
Seminar COMD Hubbard Semigroup Series
Updated on Feb 17, 2022 11:05 AM PST 
Seminar COMD Junior Seminar Series: On Karpinska’s Paradox
Updated on May 19, 2022 08:24 AM PDT 
Seminar COMD Junior Seminar Series: "Sharkovskii's ordering: From Real to Complex Polynomials"
Updated on May 19, 2022 08:34 AM PDT 
Seminar COMD Research Seminar Series
Updated on Mar 03, 2022 02:18 PM PST 
Seminar AGRS Research Seminar Series: A Gentle Introduction to KangMakarov Conformal Field Theory
Updated on May 17, 2022 02:47 PM PDT 
Seminar COMD Learning Seminar Series: Wrap Up
Updated on May 18, 2022 02:55 PM PDT 
Program Higher Categories and Categorification, Part Two
Organizers: David Ayala (Montana State University), Clark Barwick (University of Edinburgh), David Nadler (University of California, Berkeley), LEAD Emily Riehl (Johns Hopkins University), Marcy Robertson (University of Melbourne), Peter Teichner (MaxPlanckInstitut für Mathematik), Dominic Verity (Macquarie University)Though many of the ideas in higher category theory find their origins in homotopy theory — for instance as expressed by Grothendieck’s “homotopy hypothesis” — the subject today interacts with a broad spectrum of areas of mathematical research. Unforeseen descent, or localtoglobal formulas, for familiar objects can be articulated in terms of higher invertible morphisms. Compatible associative deformations of a sequence of maps of spaces, or derived schemes, can putatively be represented by higher categories, as Koszul duality for E_nalgebras suggests. Higher categories offer unforeseen characterizing universal properties for familiar constructions such as Ktheory. Manifold theory is natively connected to higher category theory and adjunction data, a connection that is most famously articulated by the recently proven Cobordism Hypothesis.
In parallel, the idea of "categorification'' is playing an increasing role in algebraic geometry, representation theory, mathematical physics, and manifold theory, and higher categorical structures also appear in the very foundations of mathematics in the form of univalent foundations and homotopy type theory. A central mission of this semester will be to mitigate the exorbitantly high "cost of admission'' for mathematicians in other areas of research who aim to apply higher categorical technology and to create opportunities for potent collaborations between mathematicians from these different fields and experts from within higher category theory.Updated on Feb 07, 2022 11:05 AM PST 
Summer Research in Mathematics 2022 Summer Research in Mathematics
MSRI's Summer Research in Mathematics program provides space, funding, and the opportunity for inperson collaboration to small groups of mathematicians, especially women and genderexpansive individuals, whose ongoing research may have been disproportionately affected by various obstacles including family obligations, professional isolation, or access to funding. Through this effort, MSRI aims to mitigate the obstacles faced by these groups, improve the odds of research project completion, and deepen their research experience.
The ultimate goal of this program is to enhance the mathematical sciences as a whole by positively affecting the research and careers of all of its participants and assisting their efforts to maintain involvement in the research community.
Updated on Nov 11, 2021 06:08 PM PST 
Summer Graduate School Integral Equations and Applications
Organizers: Fioralba Cakoni (Rutgers University), Dorina Mitrea (Baylor University), Irina Mitrea (Temple University), Shari Moskow (Drexel University)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:
 introduce graduate students to the systematic study of integral equations;
 present some of the latest theoretical advancements in the field and open problems; and
 involve participants in a handson 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).
Updated on Sep 02, 2021 04:19 PM PDT 
MSRIUP MSRIUP 2022: Algebraic Methods in Mathematical Biology
Organizers: LEAD Federico Ardila (San Francisco State University), Duane Cooper (Morehouse College), Maria Franco (Queensborough Community College (CUNY); MSRI  Mathematical Sciences Research Institute), Rebecca Garcia (Sam Houston State University), Candice Price (Smith College), Anne Shiu (Texas A & M University)The MSRIUP summer program is designed to serve a diverse group of undergraduate students who would like to conduct research in the mathematical sciences.
In 2022, MSRIUP 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.
Updated on Feb 16, 2022 09:10 AM PST 
Summer Graduate School New Directions in Representation Theory (AMSI and U. of Hawaii, Hilo)
Organizers: Angela Coughlin (Australian Mathematical Sciences Institute), Joseph Grotowski (University of Queensland), Tim Marchant (Australian Mathematical Sciences Institute), Ole Warnaar (University of Queensland), Geordie Williamson (University of Sydney)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.
Updated on Mar 25, 2022 12:20 PM PDT 
African Diaspora Joint Mathematics 2022 African Diaspora Joint Mathematics Workshop
The African Diaspora Joint Mathematics Workshop (ADJOINT) is a yearlong program that provides opportunities for U.S. mathematicians – especially those from the African Diaspora – to form collaborations with distinguished AfricanAmerican research leaders on topics at the forefront of mathematical and statistical research.
Beginning with an intensive twoweek summer session at MSRI, participants work in small groups under the guidance of some of the nation’s foremost mathematicians and statisticians to expand their research portfolios into new areas. Throughout the following academic year, the program provides conference and travel support to increase opportunities for collaboration, maximize researcher visibility, and engender a sense of community among participants. The 2022 program takes place June 20  July 1, 2022 in Berkeley, California.
Updated on Oct 13, 2021 03:27 PM PDT 
Summer Graduate School Geometric Flows (Crete, Greece)
Organizers: Nicholas Alikakos (National and Kapodistrian University of Athens (University of Athens)), Panagiota Daskalopoulos (Columbia University)[The image on this vase from Minoan Crete, dated on 15002000 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 FORTHIACM 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 nonlinear 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.
Updated on Mar 07, 2022 11:16 AM PST 
Seminar ADJOINT Professional Development Panel
Updated on Apr 27, 2022 04:01 PM PDT 
Summer Graduate School Algebraic Theory of Differential and Difference Equations, Model Theory and their Applications
Organizers: LEAD Alexey Ovchinnikov (Queens College, CUNY), Anand Pillay (University of Notre Dame), Thomas Scanlon (University of California, Berkeley), Michael Wibmer (University of Notre Dame)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 modeltheoretic 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 oneonone mentoring sessions with the lecturers and organizers.
Updated on Apr 25, 2022 11:14 AM PDT 
Summer Graduate School Random Graphs
Organizers: Louigi AddarioBerry (McGill University), Remco van der Hofstad (Technische Universiteit Eindhoven)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 realworld 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.
Updated on Sep 02, 2021 04:21 PM PDT 
Summer Graduate School Metric Geometry and Geometric Analysis (Oxford, United Kingdom)
Organizers: LEAD Cornelia Drutu (University of Oxford), Panos Papazoglou (University of Oxford)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 nonsmooth 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.
Updated on Feb 14, 2022 12:29 PM PST 
Summer Graduate School Séminaire de Mathématiques Supérieures 2022: Floer Homotopy Theory
Organizers: Kristen Hendricks (Rutgers University), Ailsa Keating (University of Cambridge), Robert Lipshitz (University of Oregon), Liam Watson (University of British Columbia), Ben Williams (University of British Columbia)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 lowdimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)equivariant SeibergWitten Floer homotopy type to resolve the Triangulation Conjecture and AbouzaidBlumberg's use of Floer homotopy theory and Morava Ktheory 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.Updated on Sep 10, 2021 11:11 AM PDT 
Summer Graduate School 2022 Joint PCMI School: Number Theory Informed by Computation
Organizers: Jennifer Balakrishnan (Boston University), Rafe Mazzeo (Stanford University), Bjorn Poonen (Massachusetts Institute of Technology), Akshay Venkatesh (Institute for Advanced Study)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 K12 Teacher Leadership Program and the Workshop for Equity in Mathematics Education. PCMI includes numerous crossprogram activities to help members from all these groups interact with one another.
Updated on Feb 02, 2022 03:52 PM PST 
Program Definability, Decidability, and Computability in Number Theory, part 2
Organizers: Valentina Harizanov (George Washington University), Barry Mazur (Harvard University), Russell Miller (Queens College, CUNY; CUNY, Graduate Center), Jonathan Pila (University of Oxford), Thomas Scanlon (University of California, Berkeley), Alexandra Shlapentokh (East Carolina University)This program is focused on the twoway interaction of logical ideas and techniques, such as definability from model theory and decidability from computability theory, with fundamental problems in number theory. These include analogues of Hilbert's tenth problem, isolating properties of fields of algebraic numbers which relate to undecidability, decision problems around linear recurrence and algebraic differential equations, the relation of transcendence results and conjectures to decidability and decision problems, and some problems in anabelian geometry and field arithmetic. We are interested in this specific interface across a range of problems and so intend to build a semester which is both more topically focused and more mathematically broad than a typical MSRI program.
Updated on Dec 21, 2021 09:51 AM PST 
Summer Graduate School MSRINCTS Joint Summer School: Recent Topic in Well Posedness
Organizers: Jungkai Chen (National Taiwan University), Mimi Dai (University of Illinois at Chicago), Yoshikazu Giga (University of Tokyo), Tsuyoshi Yoneda (Hitotsubashi University)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 NavierStokes 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.
Updated on Mar 25, 2022 12:36 PM PDT 
Summer Graduate School Mathematics of Machine Learning (INdAM and Courant Institute)
Organizers: Sebastien Bubeck (Microsoft Research), Adith Swaminathan (Microsoft Research)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.
Updated on Apr 25, 2022 11:11 AM PDT 
Summer Graduate School Topological Methods for the Discrete Mathematician
Organizers: Pavle Blagojevic (Freie Universität Berlin), Florian Frick (Carnegie Mellon University), Shira Zerbib (Iowa State University)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:
 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.
 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, simpletostate 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 wellsuited to rapidly put students in a position to approach related research questions.
Updated on Sep 07, 2021 09:52 AM PDT 
Program Simons Bridge Postdoctoral Fellowship 2022/23
Updated on Feb 10, 2022 10:34 AM PST 
Summer Graduate School Sums of Squares Method in Geometry, Combinatorics and Optimization (BIRS)
Organizers: LEAD Grigoriy Blekherman (Georgia Institute of Technology), Annie Raymond (University of Massachusetts Amherst), Cynthia Vinzant (University of Washington)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 thetabodies with applications in optimization.
The summer school will give a selfcontained 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.
Updated on Apr 07, 2022 02:41 PM PDT 
Summer Graduate School Tropical Geometry
Organizers: Renzo Cavalieri (Colorado State University), Hannah Markwig (EberhardKarlsUniversität Tübingen), Dhruv Ranganathan (University of Cambridge)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:
 Enumeration of tropical curves/ by Hannah Markwig
 Curve counting in tropical and algebraic geometry by Renzo Cavalieri
 Logarithmic geometry and stable map/s by Dhruv Ranganathan
Updated on Feb 04, 2022 09:44 AM PST 
Program Floer Homotopy Theory
Organizers: Mohammed Abouzaid (Columbia University), Andrew Blumberg (Columbia University), Kristen Hendricks (Rutgers University), Robert Lipshitz (University of Oregon), LEAD Ciprian Manolescu (Stanford University), Nathalie Wahl (University of Copenhagen)The development of Floer theory in its early years can be seen as a parallel to the emergence of algebraic topology in the first half of the 20th century, going from counting invariants to homology groups, and beyond that to the construction of algebraic structures on these homology groups and their underlying chain complexes. In continuing work that started in the latter part of the 20th century, algebraic topologists and homotopy theorists have developed deep methods for refining these constructions, motivated in large part by the application of understanding the classification of manifolds. The goal of this program is to relate these developments to Floer theory with the dual aims of (i) making progress in understanding symplectic and lowdimensional topology, and (ii) providing a new set of geometrically motivated questions in homotopy theory.
Updated on Oct 02, 2020 03:01 PM PDT 
Program Analytic and Geometric Aspects of Gauge Theory
Organizers: Laura Fredrickson (University of Oregon), Rafe Mazzeo (Stanford University), Tomasz Mrowka (Massachusetts Institute of Technology), Laura Schaposnik (University of Illinois at Chicago), LEAD Thomas Walpuski (HumboldtUniversität)The mathematics and physics around gauge theory have, since their first interaction in the mid 1970’s, prompted tremendous developments in both mathematics and physics. Deep and fundamental tools in partial differential equations have been developed to provide rigorous foundations for the mathematical study of gauge theories. This led to ongoing revolutions in the understanding of manifolds of dimensions 3 and 4 and presaged the development of symplectic topology. Ideas from quantum field theory have provided deep insights into new directions and conjectures on the structure of gauge theories and suggested many potential applications. The focus of this program will be those parts of gauge theory which hold promise for new applications to geometry and topology and require development of new analytic tools for their study.
Updated on Oct 28, 2020 09:12 AM PDT 
Program Complementary Program 202223
Updated on Feb 22, 2022 03:09 PM PST 
Workshop Connections Workshop: Analytic and Geometric Aspects of Gauge Theory
Organizers: Lara Anderson (Virginia Polytechnic Institute and State University), LEAD Laura Schaposnik (University of Illinois at Chicago)This twoday workshop will consist of various talks given by prominent female mathematicians on topics of analytic and geometric aspects of gauge theory. These will be appropriate for graduate students, postdocs, and researchers in areas related to the program. The meeting aims to support young researchers working in analytic and geometric aspects of gauge theory by facilitating mentoring from senior colleagues and helping towards the development of crucial professional skills. The format will include mentoring pairings, panel discussions, and Q&A sessions as well as the opportunity for informal discussions and connections.
Updated on May 16, 2022 11:49 AM PDT 
Workshop Introductory Workshop: Analytic and Geometric Aspects of Gauge Theory
Organizers: LEAD Aleksander Doan (Trinity College; University College London), Laura Fredrickson (University of Oregon), Michael Singer (University College London)The workshop will highlight the utility and impact of gauge theory in other areas of math. Minicourses will cover the historical utility and impact of gauge theory in areas including lowdimensional topology, algebraic geometry, and the analysis of PDE; additional talks will cover more recent directions.
Updated on May 16, 2022 11:50 AM PDT 
Workshop Connections Workshop: Floer Homotopy Theory
Organizers: Teena Gerhardt (Michigan State University), LEAD Kristen Hendricks (Rutgers University), Ailsa Keating (University of Cambridge)This workshop will feature talks by experts in Floer theory (and its applications to lowdimensional topology) and homotopy theory. It will include two expository lectures aimed at graduate students and other researchers who are new to the field, as well as a sequence of research talks and a contributed talks session. There will also be a panel discussion focusing on professional development. The majority of the speakers and panelists for this event will be women and gender minorities, and members of these groups and of other underrepresented groups are especially encouraged to attend. This workshop is open to all mathematicians.
Updated on May 16, 2022 11:50 AM PDT 
Workshop Introductory Workshop: Floer Homotopy Theory
Organizers: Sheel Ganatra (University of Southern California), Tyler Lawson (University of Minnesota Twin Cities), LEAD Robert Lipshitz (University of Oregon), Nathalie Wahl (University of Copenhagen)Over the last decade, there has been a wealth of new applications of homotopytheoretic techniques to Floer homology in lowdimensional topology and symplectic geometry, including Manolescu’s disproof of the highdimensional Triangulation Conjecture and AbouzaidBlumberg’s proof of the Arnol’d Conjecture in finite characteristic. Conversely, results in Floer theory and categorification have opened new directions of research in homotopy theory, from string topology to SLie algebras. The goal of this workshop is to introduce researchers in Floer theory to modern techniques and questions in homotopy theory and, conversely, introduce researchers in homotopy theory to ideas underlying Floer theory and its applications.
Updated on Mar 10, 2021 09:12 AM PST 
Workshop New fourdimensional gauge theories
Organizers: Andriy Haydys (Université Libre de Bruxelles), Lotte Hollands (HeriotWatt University, Riccarton Campus), LEAD ElenyNicoleta Ionel (Stanford University), Richard Thomas (Imperial College, London), Thomas Walpuski (HumboldtUniversität)This workshop will bring together researchers working on new fourdimensional gauge theories from the perspectives of differential geometry, algebraic geometry, and physics. Over the last 25 years, physicists have made tantalizing conjectures relating the Vafa–Witten equation to modular forms and the Kapustin–Witten and Haydys–Witten equations to knot theory and the geometric Langlands programme. The analytical challenges in the way of establishing these predictions are now being pursued vigorously. More recently, algebraic geometers have had enormous success in confirming and refining Vafa–Witten's predictions for projective surfaces. The workshop will serve as a platform for reporting on recent progress and exchanging ideas in all of these areas, with the aim of strengthening existing and fostering new interactions.
Created on Mar 18, 2021 02:28 PM PDT 
Workshop Floer homotopical methods in low dimensional and symplectic topology
Organizers: LEAD Mohammed Abouzaid (Columbia University), Andrew Blumberg (Columbia University), Jennifer Hom (Georgia Institute of Technology), Emmy Murphy (Northwestern University), Sucharit Sarkar (University of California, Los Angeles)The workshop will focus on the interaction between homotopy theory and symplectic topology and low dimensional topology that is mediated by Floer theory. Among the topics covered are foundational questions, applications to concrete geometric questions, and the relationship with finite dimensional approaches.
Updated on Mar 18, 2021 02:21 PM PDT 
Program Algebraic Cycles, LValues, and Euler Systems
Organizers: Henri Darmon (McGill University), Ellen Eischen (University of Oregon), LEAD Benjamin Howard (Boston College), David Loeffler (University of Warwick), Christopher Skinner (Princeton University), Sarah Zerbes (ETH Zürich), Wei Zhang (Massachusetts Institute of Technology)The fundamental conjecture of Birch and SwinnertonDyer relating the Mordell–Weil ranks of elliptic curves to their Lfunctions is one of the most important and motivating problems in number theory. It resides at the heart of a collection of important conjectures (due especially to Deligne, Beilinson, Bloch and Kato) that connect values of Lfunctions and their leading terms to cycles and Galois cohomology groups.
The study of special algebraic cycles on Shimura varieties has led to progress in our understanding of these conjectures. The arithmetic intersection numbers and the padic regulators of special cycles are directly related to the values and derivatives of Lfunctions, as shown in the pioneering theorem of GrossZagier and its padic avatars for Heegner points on modular curves. The cohomology classes of special cycles (and related constructions such as Eisenstein classes) form the foundation of the theory of Euler systems, providing one of the most powerful methods known to prove vanishing or finiteness results for Selmer groups of Galois representations.
The goal of this semester is to bring together researchers working on different aspects of this young but fastdeveloping subject, and to make progress on understanding the mysterious relations between Lfunctions, Euler systems, and algebraic cycles.
Updated on Apr 12, 2021 10:17 AM PDT 
Program Diophantine Geometry
Organizers: Jennifer Balakrishnan (Boston University), Mirela Ciperiani (University of Texas, Austin), Philipp Habegger (University of Basel), Wei Ho (University of Michigan), LEAD Hector Pasten (Pontificia Universidad Católica de Chile), Yunqing Tang (Princeton University), ShouWu Zhang (Princeton University)While the study of rational solutions of diophantine equations initiated thousands of years ago, our knowledge on this subject has dramatically improved in recent years. Especially, we have witnessed spectacular progress in aspects such as height formulas and height bounds for algebraic points, automorphic methods, unlikely intersection problems, and nonabelian and padic approaches to algebraic degeneracy of rational points. All these groundbreaking advances in the study of rational and algebraic points in varieties will be the central theme of the semester program “Diophantine Geometry” at MSRI. The main purpose of this program is to bring together experts as well as enthusiastic young researchers to learn from each other, to initiate and continue collaborations, to update on recent breakthroughs, and to further advance the field by making progress on fundamental open problems and by developing further connections with other branches of mathematics. We trust that younger mathematicians will greatly contribute to the success of the program with their new ideas. It is our hope that this program will provide a unique opportunity for women and underrepresented groups to make outstanding contributions to the field, and we strongly encourage their participation.
Updated on Feb 25, 2021 04:59 PM PST 
Workshop Connections Workshop: Algebraic Cycles, LValues, and Euler Systems
Organizers: Henri Darmon (McGill University), Ellen Eischen (University of Oregon), Benjamin Howard (Boston College), LEAD Elena Mantovan (California Institute of Technology)The Connections Workshop features presentations by both leading researchers and promising newcomers whose research has contact with the interrelated topics of algebraic cycles, Lvalues, and Euler systems. The goal is to present a variety of diverse results, so as to forge new connections, foster collaborative projects, and establish mentoring relationships. While emphasis will be placed on the work of women mathematicians, the workshop is open to all researchers. This workshop is held in honor of mathematician Bernadette PerrinRiou.
Updated on Mar 29, 2022 10:07 AM PDT 
Workshop Introductory Workshop: Algebraic Cycles, LValues, and Euler Systems
Organizers: Henri Darmon (McGill University), LEAD Ellen Eischen (University of Oregon), Benjamin Howard (Boston College), Elena Mantovan (California Institute of Technology)The Introductory Workshop aims to provide a coherent overview of current research in algebraic cycles, Lvalues, Euler systems, and the many connections between them. This includes the study of special cycles on Shimura varieties and moduli spaces of shtukas, integral representations of Lvalues and the construction of padic Lfunctions, and the construction of Euler systems from special elements in Chow groups or higher Chow groups of Shimura varieties. Workshop lectures will be organized into short lecture series, so as to allow each series to begin with expository lectures on foundational results before moving on to current research. This workshop is held in honor of mathematician Bernadette PerrinRiou.
Updated on Mar 29, 2022 10:07 AM PDT 
Workshop Connections Workshop: Diophantine Geometry
Organizers: Jennifer Balakrishnan (Boston University), LEAD Yunqing Tang (Princeton University)This workshop will highlight talks on various aspects of Diophantine Geometry. The goal of the workshop is to bring together researchers at different career stages and of various backgrounds in order to establish new collaborations and mentoring relationships. Although we will showcase the research of mathematicians who identify as women or gender minorities, this workshop is open to all.
Updated on Dec 17, 2021 02:42 PM PST 
Workshop Introductory Workshop: Diophantine Geometry
Organizers: Hector Pasten (Pontificia Universidad Católica de Chile), Yunqing Tang (Princeton University), LEAD ShouWu Zhang (Princeton University)This workshop will feature expository lectures about current developments in Diophantine geometry. This includes the uniform Mordell—Lang for rational points on curves, the Andre—Oort conjecture for special points on Shimura varieties, and effective results via Chabauty method, and related topics in Arakelov theory, unlikely intersections, arithmetic statistics, arithmetic dynamics, and padic Hodge theory.
Updated on Dec 20, 2021 09:18 AM PST 
Workshop Shimura Varieties and Lfunctions
Organizers: Michael Harris (Columbia University), David Loeffler (University of Warwick), Elena Mantovan (California Institute of Technology), Christopher Skinner (Princeton University), Sarah Zerbes (ETH Zürich), LEAD Wei Zhang (Massachusetts Institute of Technology)The topical workshop will be dedicated to Shouwu Zhang, to mark the occasion of his 60th birthday, and to honour his numerous beautiful contributions to the theory of Shimura varieties and special values of Lfunctions. It will highlight cutting edge work on topics such as the construction of Euler systems; relations between special cycles on Shimura varieties and Lfunctions, such as generalized GrossZagier formulas and the Tate conjecture; the construction of Galois representations in cohomology; and related aspects of the theory of automorphic forms.
Updated on Aug 25, 2021 03:20 PM PDT 
Workshop Degeneracy of algebraic points
Organizers: Jennifer Balakrishnan (Boston University), LEAD Mirela Ciperiani (University of Texas, Austin), Philipp Habegger (University of Basel), Wei Ho (University of Michigan), Hector Pasten (Pontificia Universidad Católica de Chile), Yunqing Tang (Princeton University), ShouWu Zhang (Princeton University)Updated on Nov 02, 2021 01:30 PM PDT 
African Diaspora Joint Mathematics 2023 African Diaspora Joint Mathematics Workshop
The African Diaspora Joint Mathematics Workshop (ADJOINT) will take place at the Mathematical Sciences Research Institute in Berkeley, CA from June 19 to June 30, 2023.
ADJOINT is a twoweek summer activity designed for researchers with a Ph.D. degree in the mathematical sciences who are interested in conducting research in a collegial environment.
The main objective of ADJOINT is to provide opportunities for inperson research collaboration to U.S. mathematicians, especially those from the African Diaspora, who will work in small groups with research leaders on various research projects.
Through this effort, MSRI aims to establish and promote research communities that will foster and strengthen research productivity and career development among its participants. The ADJOINT workshops are designed to catalyze research collaborations, provide support for conferences to increase the visibility of the researchers, and to develop a sense of community among the mathematicians who attend.
The end goal of this program is to enhance the mathematical sciences and its community by positively affecting the research and careers of AfricanAmerican mathematicians and supporting their efforts to achieve full access and engagement in the broader research community.
Each summer, three to five research leaders will each propose a research topic to be studied during a twoweek workshop.
During the workshop, each participant will:
 conduct research at MSRI within a group of four to five mathematicians under the direction of one of the research leaders
 participate in professional enhancement activities provided by the onsite ADJOINT Director
 receive funding for two weeks of lodging, meals and incidentals, and one roundtrip travel to Berkeley, CA
After the twoweek workshop, each participant will:
 have the opportunity to further their research project with the team members including the research leader
 have access to funding to attend conference(s) or to meet with other team members to pursue the research project, or to present results
 become part of a network of research and career mentors
Updated on Jan 13, 2022 11:30 AM PST 
Program Mathematics and Computer Science of Market and Mechanism Design
Organizers: Michal Feldman (TelAviv University), Nicole Immorlica (Microsoft Research), LEAD Scott Kominers (Harvard Business School), Shengwu Li (Harvard University), Paul Milgrom (Stanford University), Alvin Roth (Stanford University), Tim Roughgarden (Stanford University), Eva Tardos (Cornell University)In recent years, economists and computer scientists have collaborated with mathematicians, operations research experts, and practitioners to improve the design and operations of realworld marketplaces. Such work relies on robust feedback between theory and practice, inspiring new mathematics closely linked – and directly applicable – to market and mechanism design questions. This crossdisciplinary program seeks to expand the domains in which existing market design solutions can be applied; address foundational questions regarding our ways of developing and evaluating mechanisms; and build useful analytic frameworks for applying theory to practical marketplace design.
Updated on Feb 10, 2022 08:58 AM PST 
Program Algorithms, Fairness, and Equity
Organizers: Rediet Abebe (University of California, Berkeley), Vincent Conitzer (Duke University), Moon Duchin (Tufts University), Bettina Klaus (Université de Lausanne), Jonathan Mattingly (Duke University), LEAD Wesley Pegden (Carnegie Mellon University)This program aims to bring together researchers working at the interface of fairness and computation. This interface has been the site of intensive research effort in mechanism design, in research on partitioning problems related to political districting problems, and in research on ways to address issues of fairness and equity in the context of machine learning algorithms.
These areas each approach the relationship between mathematics and fairness from a distinct perspective. In mechanism design, algorithms are a tool to achieve outcomes with mathematical guarantees of various notions of fairness. In machine learning, we perceive failures of fairness as an undesirable side effect of learning approaches, and seek mathematical approaches to understand and mitigate these failures. And in partitioning problems like political districting, we often seek mathematical tools to evaluate the fairness of human decisions.
This program will explore progress in these areas while also providing a venue for overlapping perspectives. The topics workshop “Randomization, neutrality, and fairness” will explore the common role randomness and probability has played in these lines of work.
Updated on May 13, 2022 11:47 AM PDT 
Program Commutative Algebra
Organizers: Aldo Conca (Università di Genova), Steven Cutkosky (University of Missouri), LEAD Claudia Polini (University of Notre Dame), Claudiu Raicu (University of Notre Dame), Steven Sam (University of California, San Diego), Kevin Tucker (University of Illinois at Chicago), Claire Voisin (Collège de France; Institut de Mathématiques de Jussieu)Commutative algebra is, in its essence, the study of algebraic objects, such as rings and modules over them, arising from polynomials and integral numbers. It has numerous connections to other fields of mathematics including algebraic geometry, algebraic number theory, algebraic topology and algebraic combinatorics. Commutative Algebra has witnessed a number of spectacular developments in recent years, including the resolution of longstanding problems, with new techniques and perspectives leading to an extraordinary transformation in the field. The main focus of the program will be on these developments. These include the recent solution of Hochster's direct summand conjecture in mixed characteristic that employs the theory of perfectoid spaces, a new approach to the BuchsbaumEisenbudHorrocks conjecture on the Betti numbers of modules of finite length, recent progress on the study of CastelnuovoMumford regularity, the proof of Stillman's conjecture and ongoing work on its effectiveness, a novel strategy to Green's conjecture on the syzygies of canonical curves based on the study of Koszul modules and their generalizations, new developments in the study of various types of multiplicities, theoretical and computational aspects of Gröbner bases, and the implicitization problem for Rees algebras and its applications.
Updated on Oct 19, 2021 11:00 AM PDT 
Program Noncommutative Algebraic Geometry
Organizers: Wendy Lowen (Universiteit Antwerp), Alexander Perry (University of Michigan), LEAD Alexander Polishchuk (University of Oregon), Susan Sierra (University of Edinburgh), Spela Spenko (Université Libre de Bruxelles), Michel Van den Bergh (Universiteit Hasselt)Derived categories of coherent sheaves on algebraic varieties were originally conceived as technical tools for studying cohomology, but have since become central objects in fields ranging from algebraic geometry to mathematical physics, symplectic geometry, and representation theory. Noncommutative algebraic geometry is based on the idea that any category sufficiently similar to the derived category of a variety should be regarded as (the derived category of) a “noncommutative algebraic variety”; examples include semiorthogonal components of derived categories, categories of matrix factorizations, and derived categories of noncommutative dgalgebras. This perspective has led to progress on old problems, as well as surprising connections between seemingly unrelated areas. In recent years there have been great advances in this domain, including new tools for constructing semiorthogonal decompositions and derived equivalences, progress on conjectures relating birational geometry and singularities to derived categories, constructions of moduli spaces from noncommutative varieties, and instances of homological mirror symmetry for noncommutative varieties. The goal of this program is to explore and expand upon these developments.
Updated on May 19, 2022 01:51 PM PDT

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