Image for theorem about 9 point on cubic curve, the special case of Cayley–Bacharach theorem.

Commutative algebra is, in its essence, the study of algebraic objects, such as rings and modules over them, arising from polynomials and integral numbers.     It has numerous connections to other fields of mathematics including algebraic geometry, algebraic number theory, algebraic topology and algebraic combinatorics. Commutative Algebra has witnessed a number of spectacular developments in recent years, including the resolution of long-standing problems, with new techniques and perspectives leading to an extraordinary transformation in the field. The main focus of the program will be on these developments. These include the recent solution of Hochster's direct summand conjecture in mixed characteristic that employs the theory of perfectoid spaces, a new approach to the Buchsbaum--Eisenbud--Horrocks conjecture on the Betti numbers of modules of finite length, recent progress on the study of Castelnuovo--Mumford regularity, the proof of Stillman's conjecture and ongoing work on its effectiveness, a novel strategy to Green's conjecture on the syzygies of canonical curves based on the study of Koszul modules and their generalizations, new developments in the study of various types of multiplicities, theoretical and computational aspects of Gröbner bases, and the implicitization problem for Rees algebras and its applications.

Optical illusion staircase

Derived categories of coherent sheaves on algebraic varieties were originally conceived as technical tools for studying cohomology, but have since become central objects in fields ranging from algebraic geometry to mathematical physics, symplectic geometry, and representation theory. Noncommutative algebraic geometry is based on the idea that any category sufficiently similar to the derived category of a variety should be regarded as (the derived category of) a “noncommutative algebraic variety”; examples include semiorthogonal components of derived categories, categories of matrix factorizations, and derived categories of noncommutative dg-algebras. This perspective has led to progress on old problems, as well as surprising connections between seemingly unrelated areas. In recent years there have been great advances in this domain, including new tools for constructing semiorthogonal decompositions and derived equivalences, progress on conjectures relating birational geometry and singularities to derived categories, constructions of moduli spaces from noncommutative varieties, and instances of homological mirror symmetry for noncommutative varieties. The goal of this program is to explore and expand upon these developments. 

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 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.

Random graphs and related random discrete structures lie at the forefront of applied probability and statistics, and are core topics across a wide range of scientific disciplines where mathematical ideas are used to model and understand real-world networks. At the same time, random graphs pose challenging mathematical and algorithmic problems that have attracted attention from probabilists and combinatorialists since at least 1960, following the pioneering work of Erdős and Renyi.

Around the turn of the millennium, as very large data sets became available, several applied disciplines started to realize that many real-world networks, even though they are from various origins, share fascinating features. In particular, many such networks are small worlds, meaning that graph distances in them are typically quite small, and they are scale-free, in the sense that the number of connections made by their elements is extremely heterogeneous. 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.

The area has seen huge growth in the twenty-first century and, particularly in recent years, there has been a steady stream of solutions to important longstanding problems and many powerful new methods have been introduced. These advances include improvements in absorption techniques which have facilitated the proof of the existence of designs and related objects, the breakthrough on the sunflower conjecture whose further development eventually led to the proof of the Kahn–Kalai conjecture in discrete probability and the discovery of interactions between spectral graph theory and the study of equiangular lines in discrete geometry. These and other groundbreaking advances will be the central theme of the semester program on “Extremal Combinatorics” at SLMath.

In this program, we will bring together experts as well as enthusiastic young researchers to learn from each other, to initiate and continue collaborations, to communicate recent work, and to further advance the field by making progress on fundamental open problems and developing further connections with other branches of mathematics. We trust that younger mathematicians will greatly contribute to the success of the program with their new ideas. It is our hope that this program will provide a unique opportunity for women and underrepresented groups to make outstanding contributions to the field and we strongly encourage their participation.

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.

Probably the most classical kinetic equation is the Boltzmann equation which describes the evolutions of the phase-space density function for a dilute gas under binary molecular collisions. Other well-known classical kinetic models include the Landau equation, Vlasov equation for plasmas or other systems, Fokker-Planck equations or kinetic formulations of various macroscopic or hyperbolic systems.

In recent years, the successes of kinetic theory gave rise to an rapidly increasing variety of mathematical models beyond physics to applications in life sciences, social sciences, economy. Even more recently fascinating connections between kinetic theory and some aspects of data science have emerged.

Kinetic theory has strong and fascinating interactions with a large variety of other fields, including statistical mechanics, stochastic processes, dynamical systems...

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.

<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.

A discrete subgroup Γ of a semisimple Lie group G may be studied from many different viewpoints. On the one hand, the quotient G/Γ is a homogenous space; through the lens of homogeneous dynamics one can study flows on these spaces, their orbit closures, measure classifications, counting and equidistributions. On the other hand, the group G acts on a plethora of different geometries, including Riemannian and non-Riemannian symmetric spaces, projective spaces and flag manifolds. In many cases, this induces interesting properly discontinuous actions of Γ which can be studied using geometric methods. A flexible class of such discrete subgroups is given by Anosov groups, introduced by Labourie in his study of Hitchin representations and now accepted as the natural higher rank analogue of convex cocompact subgroups of rank one Lie groups. In recent years, their study has made tremendous advances by drawing inspiration from classical Teichmuller theory, and the theory of Kleinian groups.

When G has rank one, there has already been a fruitful interaction between the two communities, resulting in important advances in understanding the dynamics of the frame flow and unipotent flow on the frame bundle of convex cocompact and geometrically finite hyperbolic manifolds. In turn this had important applications to Apollonian circle packings, Zaremba’s conjecture, expanders, affine sieve problems for thin groups, and related problems, as well as groundberaking work in Teichmuller dynamics.

Exciting applications of the interaction between homogeneous dynamics and Anosov representations have begun to emerge in recent years, suggesting that now is very promising time to bring together these two communities. The notion of a thin subgroup, inspired from number theory, is one of the many points of convergence. Other recent advances include the study of homogeneous
dynamics in the setting of Borel Anosov groups, relations between the Hausdorff dimension of limit sets of Anosov groups with counting problems, as well as applications of the thermodynamical formalism in the study of Anosov representations. A strong link between Anosov groups and Hilbert geometry recently opened the door to a very active study of dynamics in these geometries.
This recent work seems likely to be just the first fruit of the interaction between dynamics and geometry for discrete subgroups of semisimple Lie 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 ideas, objects and techniques that interact richly with each other, and generate implications for many parts of mathematics. Geometric structures, such as hyperbolic structures on 2- and 3-manifolds, interact with dynamical properties of flows and with analysis on parameter spaces such as the Teichmuller space of a surface or a foliation by surfaces. Combinatorial objects such as complexes of curves and their generalizations give us insight into the behavior of mapping class groups, which encapsulate the topological symmetries of a surface, as well as homeomorphism and diffeomorphism groups which blend topology and dynamics.

Seminal work of Thurston in the 1970’s brought many of these ideas together in new ways, and inspired multiple lines of work in the time since then, exploring different aspects of the relationship between geometry, topology, analysis and dynamics. Recent progress in these fields has taken each in new directions, suggesting that refocusing on their interactions will yield dividends towards progress on key problems within this central area and grow outwards towards its many connections with other areas of mathematics.

As examples of structural questions in the overlap of these areas: How do we classify Anosov and pseudo-Anosov flows on 3-manifolds up to orbit equivalence? Can we relate the dynamics of pseudo-Anosov flows on hyperbolic 3-manifolds to the geometry of the underlying 3-manifolds? Can we relate the leafwise Teichmuller theory of a foliation to geometric structures on the underlying 3-manifold? How well can we understand the subgroup structure of homeomorphism and diffeomorphism groups of surfaces? Can mapping class groups of infinite-type surfaces be harnessed to study dynamical questions?

The program will bring together experts in all these fields and younger researchers, who together can address these sorts of questions and open new areas for exploration.