## Programs & Events

##### 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

##### 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 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

##### 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: one 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 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.

##### Organizing Committee

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

##### 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.

##### Organizing Committee

- Dmitri Chklovskii
- David Lipshutz

##### Summer@ICERM 2022: Computational Combinatorics

Jun 13 - Aug 5, 2022

The faculty advisers will present a variety of research projects on the combinatorics of parking functions. This overarching theme will allow participants to study and analyze parking functions by leveraging computational techniques and theory. Faculty will also guide the development of open-source computational tools for analyzing parking functions and their statistics, with time devoted to creating a database of parking functions and their generalizations.

2022 Proposed Research Project Topics Invariant and Prime Parking Sequences Statistics on parking functions and their generalizations Connection to posets and graph theory, algebraic structures, and discrete geometry

##### Lean for the Curious Mathematician 2022

Jul 11 - 15, 2022

Interactive theorem proving software can check, manipulate, and generate proofs of mathematical statements, just as computer algebra software can manipulate numbers, polynomials, and matrices. Over the last few years, these systems have become highly sophisticated and have learnt a large amount of mathematics. One has to be open to the idea these systems will change the way mathematics is done, and how it is taught in universities.

At the ICERM workshop "Lean for the Curious Mathematician 2022", experts in the Lean theorem prover will explain how to do number theory, topology, geometry, analysis, and algebra in the Lean theorem prover. This will be accessible to mathematicians without a specific background in computer-proof systems. The material covered will range from undergraduate mathematics to modern research. Participants will be invited to begin formalizing mathematical objects from their own research.

##### Organizing Committee

- Jeremy Avigad
- Kevin Buzzard
- Johan Commelin
- Yury Kudryashov
- Heather Macbeth
- Scott Morrison

##### Harmonic Analysis and Convexity

Sep 7 - Dec 9, 2022

In recent years, the interaction between harmonic analysis and convex geometry has dramatically increased, which resulted in solutions to several long-standing problems. The program will bring together leading mathematicians in both areas, along with researchers working in related applied fields, for the first-ever long-term joint program.

The main directions of the program will include: the Fourier approach to Geometric Tomography, the study of geometric properties of solids based on information about their sections and projections, Volume and Duality, Bellman technique for extremal problems of harmonic analysis, and various types of convexity of solutions of corresponding Hamilton–Jacobi–Bellman equation, as well as numerical computations and computer-assisted proofs applied to the aforementioned problems. The computational part will cover theoretical aspects (optimal algorithms, and why they work) as well as more applied ones (implementation).

##### Organizing Committee

- Javier Gomez Serrano
- Irina Holmes
- Bo'az Klartag
- Alexander Koldobskiy
- Sergei Treil
- Alexander Volberg
- Artem Zvavitch