## Programs & Events

##### Mathematical and Computational Approaches for the Einstein Field Equations with Matter Fields

Oct 26 - 30, 2020

This workshop will focus on theoretical and computational approaches to solving the Einstein field equations (the master equation of general relativity: a nonlinear, coupled, hyperbolic-elliptic PDE system) with (fluid) matter field sources, as typical of binary neutron stars and supernovae. Simulations of these systems are targets of interest to both LIGO and telescopes such as Hubble, Fermi, and CHANDRA. In this workshop, special attention will be given to the governing equations of relativistic (magneto- ) hydrodynamics and multi-scale, multi-physics modeling challenges.

##### Organizing Committee

- Stefanos Aretakis
- Manuela Campanelli
- Scott Field
- Jan Hesthaven
- Gaurav Khanna
- Luis Lehner
- Steven Liebling
- Jared Speck

##### Statistical Methods for the Detection, Classification, and Inference of Relativistic Objects

Nov 16 - 20, 2020

This workshop will focus on data analysis strategies for comparing model predictions to data. Special attention will be placed on comparing solutions to the Einstein field equations (as in workshops 2 and 3) with data collected from gravitational-wave or telescopes. The workshop will include (but will not be limited to) coverage of topics involving reduced-order models, surrogate models, machine learning, UQ, and Bayesian techniques.

##### Organizing Committee

- Sara Algeri
- Sarah Caudill
- Katerina Chatziioannou
- Alessandra Corsi
- Scott Field
- Jonathan Gair
- Jae-Hun Jung
- Gaurav Khanna

##### Introductory Workshop: Combinatorial Algebraic Geometry

Feb 1 - 5, 2021

This introductory workshop in combinatorial algebraic geometry is aimed at early career mathematicians and other mathematicians looking for an entry point into the field. The workshop will feature expository lectures on some of the basic objects of interest, together with "expert'' lectures discussing some current trends in the field. There will also be ample time for problem sessions and discussions.

##### Organizing Committee

- Anders Buch
- Melody Chan
- Thomas Lam
- Leonardo Mihalcea

##### Combinatorial Algebraic Geometry

Feb 1 - May 7, 2021

Combinatorial algebraic geometry comprises the parts of algebraic geometry where basic geometric phenomena can be described with combinatorial data, and where combinatorial methods are essential for further progress.

Research in combinatorial algebraic geometry utilizes combinatorial techniques to answer questions about geometry. Typical examples include predictions about singularities, construction of degenerations, and computation of geometric invariants such as Gromov-Witten invariants, Euler characteristics, the number of points in intersections, multiplicities, genera, and many more. The study of positivity properties of geometric invariants is one of the driving forces behind the interplay between geometry and combinatorics. Flag manifolds and Schubert calculus are particularly rich sources of invariants with positivity properties.

In the opposite direction, geometric methods provide powerful tools for studying combinatorial objects. For example, many deep properties of... (more)

##### Organizing Committee

- Anders Buch
- Melody Chan
- June Huh
- Thomas Lam
- Leonardo Mihalcea
- Sam Payne
- Lauren Williams

##### Sage/Oscar Days for Combinatorial Algebraic Geometry

Feb 15 - 19, 2021

This workshop will focus on the development of software to facilitate research in combinatorial algebraic geometry, based on the SAGE Mathematical Software System and the OSCAR Computer Algebra System. Special attention will be given to efficient computations with multi-variate polynomials, which is a critical part of many algorithms in combinatorial algebraic geometry. Aside from development of software, the workshop will feature a series of talks about polynomial computations, as well as introductory lectures about Sage and Oscar.

##### Organizing Committee

- Anders Buch
- Wolfram Decker
- Benjamin Hutz
- Michael Joswig
- Julian Rüth
- Anne Schilling

##### Geometry and Combinatorics from Root Systems

Mar 22 - 26, 2021

The purpose of the workshop is to bring together a diverse group of researchers working on combinatorial and geometric aspects related to spaces with symmetries. The workshop will cover problems arising from various flavors of Schubert Calculus and enumerative geometry on flag manifolds, and problems from geometric representation theory and combinatorial Hodge theory. The topics covered include the study of Littlewood-Richardson coefficients, quantum cohomology and quantum K theory of flag manifolds, Maulik-Okounkov stable envelopes and characteristic classes, conformal blocks, and combinatorics related to moduli spaces, Macdonald theory, and quiver polynomials, Soergel bimodules, Hodge theory of matroids. These are trends in a rapidly developing area, and our aim is to facilitate interactions among researchers who work on different problems but employ similar techniques, at the intersection of algebraic geometry, combinatorics, and representation theory.

##### Organizing Committee

- David Anderson
- Angela Gibney
- June Huh
- Thomas Lam
- Leonardo Mihalcea

##### Algebraic Geometry and Polyhedra

Apr 12 - 16, 2021

The workshop will revolve around the interplay between algebraic geometry and combinatorial structures such as graphs, polytopes, and polyhedral complexes. In particular, the workshop will foster dialogue among groups of researchers who use similar combinatorial geometric tools for different purposes within algebraic geometry and adjacent fields. The topics covered will include Newton-Okounkov bodies, Ehrhart theory, toric geometry, tropical geometry, matroids, and interactions with mirror symmetry.

##### Organizing Committee

- Federico Ardila
- Man-Wai Cheung
- Yoav Len
- Sam Payne
- Lauren Williams

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

Jul 26 - 30, 2021

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

##### D-modules, Group Actions, and Frobenius: Computing on Singularities

Aug 9 - 13, 2021

The adoption of D-module techniques has transformed the interface between commutative algebra and algebraic geometry over the last two decades. The discovery of interactions and parallels with the Frobenius morphism has been an impetus for many new results, include new invariants attached to singularities but also D- and F-module based algorithms for computing quantities that used to be unattainable.

Our goal for this workshop is to discuss computational aspects and new challenges in singularity theory, focusing on special varieties that arise from group actions, canonical maps, or universal constructions. By bringing together geometers, algebraists, and invariant theorists, we will address problems from multiple perspectives. These will include comparisons of composition chains for D- and F-modules, the impact of group actions on singularity invariants, and the structure of differential operators on singularities in varying characteristics.

##### Organizing Committee

- Christine Berkesch
- Linquan Ma
- Claudia Miller
- Claudiu Raicu
- Uli (Hans Ulrich) Walther

##### Hamiltonian Methods in Dispersive and Wave Evolution Equations

Sep 8 - Dec 10, 2021

Dispersive equations are ubiquitous in nature. They govern the motion of waves in plasmas, ferromagnets, and elastic bodies, the propagation of light in optical fibers and of water in canals. They are relevant from the ocean scale down to atom condensates. There has been much recent progress in different directions, in particular in the exploration of the phase space of solutions of semilinear equations, advances towards a soliton resolution conjecture, the study of asymptotic stability of physical systems, the theoretical and numerical study of weak turbulence and transfer of energy in systems out of equilibrium, the introduction of tools from probability and the recent incorporation of computer assisted proofs. This semester aims to bring together these new developments and to explore their possible interconnection.

Dispersive phenomena appear in physical situations, where some energy is conserved, and are naturally related to Hamiltonian systems. This semester proposes to explore... (more)

##### Organizing Committee

- Diego Cordoba
- Erwan Faou
- Patrick Gerard
- Pierre Germain
- Alexandru Ionescu
- Alex Kiselev
- Andrea Nahmod
- Kenji Nakanishi
- Benoit Pausader
- Themistoklis Sapsis
- Gigliola Staffilani

##### 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
- Joan Licata
- Anthony Licata