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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.

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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.

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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.

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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.

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Venue: MSRI, Berkeley, CA

Analytic and Geometric Aspects of Gauge Theory

The mathematics and physics around gauge theory have, since their first interaction in the mid 1970’s, prompted tremendous developments in both mathematics and physics.  Deep and fundamental tools in partial differential equations have been developed to provide rigorous foundations for the mathematical study of gauge theories.  This led to ongoing revolutions in the understanding of manifolds of dimensions 3 and 4 and presaged the development of symplectic topology.  Ideas from quantum field theory have provided deep insights into new directions and conjectures on the structure of gauge theories and suggested many potential applications.  The focus of this program will be those parts of gauge theory which hold promise for new applications to geometry and topology and require development of new analytic tools for their study.

Professor Tomasz Mrowka (MIT) has been appointed as a Clay Senior Scholar to participate in this program.

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Venue: MSRI

Floer Homotopy Theory

The development of Floer theory in its early years can be seen as a parallel to the emergence of algebraic topology in the first half of the 20th century, going from counting invariants to homology groups, and beyond that to the construction of algebraic structures on these homology groups and their underlying chain complexes.  In continuing work that started in the latter part of the 20th century, algebraic topologists and homotopy theorists have developed deep methods for refining these constructions, motivated in large part by the application of understanding the classification of manifolds. The goal of this program is to relate these developments to Floer theory with the dual aims of (i) making progress in understanding symplectic and low-dimensional topology, and (ii) providing a new set of geometrically motivated questions in homotopy theory. 

Professor Ivan Smith (Cambridge) has been appointed as a Clay Senior Scholar to participate in this program.

Image: Nathalie Wahl

CMI Enhancement and Partnership Program

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Venue: MSRI

Number Theory Informed by Computation

The IAS/Park City Mathematics Program (PCMI) is an outreach program of the Institute for Advanced Study (IAS).  Held in Park City, Utah, PCMI is an intensive three-week residential conference that includes several parallel sets of activities aimed at different groups of participants across the entire mathematics community.

Professor Hendrik Lenstra (Leiden) and Professor Maryna Viazovska (EPFL) have been appointed as Clay Senior Scholars to participate in this program.

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Venue: Park City Mathematics Institute

Thematic Program on Probability and PDEs

This research program will focus on two components of the interplay between differential equations and probability.  The first is how the introduction of randomness into differential equations affects their long-term behaviour.  The second component is how both probability and PDEs are used in mathematical models of group dynamics, and on the interplay between stochastic and PDE-based models.

Professor Kavita Ramanan (Brown) has been appointed as a Clay Senior Scholar to participate in this program.

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Venue: CRM, Université de Montréal