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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.

Tél.: [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.

Tél.: [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.

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

Institute for Computational and Experimental Research in Mathematics (ICERM)

Complex analysis is a vibrant research area—with manifold applications—in which computer experimentation has consistently proved to be a powerful and exciting mode of discovery. This program combines computational and experimental approaches in complex analysis, aiming to promote new synergies with the more theoretical facets of this rich research domain. It will include cutting-edge techniques within the field of computational complex analysis, software package tutorials and demonstrations to engage participants in the use and development of new software. The program will connect to computational and experimental mathematics via a threefold approach: 1) A discussion of modern techniques within the field of computational complex analysis; 2) Software package tutorials and demonstrations; 3) The engagement of participants, particularly long-term visitors and junior researchers in the development and deployment of software packages. This approach will directly blend theory and computation. Three Semester Program Workshops will involve participants beyond the long-term core. “Analytical Techniques and Applications” will focus on selected key themes from diverse application areas within mathematics, science and engineering. “Computational Complex Analysis” will cover recent progress on classical and emerging computational methods, and software tools. Finally, “Frontiers and Future Perspectives” will survey the broad landscape of research in applied and computational complex analysis, with a view to identifying and highlighting new developments in the field and scoping the potential for future ones. These short-term workshops will be complemented by an opening event and a range of collaborative activities organized throughout the Semester Program.

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

Complex Analysis

Institute for Computational and Experimental Research in Mathematics (ICERM)

The ongoing development of machine learning as a research tool, the rapid expansion of formalization efforts in mathematics, and recent advances in computer algebra systems are leading to a major transition in the research methods used in mathematics. This transition will be of particular significance to number theory, a field which has long had an algorithmic focus and a tradition of tabulating data. These characteristics are both embodied in the L-functions and Modular Forms Database (LMFDB), an online catalog of mathematical objects associated with the Langlands program, which has become the premier online resource for researchers in this area and a rich source of experimental data. The semester program will bring together leading researchers in computational number theory to explore the capabilities of these new tools and apply them to the vast repository of data that has been accumulated in the LMFDB, and beyond, in the hope of shedding new light on old problems, and formulating new ones. Specific focus areas of the program will include the nascent use of machine learning in number theory; the development of computer algebra systems widely used in number theory, highlighting their capabilities and future plans, and comparing their speed, correctness, and utility; and various approaches to computing with L-functions. In addition, there will be numerous opportunities for collaboration throughout the semester beginning with the opening event.

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