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Microlocal Analysis

Microlocal analysis provides tools for the precise analysis of problems arising in areas such as partial differential equations or integral geometry by working in the phase space, i.e. the cotangent bundle, of the underlying manifold. It has origins in areas such as quantum mechanics and hyperbolic equations, in addition to the development of a general PDE theory, and has expanded tremendously over the last 40 years to the analysis of singular spaces, integral geometry, nonlinear equations, scattering theory… This program will bring together researchers from various parts of the field to facilitate the transfer of ideas, and will also provide a comprehensive introduction to the field for postdocs and graduate students.

Professor Gunther Uhlmann (Washington) has been appointed as a Clay Senior Scholar to participate in this program.

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Venue: MSRI

Holomorphic Differentials in Mathematics and Physics

Holomorphic differentials on Riemann surfaces have long held a distinguished place in low dimensional geometry, dynamics and representation theory. Recently it has become apparent that they constitute a common feature of several other highly active areas of current research in mathematics and also at the interface with physics. In some cases the areas themselves (such as stability conditions on Fukaya-type categories, links to quantum integrable systems, or the physically derived construction of so-called spectral networks) are new, while in others the novelty lies more in the role of the holomorphic differentials (for example in the study of billiards in polygons, special – Hitchin or higher Teichmuller – components of representation varieties, asymptotic properties of Higgs bundle moduli spaces, or in new interactions with algebraic geometry).

It is remarkable how widely scattered are the motivating questions in these areas, and how diverse are the backgrounds of the researchers pursuing them. Bringing together experts in this wide variety of fields to explore common interests and discover unexpected connections is the main goal of our program. Our program will be of interest to those working in many different elds, including low-dimensional dynamical systems (via the connection to billiards); differential geometry (Higgs bundles and related moduli spaces); and different types of theoretical physics (electron transport and supersymmetric quantum field theory).

Professor Anna Wienhard (Heidelberg) has been appointed as a Clay Senior Scholar to participate in this program.

Image: Some holomorphic differentials on a genus 2 surface, with close up views of singular points, courtesy Jian Jiang.

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Venue: MSRI

Quantum Field Theory and Manifold Invariants

Professor Michael Hopkins (Harvard) and Professor Gregory Moore (Rutgers) have been appointed as Clay Senior Scholars to participate in this Park City Math Institute – Institute for Advanced Study program.

Image:  Skyguy414 at English Wikipedia, Wikimedia Commons

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Venue: Park City, UT

Birational Geometry and Moduli Spaces

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 Calabi-Yau Varieties in Geometry, Arithmetic and the Physics of String Theory.

Professor Claire Voisin (Collège de France) has been appointed as a Clay Senior Scholar from January to May 2019 to participate in this program.

CMI Enhancement and Partnership Program

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Venue: MSRI

Derived Algebraic Geometry

Derived algebraic geometry is an extension of algebraic geometry that provides a convenient framework for directly treating non-generic geometric situations (such as non-transverse 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.

Professor Dennis Gaitsgory (Harvard) has been appointed as a Clay Senior Scholar from January to May 2019 to participate in this program.

CMI Enhancement and Partnership Program

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Venue: MSRI

Random and Arithmetic Structures in Topology

The use of dynamical invariants has long been a staple of geometry and topology, from rigidity theorems, to classification theorems, to the general study of lattices and of the mapping class group. More recently, random structures in topology and notions of probabilistic geometric convergence have played a critical role in testing the robustness of conjectures in the arithmetic setting. The program will focus on invariants in topology, geometry, and the dynamics of group actions linked to random constructions.

Professor Uri Bader (Weizmann) has been appointed as a Clay Senior Scholar to participate in this program.

CMI Enhancement and Partnership Program

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Venue: Online from MSRI

Decidability, Definability and Computability in Number Theory

This program is focused on the two-way interaction of logical ideas and techniques, such as definability from model theory and decidability from computability theory, with fundamental problems in number theory. These include analogues of Hilbert’s tenth problem, isolating properties of fields of algebraic numbers which relate to undecidability, decision problems around linear recurrence and algebraic differential equations, the relation of transcendence results and conjectures to decidability and decision problems, and some problems in anabelian geometry and field arithmetic. We are interested in this specific interface across a range of problems and so intend to build a semester which is both more topically focused and more mathematically broad than a typical MSRI program.

Professor François Loeser (Sorbonne) has been appointed as a Clay Senior Scholar to participate in this program.

CMI Enhancement and Partnership Program

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Venue: Online from MSRI

Higher Categories and Categorification

Though many of the ideas in higher category theory find their origins in homotopy theory — for instance as expressed by Grothendieck’s “homotopy hypothesis” — the subject today interacts with a broad spectrum of areas of mathematical research. Unforeseen descent, or local-to-global formulas, for familiar objects can be articulated in terms of higher invertible morphisms. Compatible associative deformations of a sequence of maps of spaces, or derived schemes, can putatively be represented by higher categories, as Koszul duality for E_n-algebras suggests. Higher categories offer unforeseen characterizing universal properties for familiar constructions such as K-theory. Manifold theory is natively connected to higher category theory and adjunction data, a connection that is most famously articulated by the recently proven Cobordism Hypothesis.

In parallel, the idea of “categorification” is playing an increasing role in algebraic geometry, representation theory, mathematical physics, and manifold theory, and higher categorical structures also appear in the very foundations of mathematics in the form of univalent foundations and homotopy type theory. A central mission of this semester will be to mitigate the exorbitantly high “cost of admission” for mathematicians in other areas of research who aim to apply higher categorical technology and to create opportunities for potent collaborations between mathematicians from these different fields and experts from within higher category theory.

Professor Peter Teichner (MPIM) has been appointed as a Clay Senior Scholar to participate in this program.

CMI Enhancement and Partnership Program

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Venue: MSRI

Quantum Symmetries

Symmetry, as formalized by group theory, is ubiquitous across mathematics and science. Classical examples include point groups in crystallography, Noether’s theorem relating differentiable symmetries and conserved quantities, and the classification of fundamental particles according to irreducible representations of the Poincaré group and the internal symmetry groups of the standard model. However, in some quantum settings, the notion of a group is no longer enough to capture all symmetries. Important motivating examples include Galois-like symmetries of von Neumann algebras, anyonic particles in condensed matter physics, and deformations of universal enveloping algebras. The language of tensor categories provides a unified framework to discuss these notions of quantum symmetry.

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

CMI Enhancement and Partnership Program

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Venue: MSRI

Thematic Program on Toric Topology and Polyhedral Products

Toric Topology was first identified 20 years ago and has developed rapidly, with remarkably varied input from cobordism and homotopy theory, algebraic and combinatorial geometry, commutative algebra, and symplectic geometry and integrable systems.

This program is designed to give the subject a transformational push forward by bringing together experts from the many areas  currently using toric spaces, moment-angle complexes and polyhedral products. The program will  harness existing connections and create new ones. It will provide an environment favorable  for stimulating significant progress in a range of mathematical disciplines through synergy, and will be a focal point for elucidating new directions that will guide research for the next five to ten years.

Professor Gunnar Carlsson (Stanford) and Professor Shmuel Weinberger (Chicago) will give the Clay Lecutures at this event.

CMI Enhancement and Partnership Program

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Venue: Fields Institute

Universality and Integrability in Random Matrix Theory and Interacting Particle Systems

The past decade has seen tremendous progress in understanding the behavior of large random matrices and interacting particle systems. Complementary methods have emerged to prove universality of these behaviors, as well as to probe their precise nature using integrable, or exactly solvable models. This program seeks to reinforce and expand the fruitful interaction at the interface of these areas, as well as to showcase some of the important developments and applications of the past decade.

Professor Alice Guionnet (ENS Lyon) and Professor Herbert Spohn (TU München) have been appointed as Clay Senior Scholars to participate in this program.

CMI Enhancement ad Partnership Program

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Venue: MSRI

Mathematical Problems in Fluid Dynamics

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 well-known Euler equations for inviscid fluids, and the Navier-Stokes 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.

Professor Jean-Marc Delort (Paris 13) has been appointed as a Clay Senior Scholar to participate in this program.

CMI Enhancement and Partnership Program

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Venue: MSRI, Berkeley, CA