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.

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.

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.

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. We welcome participation from physics, mathematics, statistics, and astrophysics, and the speaking schedule will reflect this diversity of scientific disciplines.

The workshop will also feature a series of hands-on, computational tutorials in the afternoon. We plan to hold tutorials on (i) the SXS gravitational waveform catalog which contains important simulation outputs from the numerical relativity code SpEC, (ii) the Einstein Toolkit software platform for numerically solving the Einstein equation, the relativistic (magneto-)hydrodynamics equations,... (more)

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)

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.

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 – 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)