## Programs & Events

##### Topological and Dynamical Analysis of Brain Connectomes

May 14 - 15, 2022

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.

##### Organizing Committee

- Dmitri Chklovskii
- David Lipshutz

##### Advances in Chern-Simons Classical and Quantum Gravity

May 6 - 8, 2022

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.

##### Organizing Committee

- Stephon Alexander
- Nicolas Yunes

##### Braids in Low-Dimensional Topology

Apr 25 - 29, 2022

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.*

##### Organizing Committee

- John Etnyre
- Matthew Hedden
- Keiko Kawamuro
- Joan Licata
- Vera Vertesi

##### Braids in Symplectic and Algebraic Geometry

Mar 21 - 25, 2022

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)

##### Organizing Committee

- Inanc Baykur
- Anand Deopurkar
- Benson Farb
- Ailsa Keating
- Anthony Licata

##### Braids in Representation Theory and Algebraic Combinatorics

Feb 14 - 18, 2022

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.

##### Organizing Committee

- Anna Beliakova
- Ben Elias
- Juan González-Meneses
- Anthony Licata

##### Research Community in Algebraic Combinatorics

Feb 10 - 11, 2022

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)

##### Organizing Committee

- Susanna Fishel
- Pamela E. Harris
- Rosa Orellana
- Stephanie van Willigenburg

##### Braids

Feb 1 - May 6, 2022

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)

##### Organizing Committee

- Marc Culler
- Ben Elias
- John Etnyre
- Benson Farb
- Juan González-Meneses
- Matthew Hedden
- Keiko Kawamuro
- Anthony Licata
- Joan Licata

##### Holistic Design of Time-Dependent PDE Discretizations

Jan 10 - 14, 2022

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 â€œnaive'â€ low-order methods. However, there are fundamental challenges to achieving truly high order and full... (more)

##### Organizing Committee

- David Ketcheson
- David Keyes
- Michael Minion
- Jingmei Qiu
- Benjamin Seibold
- Carol Woodward

##### Geometric and Topological Methods in Data Science

Dec 16 - 17, 2021

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.

##### Organizing Committee

- Ian Adelstein
- Jeffrey Brock
- Smita Krishnaswamy
- Bjorn Sandstede

##### Hamiltonian Methods and Asymptotic Dynamics

Dec 6 - 10, 2021

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.

##### Organizing Committee

- Alexandru Ionescu
- Yvan Martel
- Kenji Nakanishi
- Monica Visan

##### Foam Evaluation

Nov 5 - 7, 2021

The purpose of this workshop is to bring together mathematicians interested in foams and their use in low-dimensional topology, representation theory, categorification, mathematical physics, and combinatorics. The workshop will focus on the foam evaluation formula and its applications. More concretely, we aim to:

(a) Give a more intrinsic definition of the foam evaluation, in order, for instance, to find similar formulas for the other Lie types;

(b) Understand the interplay between foams and matrix factorizations and further use foams for a unified and comprehensive approach to Khovanov-Rozansky link homology theories;

(c) Compare combinatorial foam evaluation with the geometric structures and invariants coming from gauge theory and symplectic geometry;

(d) Study potential applications of the foamy definition of link homology theories.

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

##### Organizing Committee

- Mikhail Khovanov
- Aaron Lauda
- Louis-Hadrien Robert

##### Generic Behavior of Dispersive Solutions and Wave Turbulence

Oct 18 - 22, 2021

The large-time behavior of (generic) solutions of nonlinear dispersive equations set on bounded domains is almost completely open as far as rigorous analysis goes, and fairly mysterious, even from a less rigorous viewpoint. Under the assumption of weak nonlinearity, physicists and applied mathematicians have devised a theory to approach this question, known as weak turbulence, a branch of statistical physics. Weak turbulence theory predicts that the equation will enter a chaotic regime, where the exchange of energy in phase space is governed by the so-called kinetic wave equation. Justifying the derivation of the kinetic wave equation is a fascinating mathematical task, for which some results are already known, but whose solution will likely require input from nonlinear PDEs, but also probability theory. Intimately related questions are the question of Sobolev growth (how much can or does, the Sobolev norm of a nonlinear dispersive equation grow over time), as well as the analysis of... (more)

##### Organizing Committee

- Patrick Gerard
- Pierre Germain
- Alex Kiselev
- Andrea Nahmod