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 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 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.

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)

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)