
Analytic Number Theory
Organizers: Chantal David (Concordia University), Andrew Granville (Université de Montréal), Emmanuel Kowalski (ETH Zuerich), Philippe Michel (Ecole Polytechnique Federale de Lausanne), Kannan Soundararajan (Stanford University), LEAD Terence Tao (University of California, Los Angeles)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.
This program will not only give the leading researchers in the area further opportunities to work together, but more importantly give young people the occasion to learn about these topics, and to give them the tools to achieve the next breakthroughs.
Updated on Jul 10, 2015 03:54 PM PDT 
Harmonic Analysis
Organizers: LEAD Michael Christ (University of California, Berkeley), Allan Greenleaf (University of Rochester), Steven Hofmann (University of Missouri), LEAD Michael Lacey (Georgia Institute of Technology), Svitlana Mayboroda (University of Minnesota, Twin Cities), Betsy Stovall (University of WisconsinMadison), Brian Street (University of WisconsinMadison)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 ProblemsAnalysis on Nonhomogeneous SpacesWeighted Norm InequalitiesQuantitative 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. In particular, we expect a lively interaction with the concurrent program.Updated on Aug 11, 2016 10:49 AM PDT 
Geometric Functional Analysis and Applications
Organizers: Franck Barthe (Université de Toulouse III (Paul Sabatier)), Marianna Csornyei (University of Chicago), Boaz Klartag (Tel Aviv University), Alexander Koldobsky (University of Missouri), Rafal Latala (University of Warsaw), LEAD Mark Rudelson (University of Michigan)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.
Updated on Jun 02, 2015 01:17 PM PDT 
Geometric and Topological Combinatorics
Organizers: Jesus De Loera (University of California, Davis), Vic Reiner (University of Minnesota Twin Cities), LEAD Francisco Santos (University of Cantabria), Francis Su (Harvey Mudd College), Rekha Thomas (University of Washington), Günter M. Ziegler (Freie Universität Berlin)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 Tverbergtype problems.Updated on Jul 05, 2016 08:46 AM PDT 
Group Representation Theory and Applications
Organizers: Robert Guralnick (University of Southern California), Alexander (Sasha) Kleshchev (University of Oregon), Gunter Malle (Universität Kaiserslautern), Gabriel Navarro (University of Valencia), Julia Pevtsova (University of Washington), Raphael Rouquier (University of California, Los Angeles), LEAD Pham Tiep (University of Arizona)Group Representation Theory is a central area of Algebra, with important and deep connections to areas as varied as topology, algebraic geometry, number theory, Lie theory, homological algebra, and mathematical physics. Born more than a century ago, the area still abounds with basic problems and fundamental conjectures, some of which have been open for over five decades. Very recent breakthroughs have led to the hope that some of these conjectures can finally be settled. In turn, recent results in group representation theory have helped achieve substantial progress in a vast number of applications.
The goal of the program is to investigate all these deep problems and the wealth of new results and directions, to obtain major progress in the area, and to explore further applications of group representation theory to other branches of mathematics.
Updated on Mar 16, 2016 01:25 PM PDT 
Enumerative Geometry Beyond Numbers
Organizers: Mina Aganagic (University of California, Berkeley), Denis Auroux (University of California, Berkeley), Jim Bryan (University of British Columbia), LEAD Andrei Okounkov (Columbia University), Balazs Szendroi (University of Oxford)Traditional enumerative geometry asks certain questions to which the expected answer is a number: for instance, the number of lines incident with two points in the plane (1, Euclid), or the number of twisted cubic curves on a quintic threefold (317 206 375). It has however been recognized for some time that the numerics is often just the tip of the iceberg: a deeper exploration reveals interesting geometric, topological, representation, or knottheoretic structures. This semesterlong program will be devoted to these hidden structures behind enumerative invariants, concentrating on the core fields where these questions start: algebraic and symplectic geometry.
Updated on Oct 12, 2015 03:39 PM PDT 
Hamiltonian systems, from topology to applications through analysis
Organizers: Rafael de la Llave (Georgia Institute of Technology), LEAD Albert Fathi (École Normale Supérieure de Lyon), Vadim Kaloshin (University of Maryland), Robert Littlejohn (University of California, Berkeley), Philip Morrison (University of Texas at Austin), Tere M. Seara (Universitat Politècnica de Catalunya), Serge Tabachnikov (Pennsylvania State University), Amie Wilkinson (University of Chicago)The interdisciplinary nature of Hamiltonian systems is deeply ingrained in its history. Therefore the program will bring together the communities of mathematicians with the community of practitioners, mainly engineers, physicists, and theoretical chemists who use Hamiltonian systems daily. The program will cover not only the mathematical aspects of Hamiltonian systems but also their applications, mainly in space mechanics, physics and chemistry.
The mathematical aspects comprise celestial mechanics, variational methods, relations with PDE, Arnold diffusion and computation. The applications concern celestial mechanics, astrodynamics, motion of satellites, plasma physics, accelerator physics, theoretical chemistry, and atomic physics.
The goal of the program is to bring to the forefront both the theoretical aspects and the applications, by making available for applications the latest theoretical developments, and also by nurturing the theoretical mathematical aspects with new problems that come from concrete problems of applications.
Updated on Jul 25, 2016 04:30 PM PDT 
Derived Algebraic Geometry
Organizers: Julie Bergner (University of California, Riverside), LEAD Bhargav Bhatt (University of Michigan), Dennis Gaitsgory (Harvard University), David Nadler (University of California, Berkeley), Nikita Rozenblyum (University of Chicago), Peter Scholze (Universität Bonn), Bertrand Toen (Centre National de la Recherche Scientifique (CNRS)), Gabriele Vezzosi (Università di Firenze)Derived algebraic geometry is an extension of algebraic geometry that provides a convenient framework for directly treating nongeneric geometric situations (such as nontransverse intersections in intersection theory), in lieu of the more traditional perturbative approaches (such as the “moving” lemma). This direct approach, in addition to being conceptually satisfying, has the distinct advantage of preserving the symmetries of the situation, which makes it much more applicable. In particular, in recent years, such techniques have found applications in diverse areas of mathematics, ranging from arithmetic geometry, mathematical physics, geometric representation theory, and homotopy theory. This semester long program will be dedicated to exploring these directions further, and finding new connections.
Updated on Mar 01, 2016 11:02 AM PST 
Birational Geometry and Moduli Spaces
Organizers: Antonella Grassi (University of Pennsylvania), LEAD Christopher Hacon (University of Utah), Sándor Kovács (University of Washington), Mircea Mustaţă (University of Michigan), Martin Olsson (University of California, Berkeley)Birational Geometry and Moduli Spaces are two important areas of Algebraic Geometry that have recently witnessed a flurry of activity and substantial progress on many fundamental open questions. In this program we aim to bring together key researchers in these and related areas to highlight the recent exciting progress and to explore future avenues of research.This program will focus on the following themes: Geometry and Derived Categories, Birational Algebraic Geometry, Moduli Spaces of Stable Varieties, Geometry in Characteristic p>0, and Applications of Algebraic Geometry: Elliptic Fibrations of CalabiYau Varieties in Geometry, Arithmetic and the Physics of String TheoryUpdated on Feb 29, 2016 02:50 PM PST

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