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).

Geometric tomography is the area of Mathematics dealing with the retrieval of information about solid objects based on the size of their sections or projections, or other lower dimensional data. Results from this area often find real-world applications in science and engineering.

In recent years geometric tomography has seen a rapid period of growth due to many exciting developments in harmonic analysis. The goal of the present workshop is to bring together specialists in geometric tomography, harmonic analysis, and related areas to discuss important advances and share new ideas.

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)