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)