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.
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.
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.
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.
The study of varieties with explicit combinatorial structure provides a unifying theme for this Combinatorial Algebraic Geometry semester at the Fields Institute. These varieties are surprisingly ubiquitous, arising in algebraic geometry, commutative algebra, representation theory, mathematical physics, and many other fields. The scientific aims of this program are to:
introduce the study of such ‘combinatorial varieties’ to the mathematical community,
refine the techniques used within algebraic geometry to study combinatorial varieties, and
enlarge the class of algebraic spaces which have a recognized combinatorial structure.
Local representation theory, pioneered by Richard Brauer in the 1930s had its first big successes in the classification of the finite simple groups. Since then, important and deep connections to areas as varied as topology, geometry, Lie theory and homological algebra have been discovered and used. Very recent breakthrough results have now led to the hope that some of the long standing and deep problems, some of which have been open for over five decades, can finally be settled.
Recent results relied crucially on the interplay between the theory of modular representations, the classification of finite simple groups, and Lusztig’s powerful geometric machinery built around the Weil conjectures which describes the representation theory of finite reductive groups. At the same time, this has led to a wealth of interesting new questions on finite simple groups and their representation theory, whose solution promises to be useful for many further applications. The main theme of the programme will be to exploit further this interaction with the aim of eventually solving some of the famous open conjectures, and further developing and applying the representation theory of finite simple groups.
Professor Pham Huu Tiep has been appointed as a Clay Senior Scholar from October to December 2016 to participate in this program.
This research program focuses on the interaction between constructive approximation and harmonic analysis. The aim is to facilitate broader and deeper interaction among researchers in these fields.
Approximation theory seeks to approximate complicated functions by simpler functions from certain classes and to evaluate the errors inevitably arising in such approximations. This field draws its methods from various areas of mathematics such as functional analysis, variational analysis, probability, etc. Constructive approximation strives to explicitly find the best (or nearly best) approximants in such problems which is of tremendous importance in numerous applications.
Harmonic analysis is a very old branch of mathematics which investigates the behaviour of functions using their time and frequency features as well as various (orthogonal) decompositions. Despite a long history, this field is very vital today with numerous open problems and active research areas, such as weighted inequalities, time-frequency analysis, multilinear analysis, singular integrals on rectifiable sets and many others. Harmonic analysis reveals exciting connections to other branches of mathematics, such as geometric measure theory, complex analysis, convex and discrete geometry.
Among these relations, a very special place is taken by the link between harmonic analysis and approximation theory (this interplay is well known: trigonometric polynomials, wavelets, frames, other (quasi) orthoganal systems, hyperbolic cross approximations play and important role in approximation and are a subject of active ongoing research).
CMI Enhancement and Partnership Program
Venue: Centre de Recerca Mathemàtica, Bellaterra, Spain
Differential geometry is a subject with both deep roots and recent advances. Many old problems in the field have recently been solved, such as the Poincaré and geometrization conjectures by Perelman, the quarter pinching conjecture by Brendle-Schoen, the Lawson Conjecture by Brendle, and the Willmore Conjecture by Marques-Neves. The solutions of these problems have introduced a wealth of new techniques into the field. This semester-long program will focus on the following main themes: (1) Einstein metrics and generalizations, (2) Complex differential geometry, (3) Spaces with curvature bounded from below, (4) Geometric flows, and particularly on the deep connections between these areas.
Professor Tobias Colding (MIT) has been appointed as a Clay Senior Scholar from January to May 2016 to participate in this program.