The study of tensor categories involves the interplay of representation theory, combinatorics, number theory, and low dimensional topology (from a string diagram calculation, describing the 3-dimensional bordism 2-category [arXiv:1411.0945]).

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.

Soap bubble: equilibrium solution of the mean curvature flow and constant curvature surface.

Geometry, PDE, and Relativity are subjects that have shown intriguing interactions in the past several decades, while simultaneously diverging, each with an ever growing number of branches. Recently, several major breakthroughs have been made in each of these fields using techniques and ideas from the others. 

This program is aimed at connecting various branches of Geometry, PDE, and Relativity and at enhancing collaborations across these disciplines and will include four main topics: Geometric Flows, Geometric problems in Mathematical Relativity, Global Riemannian Geometry, and Minimal Submanifolds. Specifically the program focuses on a central goal, which is to advance our knowledge toward Riemannian (sub)manifolds under geometric conditions, such as curvature lower bounds, by developing techniques in, for example, geometric flows and minimal submanifolds and further fostering new connections.

“Plateau’s Memory ” (by A. van der Net): A soap film with singularities

This program sits at the intersection between differential geometry and analysis but also connects to several other adjacent mathematical fields and to theoretical physics. Differential geometry aims to answer questions about very regular geometric objects (smooth Riemannian manifolds) using the tools of differential calculus. A fundamental object is the curvature tensor of a Riemannian metric: an algebraically complicated object that involves 2nd partial derivatives of the metric. Many questions in differential geometry can therefore be translated into questions about the existence or properties of the solutions of systems of (often) nonlinear partial differential equations (PDEs). The PDE systems that arise in geometry have historically stimulated the development of powerful new analytic methods. In most cases the nonlinearity of these systems makes ‘closed form’ expressions for a solution impossible: instead more abstract methods must be employed.

The Complementary Program has a limited number of memberships that are open to mathematicians whose interests are not closely related to the core programs; special consideration is given to mathematicians who are partners of an invited member of a core program. 

The minimum spanning tree of 100,000 uniformly random points. Colors encode graph distance from the root, which is red. Black points are those whose removal would disconnect at least 5% of the points from the rest.

This program is devoted to the study of the probabilistic and statistical properties of such networks. Central tools include graphon theory for dense graphs, local weak convergence for sparse graphs, and scaling limits for the critical behavior of graphs or stochastic processes on them. The program is aimed at pure and applied mathematicians interested in network problems.

Extremal combinatorics concerns itself with problems about how large or small a finite collection of objects can be while satisfying certain conditions. Questions of this type arise naturally across mathematics, so this area has close connections and interactions with a broad array of other fields, including number theory, group theory, model theory, probability, statistical physics, optimization, and theoretical computer science.

Top: Neutrino interactions and neutrino-atom interactions. Bottom: Collision of two "waves"

The focus of the proposed program is on so-called kinetic equations, describing the evolution of the of many-particle interacting systems. These models have the form of statistical flows, with their solutions being either a single or multiple point probability density functions or measures, supported in a space of attributes. The attributes are problem-dependent and can be molecular velocity, energy, opinion, wealth, and many others. The flow then predicts the evolution of the probability measure in time, position in space, and the interchanging of the particles' states by the transition probability.

The program will strive to give an overview of the novel mathematical tools used in kinetic theory through a broad range of classical and more recent applications.

Solution to the geometric stochastic heat equation on the sphere at a fixed time

The topic Singular Stochastic Partial Differential Equations (singular SPDE) has rapidly grown to be an active research area at the interface of Stochastic Analysis and PDEs on one hand, and Mathematical Physics on the other hand. During this decade we have witnessed a series of tremendous breakthroughs in the solution theories of SPDEs, universality problems, large-scale asymptotic behaviors of solutions, and foundational relations with quantum field theories and geometry. Many long-standing problems have been resolved via newly developed methods – notably the theories of regularity structures and paracontrolled distributions – and deep connections with other fields are quickly emerging.

It is a natural time to convene a large-scale semester program.

<p>This figure depicts dynamics of flows on convex cocompact hyperbolic 3-manifolds; where the girl is a traveller along a horocycle.</p>

This research program will bring together two intellectual communities that have made significant advances in the study of discrete subgroups of higher rank semisimple Lie groups: the homogeneous dynamics community and the community studying geometric structures and Anosov groups.

The stable and unstable foliations near a singular orbit of a pseudo- Anosov flow in 3 dimensions. Courtesy Michael Landry.

Low dimensional topology is a meeting place for many objects and ideas from diverse areas of mathematics, including foliation theory, geometry, and smooth and conformal dynamics.   For instance, many foliations on 3-manifolds admit transverse flows, connecting (local) leafwise homeomorphisms to flow dynamics and the mapping class groups of the leaves.  Leafwise conformal or hyperbolic structures can be approached through Teichmüller theory, and connect again to one-dimensional dynamics through "universal circles" organizing compactifications of all the leaves or of the flow space.  Many of these ideas originate in work of Thurston but in recent years have diverged and are ripe for reconnection.  

The program will bring together experts in all these fields together with younger researchers, who together can form new connections and open new areas for exploration.

Tropical surfaces. Images courtesy of Lars Allermann.

Tropical geometry can be viewed as a degenerate version of algebraic geometry,where the role of algebraic varieties is played by certain polyhedral complexes. As the degeneration process, called tropicalization, preserves many fundamental properties, tropical geometry provides important bridges and an exchange of methods between algebraic geometry, symplectic geometry and convex geometry; these links have been extremely fruitful and gave rise to remarkable results during the last 20 years. The main focus of the program will be on the most significant recent developments in tropical geometry and its applications. The following topics are particularly influential in the area and will be central in the program:

• real aspects of tropical geometry;
• tropical mirror symmetry and non-Archimedean geometry;
• tropical phenomena in symplectic geometry;
• matroids, combinatorial and algebraic aspects;
• tropical moduli spaces;
• tropical geometry and A1-homotopy theory.

Inverse problems (IP) arise in all fields of science and technology where a cause for an observed or desired effect is sought. In the last 50 years or so there has been substantial progress in the mathematical understanding of these problems but many questions remain open. The mathematics of these problems involves many areas in Mathematics including PDE, differential geometry, integral geometry, probability, statistics, complex analysis, numerical analysis, mathematical physics, data science, etc. Since the 2010 program at the then-Mathematical Sciences Research Institute (now Simons Laufer Mathematical Sciences Institute), there has been significant progress in inverse problems; many of the advances can be traced back to that program. However, there are still deep open questions remaining as well, some of which are discussed in this proposal. New research topics include the connection between IP and machine learning, IP for nonlinear equations, IP for nonlocal operators, and connections between statistics and IP.