## Programs & Events

##### ONLY OFFERED VIRTUALLY: Free Resolutions and Representation Theory

Aug 3 - 7, 2020

The structure of free resolutions plays an important role in analyzing singularities of varieties of low codimension. Codimension 2 Cohen-Macaulay varieties (resp. codimension 3 Gorenstein varieties) come from rank conditions on an n x (n+1) matrix (resp. a skew-symmetric (2n+1) x (2n+1) matrix).

This workshop seeks to push such results to Cohen-Macaulay varieties of codimension 3 and Gorenstein varieties of codimension 4.

This problem turns out to be related to the classification of semi-simple Lie algebras. These new methods allow one to create a â€˜mapâ€™ of free resolutions of a given format. The calculations that arise are very demanding and require new computational methods involving both commutative algebra and representation theory.

The organizers have shared two sets of notes for attendees to review before the workshop. These are downloadable here:

##### Organizing Committee

- Lars Christensen
- Claudia Miller
- Steven Sam
- Jerzy Weyman

##### ON-LINE MODULES OFFERED: GirlsGetMath: Summer Math Camp for High Schoolers

Aug 10 - 14, 2020

GirlsGetMath@ICERM is a five-day non-residential mathematics program that is open to high schoolers, regardless of gender identity, who live in or near greater Rhode Island and who will be entering the 10th or 11th grade in the fall of 2020.

GirlsGetMath occurs in an encouraging environment that builds young students' confidence in math and science.

GirlsGetMath expands participants' understanding and knowledge of mathematics through computations and experimentations.

GirlsGetMath provides expert mathematical training and mentoring.

GirlsGetMath@ICERM encourages 20-25 high schoolers to explore topics such as cryptography, the mathematics of voting, image processing, prime numbers and factoring, and fractals.

The goals of the program are:

- to show young adults that the study of mathematics can be... (more)

##### ONLY OFFERED VIRTUALLY: Symmetry, Randomness, and Computations in Real Algebraic Geometry

Aug 24 - 28, 2020

Real algebraic (and semi-algebraic) geometry studies subsets of R^n defined by a finite number of polynomial equalities and inequalities. Such sets occur ubiquitously in practice both inside and outside of mathematics. While being easy to define, semi-algebraic sets can be complicated topologically, which restricts the application of many algorithms. In recent years, there has been progress in proving much stronger results â€“ both quantitative and algorithmic -- when the problem under consideration involves the invariance under some group action.

In this workshop, we plan to focus on two situations where this phenomenon happens.

The first one is the statistical study of the topology of random real algebraic varieties as well as semi-algebraic sets, where the polynomials defining these objects are picked from a distribution invariant under the action of a certain group (usually the orthogonal group) acting on the space of variables. The behavior of the set of zeros (or more... (more)

##### Organizing Committee

- Saugata Basu
- Antonio Lerario
- Annie Raymond
- Cordian Riener

##### ONLY OFFERED VIRTUALLY: Monodromy and Galois groups in enumerative geometry and applications

Aug 31 - Sep 2, 2020

Galois groups encode the internal structure of field extensions. Less well-known is that (families) of systems of polynomial equations or geometric problems also have Galois groups that encode the internal structure of the equations or geometric problems. During the 2018 Fall program at the ICERM on Nonlinear Algebra, different groups of researchers who were studying or using Galois groups in their work became more aware of their related interests. This common thread connects recent work in enumerative geometry, statistics, computer vision, number theory, and numerical nonlinear algebra. Further connections have subsequently been realized to arithmetic enhancements of intersection theory and to random real algebraic geometry. This workshop will bring representatives of these research groups together to deepen these interactions and chart new research goals.

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

##### Organizing Committee

- Alexander Esterov
- Jose Rodriguez
- Frank Sottile

##### Advances in Computational Relativity

Sep 9 - Dec 11, 2020

The Nobel-Prize-winning detection of gravitational waves from binary black hole systems in 2015 by the Laser Interferometer Gravitational-Wave Observatory (LIGO) and the LIGO Scientific Collaboration has opened a new window on the universe. In addition, the 2017 observation of both gravitational and electromagnetic waves emitted by a binary neutron star system marked a new era of multi-messenger astronomy. While these successes are a remarkable experimental feat, they also constitute a significant computational achievement due to the crucial role played by accurate numerical models of the astrophysical sources in gravitational-wave data analysis. As current detectors are upgraded and new detectors come online within an international network of observatories, accurate, efficient, and advanced computational methods will be indispensable for interpreting the diversity of gravitational wave signals. This semester program at ICERM will emphasize the fundamental mathematical and... (more)

##### Organizing Committee

- Stefanos Aretakis
- Douglas Arnold
- Manuela Campanelli
- Scott Field
- Jonathan Gair
- Jae-Hun Jung
- Gaurav Khanna
- Stephen Lau
- Steven Liebling
- Deirdre Shoemaker
- Jared Speck
- Saul Teukolsky

##### ONLY OFFERED VIRTUALLY: Advances and Challenges in Computational Relativity

Sep 14 - 18, 2020

This kick-off workshop will seek to provide an overview of both the state-of-the-art and open challenges drawing from multiple themes (theory, analysis of the equations, computation, and data analysis) within the broad context of Einstein’s general relativity theory. The workshop will also feature a code bootcamp on the last day. The bootcamp participants will be given both an overview of the key pieces of software used in the field as well as practical instructions on installing and running example cases.

##### Organizing Committee

- Douglas Arnold
- Scott Field
- Gaurav Khanna
- Deirdre Shoemaker
- Saul Teukolsky
- Niels Warburton
- Barry Wardell

##### ONLY OFFERED VIRTUALLY: Mathematical and Computational Approaches for Solving the Source- Free Einstein Field Equations

Oct 5 - 9, 2020

This workshop will focus on theoretical and computational approaches to solving the vacuum Einstein field equations (the master equation of general relativity: a nonlinear, coupled, hyperbolic-elliptic PDE system) without matter field sources. A particular important special case is the simulation of two merging black holes, which will be emphasized throughout the workshop. Gravitational wave solutions will be another important aspect of this workshop, and special attention will be given to modeling techniques for the computation of these waves. The topics covered in this workshop will be relevant to both LIGO and LISA scientific efforts.

##### Organizing Committee

- Stefanos Aretakis
- Scott Field
- Jan Hesthaven
- Jae-Hun Jung
- Gaurav Khanna
- Stephen Lau
- Steven Liebling
- Deirdre Shoemaker
- Jared Speck
- Helvi Witek

##### ONLY OFFERED VIRTUALLY: 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

##### ONLY OFFERED VIRTUALLY: 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