PROGRAM DESCRIPTION

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.

In recent years, economists and computer scientists have collaborated with mathematicians, operations research experts, and practitioners to improve the design and operations of real-world marketplaces. Such work relies on robust feedback between theory and practice, inspiring new mathematics closely linked – and directly applicable – to market and mechanism design questions. This cross-disciplinary program seeks to expand the domains in which existing market design solutions can be applied; address foundational questions regarding our ways of developing and evaluating mechanisms; and build useful analytic frameworks for applying theory to practical marketplace design.

A graphical representation of a Markov Chain fairness analysis of a political districting in North Carolina from Chin, Herschlag, Mattingly

This program aims to bring together researchers working at the interface of fairness and computation. This interface has been the site of intensive research effort in mechanism design, in research on partitioning problems related to political districting problems, and in research on ways to address issues of fairness and equity in the context of machine learning algorithms.

These areas each approach the relationship between mathematics and fairness from a distinct perspective. In mechanism design, algorithms are a tool to achieve outcomes with mathematical guarantees of various notions of fairness. In machine learning, we perceive failures of fairness as an undesirable side effect of learning approaches, and seek mathematical approaches to understand and mitigate these failures. And in partitioning problems like political districting, we often seek mathematical tools to evaluate the fairness of human decisions.

This program will explore progress in these areas while also providing a venue for overlapping perspectives. The topics workshop “Randomization, neutrality, and fairness” will explore the common role randomness and probability has played in these lines of work.

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. 

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. 

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 Erdos 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.