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Special Geometric Structures and Analysis

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.

The aim of this program is to study geometric and analytic aspects of special holonomy metrics, instanton bundle over such spaces and the calibrated submanifolds within them. Because these objects are Einstein metrics, Yang–Mills connections and minimal submanifolds respectively they fit into a wider geometric framework. But because each of them satisfies a first-order PDE system they enjoy various special properties. This program takes the position that the natural viewpoint, both geometrically and analytically, is to consider these three classes of closely-related geometric objects together. A common feature is that weak solutions to these higher-dimensional equations may have interesting non-isolated singular sets whose detailed geometric and analytic structure is still poorly understood. Improving our understanding of such (partial) regularity of solutions is a key component of the program.

Professor Vincent Guedj (Toulouse) has been appointed as a Clay Senior Scholar to participate in this program.

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Venue: Simons Laufer Mathematical Research Institute

Commutative Algebra

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 long-standing 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 Buchsbaum–Eisenbud–Horrocks conjecture on the Betti numbers of modules of finite length, recent progress on the study of Castelnuovo–Mumford 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.

Professor Bernd Ulrich (Purdue) has been appointed as a Clay Senior Scholar to participate in this program.

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Venue: Simons Laufer Mathematical Research Institute

Noncommutative Algebraic Geometry

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 dg-algebras. 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. 

Professor Mikhail Kapranov (Kavli) has been appointed as a Clay Senior Scholar to participate in this program.

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Venue: Simons Laufer Mathematical Research Institute

Group Actions and Rigidity: Around the Zimmer Program

This term focuses on rigidity of group actions with some focus on issues that arise out of Zimmer’s groundbreaking work in the 1980’s. Areas of interest range from local and global rigidity of group actions to special rigidity phenomena in small dimensions to symmetry groups of geometric structures to measure rigidity results for quite general group actions with hyperbolicity.

This broad area has seen a very large number of dramatic breakthroughs in recent years. The goal of the term is to bring together experts and people in adjacent areas to share ideas and techniques and to foster additional breakthroughs. There will also be a significant component of the term dedicated to introducing younger participants to the broad area.

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Venue: Institut Henri Poincaré

Toric Topology, Geometry and Polyhedral Products

This Focus Program will build upon the work of the Jan-Jun 2020 Thematic Program on Toric Topology and Polyhedral Products. 

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Venue: Fields Institute

Diophantine Geometry

Number Theory concerns the study of properties of the integers, rational numbers, and other structures that share similar features. It is a central branch of mathematics with a well-known feature: it is often the case that easy-to-state problems in number theory turn out to be exceedingly difficult (e.g. Fermat’s Last Theorem), and their study leads to groundbreaking discoveries in other fields of mathematics.

A fundamental theme in number theory concerns the study of integer and rational solutions to Diophantine equations. This topic originated at least 3,700 years ago (as documented in babylonian clay tablets) and it has evolved into the highly sophisticated field of Diophantine Geometry. There are deep and fruitful interactions between Diophantine Geometry and seemingly distant fields such as representation theory, algebraic geometry, topology, complex analysis, and mathematical logic, to mention a few. In recent years, these connections have led to a large number of new results and, specially, to the partial or complete resolution of important conjectures in the field.

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 non-abelian and p-adic 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. 

Professor Mark Kisin (Harvard) has been appointed at a Clay Senior Scholar to participate in this program.

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Venue: MSRI

Algebraic Cycles, L-Values, and Euler Systems

The fundamental conjecture of Birch and Swinnerton-Dyer relating the Mordell–Weil ranks of elliptic curves to their L-functions 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 L-functions 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 p-adic regulators of special cycles are directly related to the values and derivatives of L-functions, as shown in the pioneering theorem of Gross-Zagier and its p-adic 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 fast-developing subject, and to make progress on understanding the mysterious relations between L-functions, Euler systems, and algebraic cycles.

Professor Henri Darmon (McGill) has been appointed as a Clay Senior Scholar to participate in this program.

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Venue: MSRI

Geometry and Dynamics of Integrable Systems

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Venue: Centre de Recerca Matemàtica (CRM)

Conformal Geometry and Geometric PDEs

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Venue: Centre de Recerca Matemática (CRM)

Central Configurations, Periodic Orbits and Beyond in Celestial Mechanics

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Venue: CRM

Large Cardinals and Strong Logics

This program aims to attract to the CRM a diverse group of international high-level researchers working in strong logics, large cardinals, the foundations of set theory, and the applications of set-theoretical methods in other areas of mathematics, such as algebra, set-theoretical topology, category theory, algebraic topology, homotopy theory, C*-algebras, measure theory, etc.  In all these areas there are not only direct set-theoretical applications but also new results and methods, which are amenable to the expressive power of strong logics.  

CMI Enhancement and Partnership Program

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Venue: Centre de Recerca Matemàtica

Geometric Group Theory

The field of geometric group theory emerged from Gromov’s insight that even mathematical objects such as groups, which are defined completely in algebraic terms, can be profitably viewed as geometric objects and studied with geometric techniques Contemporary geometric group theory has broadened its scope considerably, but retains this basic philosophy of reformulating in geometric terms problems from diverse areas of mathematics and then solving them with a variety of tools. The growing list of areas where this general approach has been successful includes low-dimensional topology, the theory of manifolds, algebraic topology, complex dynamics, combinatorial group theory, algebra, logic, the study of various classical families of groups, Riemannian geometry and representation theory.  The goals of this MSRI program are to bring together people from the various branches of the field in order to consolidate recent progress, chart new directions, and train the next generation of geometric group theorists.

Professor Karen Vogtmann (Warwick) has been appointed as a Clay Senior Scholar from August to December 2016 to participate in this program.

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Venue: MSRI