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Combinatorial Algebraic Geometry

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.

CMI Enhancement and Partnership Program

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Venue: Fields Institute

Local Representation Theory and Simple Groups

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.

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Venue: Centre Interfacultaire Bernoulli, EPFL

Constructive Approximation and Harmonic Analysis

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

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Venue: Centre de Recerca Mathemàtica, Bellaterra, Spain

Differential Geometry

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.

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Venue: MSRI

Geometric Functional Analysis and Applications

Geometric functional analysis lies at the interface of convex geometry, functional analysis and probability. It has numerous applications ranging from geometry of numbers and random matrices in pure mathematics to geometric tomography and signal processing in engineering and numerical optimization and learning theory in computer science.

One of the directions of the program is classical convex geometry, with emphasis on connections with geometric tomography, the study of geometric properties of convex bodies based on information about their sections and projections. Methods of harmonic analysis play an important role here. A closely related direction is asymptotic geometric analysis studying geometric properties of high dimensional objects and normed spaces, especially asymptotics of their quantitative parameters as dimension tends to infinity. The main tools here are concentration of measure and related probabilistic results. Ideas developed in geometric functional analysis have led to progress in several areas of applied mathematics and computer science, including compressed sensing and random matrix methods. These applications as well as the problems coming from computer science will be also emphasised in our program.

Professor William B. Johnson (Texas A&M) has been appointed as a Clay Senior Scholar to participate in this program.

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Venue: MSRI

Geometric and Topological Combinatorics

Combinatorics is one of the fastest growing areas in contemporary Mathematics, and much of this growth is due to the connections and interactions with other areas of Mathematics. This program is devoted to the very vibrant and active area of interaction between Combinatorics with Geometry and Topology. That is, we focus on (1) the study of the combinatorial properties or structure of geometric and topological objects and (2) the development of geometric and topological techniques to answer combinatorial problems.

Key examples of geometric objects with intricate combinatorial structure are point configurations and matroids, hyperplane and subspace arrangements, polytopes and polyhedra, lattices, convex bodies, and sphere packings. Examples of topology in action answering combinatorial challenges are the by now classical Lovász’s solution of the Kneser conjecture, which yielded functorial approaches to graph coloring, and the  more recent, extensive topological machinery leading to breakthroughs on Tverberg-type problems.

Professor Francisco Santos (Cantabria) has been appointed as a Clay Senior Scholar to participate in this program.

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Venue: MSRI

Random Matrices

This PCMI research program offers advanced scholars the opportunity to do research, collaborate with peers, meet outstanding students, and explore new teaching ideas with professional educators.  It is designed to introduce active areas of research by focusing on a specific topic.  The informal format generates lively exchanges of views and information between established and newer researchers.

Professor Craig Tracy (UC Davis) and Professor Horng-Tzer Yau have been appointed as Clay Senior Scholars to partipate in this program.

Image:  Skyguy414 at English Wikipedia, Wikimedia Commons

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Venue: Park City Math Institute - Institute for Advanced Study, Park City, Utah

Operator Algebras: Dynamics and Interactions

This intensive research program is designed simultaneously to speed up and contribute to the next game-changing developments in classification and structure theory for C^*-algebras, and to prepare for the new state of play in the area once those developments arrive. According to current developments, the IRP will strive to blend different techniques coming both from C^* and von Neumann algebras and find a common ground to maximise outputs. The program will bring together many of the world’s leading researchers working on C^*-algebras, von Neumann algebras, dynamical systems, and the interactions between these areas. There are three key objectives: 

1. To advance classification programs for C^*-algebras and for C^*-dynamical systems. 

2. To investigate further suitable models for classifiable C^*-algebras arising from dynamics. 

3. To encourage further interaction between C^*-algebras and von Neumann algebras in light of current progress in both areas

CMI Enhancement and Partnership Program

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Venue: CRM

Harmonic Analysis

The field of Harmonic Analysis dates back to the 19th century, and has its roots in the study of the decomposition of functions using Fourier series and the Fourier transform.  In recent decades, the subject has undergone a rapid diversification and expansion, though the decomposition of functions and operators into simpler parts remains a central tool and theme.  

This program will bring together researchers representing the breadth of modern Harmonic Analysis and will seek to capitalize on and continue recent progress in four major directions: Restriction, Kakeya, and Geometric Incidence Problems; Analysis on Nonhomogeneous Spaces; Weighted Norm Inequalities; and Quantitative Rectifiability and Elliptic PDE.  Many of these areas draw techniques from or have applications to other fields of mathematics, such as analytic number theory, partial differential equations, combinatorics, and geometric measure theory.  

Professor Alexander Volberg (Michigan State) has been appointed as a Clay Senior Scholar from January to May 2017 to participate in this program.

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Venue: MSRI

Analytic Number Theory

Analytic number theory, and its applications and interactions, are currently experiencing intensive progress, in sometimes unexpected directions. In recent years, many important classical questions have seen spectacular advances based on new techniques; conversely, methods developed in analytic number theory have led to the solution of striking problems in other fields.

Recent advances in analytic number theory have had repercussions in various mathematical subjects, such as harmonic analysis (including the Langlands programme), ergodic theory and dynamics (especially on homogenous spaces), additive and multiplicative combinatorics and theoretical computer science (in particular, through the theory of expander graphs).

The MSRI semester program in Spring 2017 will focus on the topic of Analytic Number Theory, with workshops and other activities focused on the most impressive recent achievements in this field. 

Professor Manjul Bhargava (Princeton) and Professor Roger Heath-Brown (Oxford) have been appointed as Clay Senior Scholars from January to May 2017 to participate in this program.

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Venue: MSRI

Non-Positive Curvature Group Actions and Cohomology

This programme is about aspects of non-positive curvature as they occur in various research areas of contemporary mathematics, such as the geometry of manifolds, including those arising from Lie group theory; synthetic geometry (e.g. CAT(0) geometry) as used notably in modern group theory; coarse geometry, a fundamental tool of contemporary topology; non-commutative geometry; and algebra.

By combining and comparing the aspects that non-positive curvature displays in various contexts, the programme aims to acquire new insight and find meaningful new directions to explore. Among the tools used to uncover new aspects of non-positive curvature one numbers cohomology, originally a tool from algebra and topology and now a research field in itself, with countless applications throughout mathematics; perturbation stability — investigating when small “errors” lead to completely new structures, or when, on the contrary, structures are robust when perturbed;fixed point and proper actions – a topic at the interface of analysis and algebra – with surprising ramifications in many other areas; and CAT(0) and coarse geometry, where CAT(0) cubical and large-scale techniques come into play to solve problems from topology and algebra. All these fields are on the cutting edge of current research, and have fruitful connections to other areas, e.g. theoretical computer science.

Professor Emmanuel Breuillard (Paris Sud) has been appointed as a Clay Senior Scholar from January to June 2017 to particpate in this program.

Image: Ma10024, Wikimedia Commons

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Venue: Isaac Newton Institute

Homological Mirror Symmetry

Homological Mirror Symmetry (HMS) was initiated by Maxim Kontsevich. It benefits from a close relationship with string theory, and has developed into a powerful and versatile idea. During the program, we will consider the core conjectures of HMS, and its role as a framework within which wider questions from mirror symmetry and other parts of mathematics can be studied. This is still a developing subject, and the program is open to a variety of approaches and viewpoints.

Professor Dmitry Orlov (Steklov) has been appointed as a Clay Senior Scholar from September 2016 to April 2017 to participate in this program.

Image: Eecc, Wikimedia Commons

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Venue: Institute for Advanced Study