In recent years, the interaction between harmonic analysis and convex geometry has dramatically increased, which resulted in solutions to several long-standing problems. The program will bring together leading mathematicians in both areas, along with researchers working in related applied fields, for the first-ever long-term joint program.

The main directions of the program will include: the Fourier approach to Geometric Tomography, the study of geometric properties of solids based on information about their sections and projections, Volume and Duality, Bellman technique for extremal problems of harmonic analysis, and various types of convexity of solutions of corresponding Hamilton–Jacobi–Bellman equation, as well as numerical computations and computer-assisted proofs applied to the aforementioned problems. The computational part will cover theoretical aspects (optimal algorithms, and why they work) as well as more applied ones (implementation).

Probabilistic methods have long played an important role in various areas of geometry and analysis. Notable applications of probabilistic methods appear, for example, in geometric functional analysis, in harmonic analysis, and in discrete mathematics. Conversely, mathematical phenomena of fundamentally geometric and analytic origin, such as the concentration of measure phenomenon, play a central role in modern probability theory. Novel interactions between probability, geometry and analysis continue to drive important innovations in these fields.

The aim of this workshop is to bring together a diverse range of experts from probability, geometry, and analysis, in order to promote further dialogue between these fields and to catalyze the creation of new interactions.

Extremal problems in harmonic analysis recently acquired prominence in questions ranging from optimizers in Fourier restriction results to sharp geometric inequalities to sharp estimates of various singular operators of Calderón–Zygmund type. Sharp inequalities and their stability versions reveal new connections between harmonic analysis, geometric measure theory, additive combinatorics, and stochastic optimal control. There are many examples of sharp estimates by stochastic control approach and the use of special types of convexity and Monge–Ampére equation. There are interesting examples of using the computational tools in proving sharp geometric inequalities for martingales and on Hamming cube and for Fourier restriction inequalities.

Discrete optimization is a vibrant area of computational mathematics devoted to efficiently finding optimal solutions among a finite or countable set of possible feasible solutions.

A famous and classical example of a problem in discrete optimization is the traveling salesperson problem: For given cities and distances of traveling from one city to another, we seek to find the shortest route that visits each city once and returns to the starting city. Discrete optimization problems naturally arise in many kinds of applications including bioinformatics, telecommunications network design, airline scheduling, circuit design, and efficient resource allocation. The field also connects to a variety of areas in mathematics, computer science, and data analytics including approximation algorithms, convex and tropical geometry, number theory, real algebraic geometry, parameterized complexity theory, quantum computing, machine learning, and mathematical logic.

The semester program... (more)

Discrete optimization is a vibrant area of computational mathematics devoted to efficiently finding optimal solutions among a finite or countable set of possible feasible solutions.

This Boot-camp will be the opening event of the semester and it aims to attract young researchers to this topic.The four mini courses, presented by four speakers known for high-quality exposition, will cover various subjects such as new advances in approximation algorithms, mixed integer non-linear programming, algebraic techniques in optimization and applications to social sciences. The event provides a taste of the many methods and hot topics to be discussed during the semester. The event will also include a poster session to allow graduate students to present their work and other community building activities.

Discrete periodic Schrodinger operators describe the behavior of individual electrons in "ideal" crystals in the tight-binding model of solid state physics. Spectra of such operators have the usual band-gap structure, and the corresponding dispersion relations are algebraic varieties. In the 1990's Gieseker, Knorrer, and Trubowitz used toroidal compactifications to solve questions such as irreducibility of Bloch and Fermi varieties and the density of states, for a class of mono-atomic models. Their work showed that while spectral theory is focused on the real part of the Bloch variety, the study of complex singularities and compactifications is crucial for describing formation of bands and gaps.

After a gap of 30 years, spectral theory is again interacting with algebraic geometry. Recently, W. Liu gave an algebraic method to obtain more general proofs of irreducibility for Fermi surfaces, Kravaris used free resolutions to study density of states, and Kuchment and coauthors used... (more)

Mixed-Integer Linear Optimization has been an important topic in optimization theory and applications since the 1960s. As a mathematical subject, it is a rich combination of aspects of geometry, algebra, number theory, and combinatorics. The interplay between the mathematics, modeling, and algorithmics makes it a deep and fascinating subject of applied mathematics, which has had an enormous impact on real-world applications. But many physical systems have nonlinear aspects and further discrete design aspects. So we are naturally led to the paradigm of Mixed-Integer Non-Linear Optimization. But the mathematics and effective algorithmics of this subject are far more daunting than the linear case, and so there is a focus on broad sub-classes where results from the linear world can be lifted up. Furthermore, effective modeling techniques are much more subtle and are intertwined with state-of-the-art algorithmics and software which are rapidly evolving.

This workshop focuses on the latest... (more)

Combinatorial optimization is an active research field in mathematics, with an immense range of applications. This workshop will bring together researchers and leading experts interested in the mathematical foundations of combinatorial optimization algorithms to discuss new tools and methods, in particular regarding the use of algebraic, analytical, and geometric techniques. Special emphasis will be given on polyhedral methods, since they are at the core of several groundbreaking combinatorial optimization results developed in recent years.

The aim of this workshop is to discuss many exciting recent developments on the computational side of discrete optimization. The workshop has three main themes. The first theme is that of commercial and academic/open-source solvers that have allowed the solution of very large-scale problems, and of recent developments in exact solvers that have allowed for proofs of results in logic, knot theory, and combinatorics. The second theme is the interaction between optimization and machine learning: these two areas complement each other in several ways. The third theme is quantum computing and unconventional computing architectures: quantum computing has been used to tackle combinatorial optimization problems, and quantum algorithms exist for other related optimization problems such as linear and semidefinite relaxations.

This workshop will focus on the intersection of mathematics, statistics, machine learning, and computation, when viewed through the lens of optimal transport (OT). Mathematical topics will include low-dimensional models for OT, linearizations of OT, and the geometry of OT including gradient flows and gradient descent in the space of measures. Relevant statistical topics will include reliable and efficient estimation of OT plans in high dimensions, the role of regularization in computing OT distances and plans, with applications to robust statistics, uncertainty quantification, and overparameterized machine learning. Computation will be a recurring theme of the workshop, with emphasis on the development of fast algorithms and applications to computational biology, high energy physics, material science, spatio-temporal modeling, natural language processing, and image processing.

This workshop focuses on the interplay between dynamics, rigidity, and arithmetic in hyperbolic geometry and related areas. There have been many striking developments in recent years, particularly related to totally geodesic submanifolds in both finite and infinite volume hyperbolic and even complex hyperbolic manifolds.

One aim of this workshop is to expose young researchers to these breakthroughs providing them with the necessary background from dynamics, and geometry to allow them to appreciate some of these recent advances, and prepare them to make new original contributions. For this purpose, we will have minicourses on "Arithmeticity, Superrigidity and totally geodesic manifolds", and "Rigidity and geodesic planes in infinite volume hyperbolic manifolds". These courses will be preceded by an introductory minicourse on Hyperbolic geometry. We will also have a minicourse on "Understanding of geodesic planes in hyperbolic 3-manifolds via computations and visualization". In... (more)