
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 
Diophantine Geometry
Organizers: Jennifer Balakrishnan (Boston University), Mirela Ciperiani (University of Texas, Austin), Philipp Habegger (University of Basel), Wei Ho (Institute for Advanced Study), LEAD Hector Pasten (Pontificia Universidad Católica de Chile), Yunqing Tang (University of California, Berkeley), 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 
Mathematical Problems in Fluid Dynamics, part 2
PROGRAM DESCRIPTION
Fluid dynamics is one of the classical areas of partial differential equations, and has been the subject of extensive research over hundreds of years. It is perhaps one of the most challenging and exciting fields of scientific pursuit simply because of the complexity of the subject and the endless breadth of applications.
The focus of the program is on incompressible fluids, where water is a primary example. The fundamental equations in this area are the wellknown Euler equations for inviscid fluids, and the NavierStokes equations for the viscous fluids. Relating the two is the problem of the zero viscosity limit, and its connection to the phenomena of turbulence. Water waves, or more generally interface problems in fluids, represent another target area for the program. Both theoretical and numerical aspects will be considered.
Updated on Oct 04, 2022 04:03 PM PDT 
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 Nov 11, 2022 01:37 PM PST 
Algorithms, Fairness, and Equity
Organizers: Vincent Conitzer (Duke University), Moon Duchin (Tufts University), Bettina Klaus (University of 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 Nov 11, 2022 01:41 PM PST 
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 May 24, 2022 10:29 AM PDT 
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 (Vrije Universiteit Brussel), Michel Van den Bergh (Hasselt University)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 
New Frontiers in Curvature: Flows, General Relativity, Minimal Submanifolds, and Symmetry
Organizers: LEAD Ailana Fraser (University of British Columbia), LanHsuan Huang (University of Connecticut), Richard Schoen (University of California, Irvine), LEAD Catherine Searle (Wichita State University), Lu Wang (Yale University), Guofang Wei (University of California, Santa Barbara)Geometry, PDE, and Relativity are subjects that have shown intriguing interactions in the past several decades, while simultaneously diverging, each with an ever growing number of branches. Recently, several major breakthroughs have been made in each of these fields using techniques and ideas from the others.
This program is aimed at connecting various branches of Geometry, PDE, and Relativity and at enhancing collaborations across these disciplines and will include four main topics: Geometric Flows, Geometric problems in Mathematical Relativity, Global Riemannian Geometry, and Minimal Submanifolds. Specifically the program focuses on a central goal, which is to advance our knowledge toward Riemannian (sub)manifolds under geometric conditions, such as curvature lower bounds, by developing techniques in, for example, geometric flows and minimal submanifolds and further fostering new connections.
Updated on Nov 17, 2022 10:10 AM PST 
Special Geometric Structures and Analysis
Organizers: Eleonora Di Nezza (Institut de Mathématiques de Jussieu), LEAD Mark Haskins (Imperial College, London), Tristan Riviere (ETH Zurich), Song Sun (University of California, Berkeley), Xuwen Zhu (Northeastern University)This program sits at the intersection between differential geometry and analysis but also connects to several other adjacent mathematical fields and to theoretical physics. Differential geometry aims to answer questions about very regular geometric objects (smooth Riemannian manifolds) using the tools of differential calculus. A fundamental object is the curvature tensor of a Riemannian metric: an algebraically complicated object that involves 2nd partial derivatives of the metric. Many questions in differential geometry can therefore be translated into questions about the existence or properties of the solutions of systems of (often) nonlinear partial differential equations (PDEs). The PDE systems that arise in geometry have historically stimulated the development of powerful new analytic methods. In most cases the nonlinearity of these systems makes ‘closed form’ expressions for a solution impossible: instead more abstract methods must be employed.
Updated on Nov 10, 2022 04:20 PM PST 
Probability and Statistics of Discrete Structures
Organizers: Louigi AddarioBerry (McGill University), Christina Goldschmidt (University of Oxford), PoLing Loh (University of Cambridge), Gabor Lugosi (Universitat Pompeu Fabra), Dana Randall (Georgia Institute of Technology), LEAD Remco van der Hofstad (Technische Universiteit Eindhoven)Random graphs and related random discrete structures lie at the forefront of applied probability and statistics, and are core topics across a wide range of scientific disciplines where mathematical ideas are used to model and understand realworld networks. At the same time, random graphs pose challenging mathematical and algorithmic problems that have attracted attention from probabilists and combinatorialists since at least 1960, following the pioneering work of Erdos and Renyi.
Around the turn of the millennium, as very large data sets became available, several applied disciplines started to realize that many realworld networks, even though they are from various origins, share fascinating features. In particular, many such networks are small worlds, meaning that graph distances in them are typically quite small, and they are scalefree, in the sense that the number of connections made by their elements is extremely heterogeneous. This program is devoted to the study of the probabilistic and statistical properties of such networks. Central tools include graphon theory for dense graphs, local weak convergence for sparse graphs, and scaling limits for the critical behavior of graphs or stochastic processes on them. The program is aimed at pure and applied mathematicians interested in network problems.Updated on Nov 21, 2022 03:40 PM PST

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