The 2020 Summer@ICERM program, held virtually due to the COVID-19 pandemic, involved 19 students from across the US in research projects investigating large-scale linear algebra, model reduction, randomized algorithms, and deep learning. Since the program, some students have begun successful technical careers in mathematics and computation, and some have matriculated in graduate school programs. This in-person reunion event, to be held from June 9-10, 2022 at ICERM, aims to rekindle professional relationships and possibly spark new directions for research.

Mathematical models arising from scientific applications frequently have a large number of degrees of freedom, and modern observational or empirical datasets have high-dimensional features. Such high-dimensional realities from either simulation or experimental data makes direct computational analysis, compression, and/or probing tasks such as outer-loop optimization, design, and/or uncertainty quantification computationally infeasible. One paradigm for addressing such a challenge is mathematics-based model reduction, which aims to find and exploit low-dimensional structure in high-dimensional models to generate a computationally efficient emulator, often with provable accuracy guarantees. A complementary class of approaches is found in low-rank approximation and statistics where data reduction techniques can efficiently explore and mine parsimonious summarizations of high-dimensional datasets. One major goal of the Spring 2020 program, and the foundational theme for this proposed... (more)

With the substantial recent progress in connectomics, the study of comprehensive maps of nervous systems, much more is known about the connectivity structure of brains. This has led to a multitude of new questions about the relationship between connectivity patterns, neural dynamics and brain function, many of which lead to new mathematical problems in graph theory and dynamics on graphs. The goal of this workshop is to bring together a broad range of researchers from neuroscience, physics, mathematics, and computer science to discuss new challenges in this emergent field and promote new collaborations.

This workshop is fully funded by a Simons Foundation Targeted Grant to Institutes.

This two-day interactive and research-oriented workshop brings together researchers and leaders at the interface of general relativity, quantum gravity, and mathematics with a focus on Chern-Simons Classical and Quantum Gravity. A main goal of the workshop is to find new synergies across sub-disciplines with an eye towards observational signatures.

Braids are deeply entwined with low-dimensional topology. Closed braids are knots and links, while viewing braid groups as surface mapping class groups connects the topic to fundamental constructions of three- and four-manifolds. The question of how properties of braids or mapping classes reflect the associated manifolds arises in Dehn surgery, link invariants, and contact and symplectic geometry. The workshop will highlight recent advances in these and other areas of low-dimensional topology where braids and mapping classes play a significant role. The workshop will also explore related algorithms, with an eye towards their (efficient) implementation.

Please note that this program is nearing capacity for in-person participation. Consider applying for this program virtually.

Incarnations of braid groups, or generalizations thereof, naturally arise in a range of active research areas in symplectic and algebraic geometry. This is a rich and diverse ecosystem, and the workshop will aim to bring together speakers from all corners of it. A unifying theme is monodromy: on the one hand, generalized braid groups arise in symplectic and algebraic geometry as fundamental groups of moduli spaces, loosely construed -- for instance, of complements of discriminant loci of singularities or of hyperplane arrangements, or moduli spaces of deformations of complex or symplectic structures. On the other hand, monodromy ideas motivate representations of generalized braid groups as various flavors of geometric automorphisms -- for instance, as (framed) mapping class group elements, symplectic Dehn twists, spherical twists in derived categories, or flop functors for 3-folds. These perspectives lead in turn to a wide array of further geometric applications, from classifications... (more)

Braid groups and their generalizations play a central role in a number of places in 21st-century mathematics. In modern representation theory, braid groups have come to play an important organizing role, somewhat analogous to the role played by Weyl groups in classical representation theory. Recent advances have established strong connections between homological algebra (t-structures and stability conditions), geometric representation theory (Hilbert schemes, the Hecke category, and link homologies), and algebraic combinatorics (shuffle algebras, symmetric functions, and also Garside theory). Braid groups appear prominently in many of these connections. The goal of this workshop will be to bring experts in these different areas together to both communicate recent advances and also to formulate important questions for future work.

The Women in Algebraic Combinatorics Research Community will bring together researchers at all stages of their careers in algebraic combinatorics, from both research and teaching-focused institutions, to work in groups of 4-6, each directed by a leading mathematician. The goals of this program are: to advance the frontiers of cutting-edge algebraic combinatorics, including through explicit computations and experimentation, and to strengthen the community of women working in algebraic combinatorics.

Successful applicants will be assigned to a group based on their research interests. The groups will work on open problems in algebraic combinatorics and closely related areas, including representation theory, special functions, and discrete geometry. Several of the proposed projects will extensively involve experimentation and computation, which will increase the likelihood that concrete progress is made over the course of the initial workshop and following 6 months, and provide useful... (more)

Braid groups were introduced by Emil Artin almost a century ago. Since then, braid groups, mapping class groups, and their generalizations have come to occupy a significant place in parts of both pure and applied mathematics. In the last 15 years, fields with an interest in braids have independently undergone rapid development; these fields include representation theory, low-dimensional topology, complex and symplectic geometry, and geometric group theory. Braid and mapping class groups are prominent players in current mathematics not only because these groups are rich objects of study in their own right, but also because they provide organizing structures for a variety of different areas. For example, in modern representation theory, important equivalences of categories are organized into 2-representations of braid groups, and these same 2-representations appear prominently in parts of geometry and mathematical physics concerned with mirror dualities; in low-dimensional topology,... (more)

The workshop aims to spur a holistic approach to the design of time-dependent PDE discretizations, particularly in terms of developing time integration techniques that are intertwined with spatial discretization techniques, focusing on: generalized ImEx methods, asymptotic-preserving and structure-preserving methods, methods that exploit low-rank dynamics, analysis of order reduction, parallel in time methods, and performant, maintainable, extensible software implementations.

Recent decades have seen increasing use of first-principles-based simulations via time-dependent partial differential equations (PDE), with applications in astrophysics, climate science, weather prediction, marine science, geosciences, life science research, defense, and more. Growing computational capabilities have augmented the importance of sophisticated high-order and adaptive methods over “naive'” low-order methods. However, there are fundamental challenges to achieving truly high order and full... (more)

The goal of this meeting is to bring together researchers using geometric and topological methods to study data. Fields of interest include manifold learning, topological data analysis, neural networks, and machine learning. While this plan is to focus on the mathematics, applications to neuroscience and quantitative biology will also be explored.

Recent progress in the analysis of dispersive PDE's has revealed various aspects of long-time dynamics or behavior of solutions, from the basic three types (scattering, blow-up, and solitons) to more complicated combinations, transitions, and oscillations among them, and so on. The goal of this workshop is for the participants to draw integrated landscapes of those diverse phenomena, aiming towards more a complete description, classification, and prediction of global dynamics, as well as new phenomena and methods.