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)

In spite of their omnipresence and importance, a number of questions about knots remain elusive. Addressing them solicits techniques from a range of mathematical disciplines at the interface of algebra, analysis, geometry, modeling, and low-dimensional topology. Some of the most exciting recent avenues of research include optimizing geometry, quantum knot invariants, and applications in material sciences, physics, and molecular biology.

This workshop emphasizes bridging the gap between theoretical, computational, and experimental approaches in knot theory and its applications, including artificial intelligence.

MSML2023 is the forth edition of a newly established conference, with emphasis on promoting the study of mathematical theory and algorithms of machine learning, as well as applications of machine learning in scientific computing and engineering disciplines. This conference aims to bring together the communities of machine learning, applied mathematics, and computational science and engineering, to exchange ideas and progress in the fast-growing field of scientific machine learning (SciML). The objective of this annual conference series is to promote the study of:

• Theory and algorithms of machine learning.
• Applications in scientific and engineering disciplines such as physics, chemistry, material sciences, fluid and solid mechanics, etc.
• To provide hands-on tutorials for students and new researchers in the field.

Previous MSML Conferences:
First MSML: (more)

The Summer@ICERM faculty advisers will present a variety of research projects on the combinatorial and graph theoretical properties of DNA self-assembly. By modeling nanostructures with discrete graphs, efficient DNA self-assembly becomes a mathematical puzzle. Faculty will also guide the development of computational tools which can be used to aid in answering fundamental questions that arise in this field.

Throughout the eight-week program, students will be introduced to the research topic with interactive lectures. Afterward, students will work on their projects in assigned groups of two to four, supervised by faculty advisors and aided by teaching assistants. Students will meet daily, give regular talks about their findings, attend mini-courses, guest talks, and professional development seminars, practice coding, and Tex typesetting, and will acquire skills in free software development. Students will learn how to collaborate mathematically, working closely in their teams to write... (more)

The field of mathematical and computational biology is rapidly growing. The most applicable computational models have been developed in collaboration between computational and life science researchers. This workshop aims to bring these groups together to facilitate and promote collaborations among them.

A mathematical model for one disease might also be useful in modeling another disease. Some researchers are working on theoretical mathematical & statistical problems related to biological and biomedical applications, while others are developing computational methodologies to address fundamental life science knowledge gaps.

This workshop fosters and features collaborations among these groups along with experimentalists and physicians. Theoreticians will be exposed to a variety of open biological questions in need of state-of-the-art and efficient mathematical methods. Computational scientists will learn about more robust and efficient methods that could be tailored to answer... (more)