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)

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 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)

The collection and analysis of large-scale population level and individual mobility and social mixing data raises fundamental ethical questions related to privacy, individual autonomy, consent, and the distribution of power in society. Balancing those concerns with the desires of public health researchers and policy makers to learn what they need from the data is a central challenge. Ethics is a fundamentally discursive discipline and useful guidance on any of the challenges mentioned above can only result from actively engaging with a variety of perspectives and openly discussing their implications for the design and implementation of the big data-driven methods and technologies used in public health research. At the same time, ethicists must gain substantive insight into the technical details of these means if they are to identify and discuss specific concerns, and provide targeted recommendations.

In this multidisciplinary workshop, we will brainstorm new ethical challenges... (more)

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.

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.

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.

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).

GirlsGetMath@ICERM is a five-day non-residential mathematics program that is open to high schoolers, regardless of gender, who live in or near greater Rhode Island and who will be entering the 10th or 11th grade in the fall of 2022.

GirlsGetMath occurs in an encouraging environment that builds young students' confidence in math and science.

GirlsGetMath expands participants' understanding and knowledge of mathematics through computations and experimentations.

GirlsGetMath provides expert mathematical training and mentoring.

GirlsGetMath@ICERM encourages 20-25 high schoolers to explore topics such as cryptography, the mathematics of voting, image processing, prime numbers and factoring, and fractals.

The goals of the program are:

• to show young adults that the study of mathematics can be exciting,... (more)

The Numerical Relativity community summer school aims to provide graduate students and postdocs with the tools needed to master and understand cutting-edge numerical relativity simulations. Our one-week lecture series is geared toward providing a comprehensive introduction to the numerical study of gravitational waves, relativistic fluids, and modified gravity. In addition, we hope to strengthen the ties between different code communities and plan hand-on sessions and hackathons to foster interaction amongst the participants. There will also be the opportunity for students and postdocs to interact with senior participants and lecturers of the ICERM workshop program.

As part of your application, please briefly answer the following questions in your personal statement:

1. How does your research involve numerical relativity, or how do you plan to involve numerical relativity in the future?
2. Do you have experience in computational physics and/or numerical relativity? Which,... (more)