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Institute for Computational and Experimental Research in Mathematics, Brown University, Providence, RI (ICERM)

This workshop encompasses three major aspects of computation within Representation Theory and Algebraic Combinatorics. One concerns the development of efficient algorithms to compute important quantities in order to understand and classify them better. Such problems include structure constants and representation theoretic multiplicities, mutation invariants in cluster algebras, computing dimensions of coinvariant rings, characters of finite-dimensional representations, coefficients of Kazhdan-Lusztig polynomials, web bases etc. This is closely related to understanding what optimality we could expect and in particular the computational complexity aspects of those problems. Their computational complexity class can also be used to understand the existence of combinatorial interpretations, in particular for major structure constants lacking positive formulas like Kronecker and plethysm coefficients. On the other hand, representation theory has seen important applications within computational complexity theory, in the context of Geometric Complexity Theory and Quantum Information Theory.

ICERM/Brown University

Statistical models are almost never right. All models involve certain parametric and structural assumptions. Bayesian nonparametric inference (BNP) is an increasingly widely used approach to mitigate the dependence on such assumptions. After a rapid expansion of BNP research over the past 30 years, BNP is now a mature research area in statistics and machine learning. The current main challenges are related to computational hurdles and bottlenecks and the closely related need to tackle more complex and highly structured problems. This program bring together researchers working in BNP, including computation, foundations, methodology and application of BNP methods, with the goal of identifying newly emerging computational strategies and inference approaches. The program and invited talks are planned to balance theoretical expertise, interest and prowess in computational methods, and exposure to selected substantial application areas. The intended nature of the program as identifying synergies of different approaches and potentially new research directions naturally leads to favoring breath over depth, with more emphasis on covering diverse areas rather than on in-depth discussions of a single specific theme.

Tel.: [4018635030];     Email: info@icerm.brown.edu

Institute for Computational and Experimental Research in Mathematics

Spheres are the basic building blocks of geometry. More complicated geometric objects can be built by attaching spheres to each other along continuous maps. For many purposes, such constructions depend only on the homotopy classes of these continuous maps. A fundamental problem in algebraic topology is to compute these homotopy classes, i.e., to compute the homotopy groups of spheres. Machines can be used to great effect in the exhaustive computation of these fundamental invariants of homotopy theory. The workshop will study several software packages that are specifically designed for this purpose. Participants will have the opportunity to interact directly with codebases in work groups led by experienced programmers. The workshop will also introduce a variety of projects in homotopy theory that rely on computers.

Tel.: [4018635030];     Email: info@icerm.brown.edu

Institute for Computational and Experimental Research in Mathematics

3D reconstruction of scenes from image data is a classic problem in computer vision. Interest in this problem is at an all-time high, thanks to a wealth of applications which impact society in powerful ways---augmented reality, autonomous driving, cultural heritage, localization, mapping, and space exploration, to name a few. Some of these applications demand reliability of 3D reconstructions, e.g., for human safety, which translates to difficult and unsolved problems in mathematics. Other applications require efficiency at an ever-increasing scale, which calls for novelty in computational methods. This workshop will bring together experts working at an exciting crossroads where engineering, geometry, and optimization impact 3D reconstruction.

Tel.: [4018635030];     Email: info@icerm.brown.edu

Institute for Computational and Experimental Research in Mathematics

Higher category theory, also known as ∞-category theory, is now a fundamental area in modern mathematics, playing a crucial role in many areas of science, such as algebraic topology, algebraic geometry, mathematical physics, and theoretical computer science. Despite its importance, it is still known to be very abstract and technical, and hence difficult to learn without direct access to relevant experts. Formalization of mathematics is a modern approach that uses computers to precisely formulate mathematical statements and proofs via proof assistants. Initially formalization helped verify complicated mathematical results, such as the four color theorem or the liquid tensor experiment. However, in recent years proof assistants have also been used to teach mathematics and design exercises.

This workshop aims to teach participants the fundamentals of higher category theory using the proof assistant Rzk. The participants will learn both the classical point of view and the type theoretic point of view in two lecture series, and, in the exercise sessions, will learn how to use the proof assistant Rzk to prove basic higher categorical results.

Tel.: [4018635030];     Email: info@icerm.brown.edu