modeling as learners of mathematics or in their teacher preparation.  In some cases, mathematics teacher educators have limited experience with mathematical modeling while being largely responsible for preparing future teachers.

Currently, the knowledge in teaching and learning MM is underdeveloped and underexplored.  Very few MM resources seem to reach the K-16 classrooms.  Collective efforts to build a cohesive curriculum in MM and exploration of effective teaching practices based on research are necessary to make mathematical modeling accessible to teacher educators, teachers and students.

At the undergraduate level, mathematical modeling has traditionally been reserved for university courses for students in STEM majors beyond their sophomore year.  Many of these courses introduce models but limit the students’ experience to using models that were developed by others rather than giving students the opportunity to generate their own models as is common in everyday life, in modeling competitions and in industry.

The CIME workshop on MM will bring together mathematicians, teacher educators, K-12 teachers, faculty and people in STEM disciplines.  As partners we can address ways to realize mathematical modeling in the K-12 classrooms, teacher preparation, and lower and upper division coursework at universities.  The content and pedagogy associated with teaching mathematical modeling needs special attention due to the nature of modeling as a process and as a body of content knowledge.

The NSF Mathematical Sciences Institutes Diversity Committee hosts the 2018 Blackwell-Tapia Conference and Awards Ceremony. This is the ninth conference since 2000, held every other year, with the location rotating among NSF Mathematics Institutes. The conference and prize honors David Blackwell, the first African-American member of the National Academy of Science, and Richard Tapia, winner of the National Medal of Science in 2010, two seminal figures who inspired a generation of African-American, Native American and Latino/Latina students to pursue careers in mathematics. The Blackwell-Tapia Prize recognizes a mathematician who has contributed significantly to research in his or her area of expertise, and who has served as a role model for mathematical scientists and students from underrepresented minority groups, or has contributed in other significant ways to addressing the problem of underrepresentation of minorities in math.

The 2018 recipient of the Blackwell-Tapia Prize is Dr. Ronald E. Mickens, the Distinguished Fuller E. Callaway Professor in the Department of Physics at Clark Atlanta University.

The conference will include scientific talks, poster presentations, panel discussions, ample opportunities for networking, and the awarding of the Blackwell-Tapia Prize. Participants are invited from all career stages and will represent institutions of all sizes across the country, including Puerto Rico.

The Mathematical Sciences Diversity Initiative holds a Modern Math Workshop (MMW) prior to the SACNAS National Conference each year. The 2018 MMW will be hosted by SAMSI at the Henry B. Gonzalez Convention Center, San Antonio, Texas on October 10th and 11th, 2018. This workshop is intended to encourage undergraduates, graduate students and recent PhDs from underrepresented minority groups to pursue careers in the mathematical sciences and build research and mentoring networks. The Modern Math Workshop is a pre-conference event at the SACNAS National Conference. The MMW includes a keynote lecture, mini-courses, research talks, a question and answer session and a reception.

The purpose of the summer school is to introduce graduate students to state-of-the-art methods and results in Hamiltonian systems and symplectic geometry. We focus on recent developments on the study of chaotic motion in Hamiltonian systems and its applications to models in Celestial Mechanics.

Harmonic analysis is a central field of mathematics with a number of applications to geometry, partial differential equations, probability, and number theory, as well as physics, biology, and engineering. The Graduate Summer School will feature mini-courses in geometric measure theory, homogenization, localization, free boundary problems, and partial differential equations as they apply to questions in or draw techniques from harmonic analysis. The goal of the program is to bring together students and researchers at all levels interested in these areas to share exciting recent developments in these subjects, stimulate further interactions, and inspire the new generation to pursue research in harmonic analysis and its applications.

The image of a large sphere isometrically embedded into a small space through a C^1 embedding. (Attributions: E. Bartzos, V. Borrelli, R. Denis, F. Lazarus, D. Rohmer, B. Thibert)

This two week summer school will introduce graduate students to the theory of h-principles.  After building up the theory from basic smooth topology, we will focus on more recent developments of the theory, particularly applications to symplectic and contact geometry, and foliation theory.

Explore Outstanding Phenomena in Animal Behavior

Jointly hosted by Janelia and the Mathematical Sciences Research Institute (MSRI), this program will bring together 15-20 advanced PhD students with complementary expertise who are interested in working at the interface of mathematics and biology. Emphasis will be placed on linking behavior to neural dynamics and exploring the coupling between these processes and the natural sensory environment of the organism. The aim is to educate a new type of global scientist that will work collaboratively in tackling complex problems in cellular, circuit and behavioral biology by combining experimental and computational techniques with rigorous mathematics and physics.

The MSRI-UP summer program is designed to serve a diverse group of undergraduate students who would like to conduct research in the mathematical sciences.

In 2018, MSRI-UP will focus on the core role of (linear) algebra in current research and application areas of Data Science ranging from unsupervised learning, clustering and networks, to algebraic signal processing and feature extraction, to the central role linear algebra plays in deep machine learning.  The research program will be led by Dr. David Uminsky, Associate Professor of Mathematics and Statistics at the University of San Francisco.

This Summer Graduate School will introduce students to the modern theory of the  inhomogeneous Cauchy-Riemann equation, the fundamental partial differential equation of Complex Analysis. This theory uses powerful tools of partial differential equations, differential geometry and functional analysis to obtain a refined understanding of holomorphic functions on complex manifolds. Besides students planning to work in complex analysis, this course will be valuable to those planning to study partial differential equations, complex differential and algebraic geometry, and operator theory. The exposition will be self-contained and the prerequisites will be kept at a minimum

Higher categorical structures and homotopy methods have made significant influence on geometry in recent years. This summer school is aimed at transferring these ideas and fundamental technical tools to the next generation of mathematicians.

The summer school will focus on the following four topics:  higher categorical structures in geometry, derived geometry, factorization algebras, and their application in physics.  There will be eight to ten mini courses on these topics, including mini courses led by Chirs Brav, Kevin Costello, Jacob Lurie, and Ezra Getzler. The prerequisites will be kept at a minimum, however, a introductory courses in differential geometry, algebraic topology and abstract algebra are recommended.

The Infinite Possibilities Conference (IPC) is a national conference that is designed to promote, educate, encourage and support women of color interested in mathematics and statistics, as a step towards addressing the underrepresentation of African-Americans, Latinas, Native Americans, and Pacific Islanders in these fields. 

IPC aims to:

• fulfill a need for role models and community-building
• provide greater access to information and resources for success in graduate school and beyond
• raise awareness of factors that can support or impede underrepresented women in the mathematical sciences

A unique gathering, the conference brings together participants from across the country, at all stages of education and career, for mentoring and mathematics.

On March 8-10, 2018, IPAM will host a conference showcasing the achievements of Latinx in the mathematical sciences. The goal of the conference is to encourage Latinx to pursue careers in the mathematical sciences, to promote the advancement of Latinx currently in the discipline, to showcase research being conducted by Latinx at the forefront of their fields, and, finally, to build a community around shared academic interests. The conference will be held on the UCLA campus in Los Angeles, CA. It will begin at noon on Thursday, March 8.

This conference is sponsored by the Mathematical Sciences Institutes Diversity Initiative, with funding from the National Science Foundation Division of Mathematical Sciences.

Our mathematics education system is inequitable. It operates in ways that leave a significant proportion of students with negative mathematics experiences and inadequate mathematical preparation. The problem is historical and systemic, and the students most disaffected by the current system are overwhelmingly Black and Latino, Indigenous, poor, women, immigrant or first generation college students. If our mathematics community is to sustainably grow and thrive, mathematics education at all levels must be transformed.

This workshop focuses on students for whom we do not yet successfully ensure access to and advancement in mathematics. Sessions will share relevant programmatic efforts and innovative research that have been shown to maintain or increase students’ engagement and interests in mathematics across k-12, undergraduate and graduate education. The sessions will focus particularly on reproducible efforts that affirm those students’ identities and their diverse intellectual resources and lived experiences. These efforts at various levels of mathematics education will highlight ways in which meaningful experiences in mathematics can disrupt ongoing systemic oppression. Participants will leave with conceptual and practical ways to open up and elevate mathematics education where all students thrive.

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As part of the Mathematical Sciences Collaborative Diversity Initiatives, nine mathematics institutes are pleased to offer their annual SACNAS pre-conference event, the 2017 Modern Math Workshop (MMW). The Modern Math Workshop is intended to encourage minority undergraduates to pursue careers in the mathematical sciences and to assist undergraduates, graduate students and recent PhDs in building their research networks. The Modern Math Workshop is part of the SACNAS National Conference; the workshop and the conference take place in the Salt Palace Convention Center in Salt Lake City, Utah. The MMW starts at 1:00 pm on Wednesday, October 18 with registration beginning at noon.

The summer school will be an introduction to the more algebraic aspects of the theory of automorphic forms and representations. One of the goals will be to understand the statements of the main conjectures in the Langlands programme. Another will be to gain a good working understanding of the fundamental definitions in the theory, such as principal series representations, the Satake isomorphism, and of course automorphic forms and representations for groups such as GL_n and its inner forms.

The purpose of the summer school is to introduce graduate students to the recent developments in the area of dispersive partial differential equations (PDE), which have received a great deal of attention from mathematicians, in part due to ubiquitous applications to nonlinear optics, water wave theory and plasma physics.

Recently remarkable progress has been made in understanding existence and uniqueness of solutions to nonlinear Schrodinger (NLS) and KdV equations, and properties of those solutions. We will outline the basic tools that were developed to address these questions. Also we will present some of recent results on derivation of NLS equations from quantum many particle systems and will discuss how methods developed to study the NLS can be relevant in the context of the derivation of this nonlinear equation.

McMullen’s g-Conjecture from 1970 is a shining example of mathematical foresight that combined all results available at that time to conjure a complete characterization of face numbers of convex simple/simplicial polytopes. The key statement in its verification is that certain combinatorial numbers associated to geometric (or topological) objects are non-negative. The aim of this workshop is to introduce graduate students to selected contemporary topics in geometric combinatorics with an emphasis on positivity questions. It is fascinating that the dual notions of simple and simplicial polytopes lead to different but equally powerful algebraic frameworks to treat such questions. A key feature of the lectures will be the simultaneous development of these algebraic frameworks from complementary perspectives: combinatorial-topological and convex-geometric.  General concepts (such as Lefschetz elements, Hodge–Riemann–Minkowski inequalities) will be developed side-by-side, and analogies will be drawn to concepts in algebraic geometry, Fourier analysis, rigidity theory and measure theory. This allows for entry points for students with varying backgrounds.  The courses will be supplemented with guest lectures highlighting further connections to other fields.

The theory of dynamical systems has witnessed very significant developments in the last decades, includi​n​g the work of two 2014 Fields medalists, Artur Avila and Maryam Mirzakhani. ​The school will concentrate on the recent significant developments in the field of dynamical systems and present some of the present main streams of research. Two central themes will be those of partial hyperbolicity on one side, and rigidity, group actions and renormalization on the other side.​ ​Other themes will ​include homogeneous dynamics and geometry and dynamics on infinitely flat surfaces (both providing connections to the work of Maryam Mirzakhani), topological dynamics, thermodynamical formalism, singularities and bifurcations in analytic dynamical systems.  

We will give an introduction to categorical representation theory, focusing on the example of Soergel bimodules, which is a categorification of the Iwahori-Hecke algebra. We will give a comprehensive introduction to the "tool box" of modern (higher) representation theory: diagrammatics, homotopy categories, categorical diagonalization, module categories, Drinfeld center, algebraic Hodge theory.

Subfactor theory is a subject from operator algebras, with many surprising connections to other areas of mathematics. This summer school will be devoted to understanding the representation theory of subfactors, with a particular emphasis on connections to quantum symmetries, fusion categories, planar algebras, and random matrices

This workshop will focus on preparing each participant for a successful career as a mathematician at a college or university. Beginning with the hiring process, a thorough discussion of the various elements of the application packet will take place in the context of each participant's materials. Working individually with experienced faculty, participants will review and refine their cover letters, C.V., research, and teaching statements. This will be followed by activities related to the interview. The primary goals of the workshop are to develop an understanding of the hiring process from the institutions' perspective, to refine the application packet, to learn what to expect during the interview process (including the job talk), and to prepare for negotiating salary and start-up packages.

Additional time will be spent on aspects of the pre-tenure years including the development of a research program, writing grant proposals, and mentoring research students. The three-day workshop will consist of one-on-one work with experienced mentors, small group discussions, critique of written materials, plenary sessions, and time for individual work and consultation.

The purpose of the school will be to introduce graduate students to foundational results in commutative algebra, with particular emphasis of the diversity of the related topics with commutative algebra. Some of these topics are developing remarkably in this decade and through learning those subjects the graduate students will be stimulated toward future research. 

Success rates in mathematics as well as recruitment and retention rates in the mathematics pipeline are low at all education levels and are, across predictable demographics, disproportionately low for students who are women, Latin@, Black, American Indian, recent immigrants, emergent bilinguals/multilinguals, and poor. Efforts to address these low rates often focus on programmatic solutions such as creating mentoring or bridge programs to address perceived deficiencies. While these programs achieve some success, evidence suggests that they may not substantially improve students’ subsequent success in mathematics or meaningfully address the ways that students experience mathematics instruction.

The 2017 CIME workshop will focus on observations of mathematics classrooms through the lens of equity. Specifically, we will use observation as a tool for understanding and improving imbalances of access, participation, and power in mathematics teaching and learning. In doing so, we seek to better understand students’ experiences in mathematics classrooms in order to improve academic success, recruitment and retention, and meaningful experiences for historically marginalized populations.

Five questions structure the highly interactive design of the workshop:

1. What does it mean to create an equitable classroom environment? How can the structure of classroom interactions lead to imbalances of access, identity, and power in mathematics teaching and learning? How can such structures be rebuilt to better serve all students?
2. How might observations of mathematics instruction help us to identify power dynamics in classrooms? What language is helpful to describe interactions in mathematics classrooms? What might we learn from observations about how culture and identity are developed for some students but not others? What do classroom observations reveal about how instruction supports or discourages engagement in mathematics for students of different backgrounds?
3. What does it mean to observe interactions in a mathematics classroom with an eye towards equity? What language is helpful to describe interactions in mathematics classrooms? How do we observe and describe interactions among students, between students and mathematics, between students and instructors, and between students and resources (i.e., textbooks, computers, chalkboards, manipulatives)?
4. What professional experiences can support mathematics instructors to learn how to observe for, describe, interpret, and productively address interactions in the mathematics classroom from the lens of equity? What professional experiences can support mathematics instructors to increase the number of equitable interactions and decrease the number of inequitable ones in their classrooms?
5. What measures might be useful in tracking our progress in learning to see, describe, interpret, and productively address (in)equitable interactions in mathematics classrooms? What measures and tools might be useful in tracking the impacts on instruction and student learning? How might we develop infrastructure to help with this work (video library, faculty resources, etc.)?

Bringing together new and experienced leaders of Math Circles and other similar outreach programs, this year’s Circle on the Road will include discussions, presentations, and opportunities to facilitate different mathematical problems. In addition, some informal STEM education researchers will join us to further our research and evaluation efforts.

Tropical geometry uses a combination of techniques from algebraic geometry, combinatorics, and convex polyhedral geometry to study degenerations of algebraic varieties; the simplest tropical objects are tropical curves, which one can think of as "shadows" of algebraic curves.  Linear equivalence of divisors on an abstract tropical curve is determined by a simple but rich combinatorial process called "chip firing", which was discovered independently in the discrete setting by physicists and graph theorists.  From a pedagogical point of view, one can view tropical curves as a combinatorial model for the highly analogous but more abstract theory of algebraic curves, but there is in fact much more to the story than this: one can use tropical curves and chip firing to prove theorems in algebraic geometry and number theory.  This field is relatively new, so participants will have the opportunity to start from scratch and still get a glimpse of the cutting edge in this active research area.

Ab initio or first principle electronic structure theories, particularly represented by Kohn-Sham density functional theory (KS-DFT), have been developed into workhorse tools with a wide range of scientific applications in chemistry, physics, materials science, biology etc. What is needed are new techniques that greatly extend the applicability and versatility of these approaches. At the core, many of the challenges that need to be addressed are essentially mathematical. The purpose of the workshop is to provide graduate students a self-contained introduction to electronic structure theory, with particular emphasis on frontier topics in aspects of applied analysis and numerical methods. 

Character Theory of Finite Groups provides one of the most powerful tools to study groups. In this course we will give a gentle introduction to basic results in the Character Theory, as well as some of the main conjectures in Group Representation Theory, with particular emphasis on the McKay Conjecture.

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This school is oriented to the presentation of theory, algorithms and applications for the solution of mixed integer nonlinear problems (MINLP). This type of problems appears in numerous application areas where the modelization of nonlinear phenomena with logical constraints is important; we must remember here the memorable phrase “the world is nonlinear”. Nowadays the theoretical aspects of this area are spread in a number of recent papers which makes it difficult, for non-specialist, to have a solid background of the existing results and new advances in the field. This school aims to organize and present this material in an organized way. Moreover, it also pursues to link theory with actual applications. In particular, remarkable applications can be found in air traffic control agencies, the air companies, the electric power generation companies, the chemical complex units, the analysis of financial products usually associated with risk dealing and in the algorithms in the statistical field and artificial intelligence as for instance artificial neural networks, or supporting vector machines, among many others.

The goal of the workshop is to present harmonic analysis techniques in $R^n$ (the ``flat" setting), and then to show how those techniques extend to much rougher settings, with application to the theory of elliptic equations. Thus, the subject matter of the workshop will introduce the students to an active, current research area:  the interface between harmonic analysis, elliptic PDE, and geometric measure theory.

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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 2016 program will be led by Prof. Luis Garcia-Puente of Sam Houston State University.

The purpose of this summer school is to focus on the interplay of dynamical and biological systems, developing the rich connectionbetween science and mathematics that has been so successful to date. Our focus will be on understanding the mathematical structure of dynamical systems that come from biological problems, and then relating the mathematical structures back to the biology to provide scientific insight. We will focus on five key areas: complex bio-networks, multi scale biological dynamics, biological waves, nonlinear dynamics of pattern formation, and disease dynamics. For each of the five key areas, we will invite 2-3 world leaders who are also excellent communicators to deliver a series of 2-4 one-hour lectures. We expect an average of eight hours of lecture per subject area, spread over approximately two weeks.

Bay Area Discrete Math Days are one-day meetings aimed at facilitating communication between researchers and graduate students of discrete mathematics around the San Francisco Bay Area.These days happen semi-annually and strive to create an informal atmosphere to talk about discrete mathematics. The term "discrete mathematics" is chosen to include at least the following topics: Algebraic and Enumerative Combinatorics, Discrete Geometry, Graph Theory, Coding and Design Theory, Combinatorial Aspects of Computational Algebra and Geometry, Combinatorial Optimization, Probabilistic Combinatorics, and Combinatorics in Mathematical Physics

The 2016 CIME workshop focuses directly on the teaching of mathematics at the university and precollege levels. Teaching is not easy to examine in disciplined ways because it is so familiar and seems so obvious.  Although teaching shapes students’ opportunities to learn, what teachers are actually doing is difficult to observe and describe. This impedes work on improving teaching.
 
This workshop will offer the opportunity to study and talk closely about mathematics teaching through close observation and discussion of video tapes in a setting that will bring together professionals with a range of perspectives, knowledge, experience, and orientations. The goal of the workshop is to develop language and methods for describing, analyzing and evaluating what can be seen in the classroom, with the ultimate goal of helping us shape and improve teaching — our own and more broadly.
 
Four questions structure the highly interactive design of the workshop:

1. What skills are needed for observing teaching in ways that inform improvement efforts? What is involved in observing teaching? What is the teacher saying and doing? What are students saying and doing? What is the mathematics at play? What else is happening? And what do these imply for teaching?
2. How can the practice and use of observation be structured in order to improve mathematics teaching? What approaches are available? What are their strengths and weaknesses?
3. Observation-based assessment of teaching: Why, what, and how? What are the risks?
4. How can we develop and sustain a cross-professional community that observes and evaluates teaching in such a way that different communities communicate with and learn from each other to support a cycle of improvement in the teaching of mathematics at all levels?

The workshop will provide a library of videos of mathematics teaching for study. In addition, participants are encouraged to submit a short video clip of their own teaching, together with a brief background commentary.  These videos will provide a central text for our collective work on discussing and assessing mathematics teaching.

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As part of the Mathematical Sciences Collaborative Diversity Initiatives, nine mathematics institutes are pleased to host their annual SACNAS pre-conference event, the 2015 Modern Math Workshop (MMW). The Modern Math Workshop is intended to encourage minority undergraduates to pursue careers in the mathematical sciences and to assist undergraduates, graduate students and recent PhD’s in building their research networks.

The purpose of this two week workshop is to introduce graduate students to state-of-the-art methods and results in mathematical fluid dynamics. In the first week, we will discuss the mathematical foundations and modern analysis aspects of the Navier-Stokes and Euler equations. In the second week, we will run two courses concurrently on the topics of inviscid limits and hydrodynamic stability. Specifically, one course will focus on boundary layers in high Reynolds number flows and the Prandtl equations while the other will focus on mixing and connections to turbulence. Through the lectures and associated problem sessions, the students will learn about a number of new analysis tools and principles of fluid mechanics that are not always taught in a graduate school curriculum.

These courses will give students a full overview of the results of Zhang and Maynard on gaps between primes, and will provide them will a clear understanding of the tools involved. This will make accessible a significant part of modern analytic number theory. The lecturers will also make sure to include, within their course, examples and discussions going further than is strictly required to understand the proofs of Zhang and Maynard, e.g., in the direction of automorphic forms and the Riemann Hypothesis over finite fields.

This course is for students and researchers with backgrounds in mathematics and computational sciences who are
interested in applying their skills toward problems in neuroscience. It will introduce the major open questions of
neuroscience and teach state-of–the-art techniques for analyzing and modeling neuroscience data sets. The course is designed for students at the graduate level and researchers with background in a quantitative field such as
engineering, mathematics, physics or computer science who may or may not have a specific neuroscience
background. The goal of this summer course is to help researchers find new exciting research areas and at the same time to strengthen quantitative expertise in the field of neuroscience. The course is sponsored by the National Science Foundation from a grant supporting activities at the data sharing repository CRCNS.org, the Helen Wills
Neuroscience Institute, the Simons Institute for the Theory of Computing and the Mathematical Science Research
Institute.

This Summer Graduate School will introduce mathematics graduate students to the rapidly emerging area of systems biology. In particular, we will focus on the design and emergent behaviors of molecular networks used by cells to interpret their environments and create robust temporal-spatial behaviors. This will be a very hands-on workshop with students working alone and in teams to program and present key ideas.

This summer graduate school will take place at the National Institute for Mathematical Sciences in Daejeon, South Korea.  The purpose of this summer school is to introduce some of the basic ideas and methods of random matrix theory to graduate students.  In particular there will be three lecture series on random matrix theory from three different perspectives: from the view points of the integrable structures, the moment method, and the Stieltjes transorm technique.  In addition to the lectures, there will be discussion sessions, and the students will also have plenty of time to interact with the lecturers and with other students.

Please note that accepted students will be provided up to $1700 in travel reimbursement, in addition to meals and accommodation.

The lectures will focus on the following four topics: geometry of eigenvalues, geometry of eigenfunctions, spectral theory on manifolds with singularities and computational spectral theory. There has been a number of remarkable recent developments in these closely related fields. The goal of the school is to shed light on different facets of modern spectral theory and to provide a unique opportunity for graduate students and young researchers to get a “big picture” of this rapidly evolving area of mathematics. A particularly novel aspect of the school is the emphasis on the interactions between spectral geometry and computational spectral theory.

The aim of this workshop is to introduce graduate students to some specific core topics which will be under study at the upcoming MSRI program on Geometric Group Theory (GGT) in 2016.  GGT encompasses a wide range of topics. The four minicourse topics have been chosen because they are central themes in GGT and in the upcoming MSRI program. Moreover, each topic is accessible to students with a range of backgrounds: the basic definitions are straightforward, with many simple and illuminating examples to work through, yet lead through to important questions in current research.

The 2015 CRM-PIMS Summer School in Probability will take place in Montreal, Canada, from June 15-July 11, 2015. The school is built around two principal 24-hour lecture courses, which will be delivered by Alice Guionnet (random matrices, free probability and the enumeration of maps) and Remco van der Hofstad (high-dimensional percolation and random graphs). There will additionally be mini-courses by Louigi Addario-Berry (random minimum spanning trees), Shankar Bhamidi (dynamic random network models) and Jonathan Mattingly (stabilization by noise). Some time is reserved for participants to present their own work.

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 2015 program will be led by Prof. Francis Su from Harvey Mudd College.

The National Science Foundation (NSF) and non-profit organizations each provide critical support to the U.S. basic research enterprise in the mathematical and physical sciences. While the missions of these funders differ, many of their goals align and the grantee communities have significant overlap. With the ultimate aim of helping to advance the scientific frontier in the most effective way, we propose to hold a workshop to examine partnerships between the Directorate of Mathematical and Physical Sciences (MPS) at NSF and non-profit funders in MPS-related disciplines to
•       understand different models of collaboration (the “how”);
•       understand different motivations for collaboration (the “why”); and
•       develop opportunities for future communication and/or collaboration.

This workshop will address the critical issue of developmental mathematics at two- and four-year colleges and universities and the broader dynamic of mathematics remediation that occurs at all levels. It will engage mathematicians, K-12 teachers, mathematics educators, and administrators in a conversation about the goals of developmental mathematics and the contributions that our different professional communities make to this work. Key questions that will be addressed are:

1. How do we teach content in ways that acknowledge and leverage each student's prior learning experiences? In particular, how do we take advantage of a student's maturity while refining his or her learning habits where necessary?

2. How can developmental mathematics instruction move students through mathematics which must be relearned while simultaneously gaining momentum on more advanced mathematics (including the development of mathematical practices needed for meaningful mathematical work)?

3. What are strategies for supporting the needs of the wide range of students in developmental mathematics programs--those developing mathematical skills for life in general as well as those developing the foundation necessary to proceed towards a STEM major?  How can we successfully address equity issues raised for students from groups underrepresented in STEM fields? How can developmental mathematics instruction blend synchronous and asynchronous instruction to achieve maximal efficiency and impact?

4. What is the proper balance between addressing the needs of the wide range of students mentioned in the preceding point and keeping instruction and course offerings concise?

5. What are the characteristics, training, and practices of a successful developmental mathematics teacher?

6. What support services enhance the success of a developmental mathematics program?

Geometric and complex analysis is the application of tools from analysis to study questions from geometry and topology. This two week summer course will provide graduate students with the necessary background to begin studies in the area. The first week will consist of introductory courses on geometric analysis, complex analysis, and Riemann surfaces. The second week will consist of more advanced courses on the regularity theory of Einstein manifolds, Kahler-Einstein manifolds, and the analysis of Riemann surfaces.

Stochastic Partial Differential Equations (SPDEs) serve as fundamental models of physical systems subject to random inputs, interactions or environments. It is a particular challenge to develop tools to construct solutions, prove robustness of approximation schemes, and study properties like ergodicity and fluctuation statistics for a wide variety of SPDEs. 

The purpose of this two week workshop is to educate graduate students on the state-of-the-art methods and results in SPDEs. The three courses which will be run simultaneously will highlight different (though related) aspects of this area including (1) Fluctuation theory of PDEs with random coefficients (2) Ergodic theory of SPDEs and (3) Exact solvability of SPDEs

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.

Videos of selected lectures may be found here.

The program in 2014 will bring together a diverse group of mathematicians and scientists with interests in fundamental questions in mathematics and the behavior of materials. The meeting addresses several themes including computational investigations of material properties, the emergence of long- range order in materials and self-assembly, the geometry of soft condensed matter and the calculus of variations, phase transitions and statistical mechanics. The program will cover several topics in discrete and differential geometry that are motivated by questions in materials science. Many central topics, such as the geometry of packings, problems in the calculus of variations and phase transitions, will be discussed from the complementary points of view of mathematicians and physicists.

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 2014 program will be led by Dr. Victor Moll from Tulane University.

The purpose of the workshop is to introduce graduate students to the recent developments in the area of dispersive partial differential equations (PDE).

Dispersive equations have received a great deal of attention from mathematicians because of their applications to nonlinear optics, water wave theory and plasma physics. We will outline the basic tools of the theory that were developed with the help of multi-linear Harmonic Analysis techniques. The exposition will be as self-contained as possible.

The 2014 CIME workshop will focus on the role played by mathematics departments in preparing future teachers.  As part of this focus, the workshop will consider two broad questions: What mathematics should teachers know, and how should they come to know this mathematics?

The Conference Board of the Mathematical Sciences publication, The Mathematical Education of Teachers II, recommends that, at institutions that prepare teachers, teacher education should be “an important part of a mathematics department’s mission” (p.19). Certainly, at some universities, mathematicians are significantly involved in the mathematical experiences of students who are planning become teachers. But there are many other departments where this is not true. Future mathematics teachers are enrolled in the department’s mathematics classes, but no one is attending to the fact that this is where they are developing mathematical knowledge and (from watching their instructors) ideas about how teach mathematics.  This role – whether deliberate or latent –– is vitally important for the mathematical preparation of beginning teachers.  

The CIME workshop has three core aims: (A) to acquaint mathematicians with basic facts about teacher education and how teacher education intersects with the math department even when no one is taking special note of the department’s role; (B) to explore a set of key questions and best practices central to taking advantage of the role that mathematics departments do – or could – play in the mathematical preparation of teachers:

1. What is known about effective mathematical preparation of teachers, including curriculum, instructional approaches, and assessments?  
2. What supports do mathematicians and mathematics departments need to carry out this important role effectively?  What are examples of successful models and what evidence exists about their effects?  
3. What are some of the persistent problems or challenges and what are promising examples of addressing these? 

and (C) to identify possible action steps to provide more collective capacity for math departments to contribute to teachers’ mathematical education.

In cooperation with INdAM (Istituto Nazionale di Alta Matematica) and the CMI (Clay Mathematical Institute), MSRI will sponsor a summer graduate workshop on Mathematical General Relativity in Cortona during the summer of 2013; the school will reprise the very successful school of Mathematical General Relativity held at MSRI in 2012.

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 summer school 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.

This is a three-week institute on the mathematics of grades 6-8 in direct response to the recent adoption of the Common Core Mathematics Standards (CCMS) by California.

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.

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.

This workshop will bring together new and experienced leaders of math circles for students and teachers. We welcome anyone who is interested in learning more about math circles, especially teachers. Workshop activities will include discussions, presentations, and a mathematics festival.
Participants will begin collaborating before the workshop to develop sample math circle sessions that they will present during the festival. These activities will be collaboratively evaluated and refined during the workshop.

This is a three-week institute on the mathematics of grades 6-8 in direct response to the recent adoption of the Common Core Mathematics Standards (CCMS) by California.

This workshop will bring together new and experienced leaders of math circles for students and teachers.

Workshop activities will include discussions, presentations, and a mathematics festival to be held outside of the MathAlive! exhibit that will be in the Smithsonian Institution.

Participants will begin collaborating before the workshop to develop sample math circle sessions that they will present during the festival. These activities will be collaboratively evaluated and refined during the workshop.

The wide adoption of the Common Core State Standards for Mathematics (CCSSM) offers a helpful curricular coherence to the environment of teacher education. And so the CCSSM present both an opportunity and a challenge to teacher education. An opportunity because of the greater focus made possible. A challenge because not only of the ambitious level of the CCSSM, but also of the prominent role in them of Mathematical Practices. While most mathematicians will find these congenial, much needs to be done to make them meaningfully understood by teachers and teacher educators, and, still more, how to enact them as an organic aspect of instruction. The CIME workshop aims to gather and stimulate ideas for how to meet this opportunity and challenge.

This first workshop in a series addresses the professional advancement of underrepresented minorities in the mathematical sciences. It will include an introduction to mathematics represented in the MSRI research programs aimed at faculty in minority serving and primarily undergraduate institutions. Anyone who will be seeking employment in mathematics within the next couple of years would benefit from attending this workshop.

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 BACT Summer Workshop supports teachers in their development of problem solving skills as well as supporting the incorporation of problem solving into their teaching curriculum. During the earlier part of the week teachers will gain experience with a variety of problem solving techniques such as symmetry, mathematical patterns, and parity. Subsequent sessions will focus on particular topics such as geometry, sequences, counting, and number theory.

Note: for 2011 Workshop there will be two parallel sessions: one for elementary teachers and one for secondary teachers.

This workshop will showcase materials and successful teacher education programs, examine the Common Core State Standards and its implications, and explore how mathematics education research can improve practice.

This workshop will bring together people who have experience running math circles and teams of people who wish to start a math circle. The workshop will begin on Friday, with discussions and presentations related to math circles. On Saturday several sample math circle sessions will be offered, and the workshop will conclude on Sunday with more discussions and presentations.









 

This workshop is intended to introduce to graduate students the main ideas of algebraic, geometric and combinatorial methods in global optimization. We emphasize the major developments in the past few years from two viewpoints. The first one is that of the interaction of semidefinite programming and real algebraic geometry and includes topics such as linear matrix inequalities, positive polynomials, and sums of squares. The second viewpoint is that of primal methods and generating function methods in integer linear and nonlinear optimization.

This is a fourteen-day institute (July 6 to July 23, 2010) on algebra together with five Saturday sessions spread over the 2010-2011 school year. The main target is middle school teachers; preference will be given to teachers who attended the 2009 Pre-Algebra Institute and teams from the same school or same district. However, high school teachers and upper elementary school teachers will also be considered. There is a limited number of seats, so get your application in as soon as possible.

The National Council of Teachers of Mathematics has just released a new document, Focus in High School Mathematics: Reasoning and Sense-Making. The Council of Chief State School Officers and the National Governor’s Association have initiated a state led effort to produce Common Core State Standards, which they hope will move states toward national curricular coherence. The national scene is being transformed through stimulus money aimed at having states adopt common standards. This is a significant time for mathematicians to weigh in for coherence and a focus on thinking, understanding and sense-making. For this reason MSRI will host the seventh Critical Issues in Mathematics Education Workshop on this topic. Themes of the workshop will include international comparisons, the role of a coherent national curriculum in the teaching of mathematics, and the ways in which technology can be used to support reasoning and sense-making.

This conference will bring together people who have experience running math circles with *teams* of people who wish to start a math circle. The workshop will begin on Saturday with a Math Festival for school children in the Phoenix area. The following two days will have panel discussions and presentations on various topics of interest to people who run or wish to run a math circle.

The Winter workshop supports teachers in their development of problem solving skills as well as sharing with them information about upcoming mathematical opportunities for students and teachers. This will be a great opportunity for teachers new to the Math Circle program and experienced Math Circle teachers.

The aim of the Circle for Teachers is to equip educators with an effective problem-solving approach to teaching mathematics.

Teaching Undergraduates Mathematics will be the sixth in a series of Critical Issues in Education workshops hosted by the Mathematical Sciences Research Institute (MSRI) in Berkeley, CA. Whereas previous workshops focused on K-12 education and teacher education, this workshop will focus on undergraduate education.

This three-week course in elementary mathematics directly addresses the mathematics needed for teaching Number Sense in K-7 classrooms. It develops whole numbers and fractions from the beginning, with an excursion into some elementary number theory about divisibility properties of whole numbers and the algorithm that yields the HCF of two numbers. Each day begins with mathematics lectures by a Berkeley mathematician and concludes with small group sessions guided by MPDI graduates.

For over two decades, the teaching and learning of algebra has been a focus of mathematics education at the precollege level. This workshop will examine issues in algebra education at two critical points in the continuum from elementary school to undergraduate studies: at the transitions from arithmetic to algebra and from high school to university. In addition, the workshop will involve participants in discussions about various ways to structure an algebra curriculum across the entire K-12 curriculum.

Building on the issues investigated in these previous workshops, this workshop will focus concretely on courses, programs and materials that aim to increase teachers’ mathematical knowledge for teaching. Both courses and programs that lead to initial certification and professional development of current teachers will be examined at the workshop. In addition, the workshop will examine efforts by colleges, universities, school districts, professional organizations and funding agencies to support people who teach these courses or lead these workshops.

Using Math to Teach Math (PDF 5.5MB) Second conference in the MSRI series "Critical Issues in Mathematics Education" This workshop will be held at Asilomar in Pacific Grove, California. The conference takes as a premise that improving students’ mathematics learning depends on improving mathematics teaching, for which teachers’ knowledge of mathematics is a key factor. It will bring together different groups for whom issues of teachers’ mathematical knowledge are of critical concern, and explore current perspectives, evidence, and programs. Three questions structure its highly interactive design: 1. Why should K-8 teachers know mathematics? 2. What is the nature of the knowledge of mathematics needed for effective teaching? 3. What can mathematics departments and schools of education do to help teachers develop such knowledge? The conference will foster productive partnerships among research mathematicians, mathematics educators, educational researchers, teachers of school mathematics, and policymakers that will support them in their efforts. This conference is made possible by generous support from The National Science Foundation (www.nsf.gov), MfA Math for America(www.mathforamerica.org),Texas Instruments (www.ti.com). Noyce Foundation, and Center for Proficiency in Teaching Mathematics