LOW REGULARITY PHYSICS AND GEOMETRY SEMINAR:

This seminar, which is part of the Thematic Program on Nonsmooth Riemannian and Lorentzian Geometry, meets in person at the Fields Institute in FL210 on Wednesday afternoons starting at 14h00 Toronto Time and is also open to watch online via zoom. Register here. Videos of prior talks are here.

To encourage collaboration between postdocs and senior participants, the speakers will be paired:

* a senior speaker presents an introductory talk

* a junior speaker presents a related talk

All talks will be related in some way to the geometry and physics of general relativity in settings where there is low regularity and singularities may occur.

ORGANIZERS:

Ghazal Geshnizjani, Robert McCann, Christina Sormani, and Mathias Braun

FALL 2022: Each talk will be 45 minutes with 5 minutes questions

14h00-14h50: a senior speaker presents an introductory talk

15h10-16h00: a junior speaker presents a related talk

FALL SCHEDULE:

September 7 Theo Sturm and Daniele Semola

14h00 Distribution-valued Ricci Bounds for Metric Measure Spaces

Theo Sturm (Uni Bonn)

Abstract: We will study metric measure spaces (X, d, m) beyond the scope of spaces with synthetic lower Ricci bounds. In particular, we introduce distribution- valued lower Ricci bounds BE_1(κ, ∞)

* 􏰀 for which we prove the equivalence with sharp gradient estimates,

* 􏰀 the class of which will be preserved under time changes with arbitrary ψ ∈ Lip_b(X), and

* which are satisfied for the Neumann Laplacian on arbitrary semi-convex subsets Y ⊂ X.

In the latter case, the distribution-valued Ricci bound will be given by the signed measure

κ = k m_Y + l σ_{∂Y}

where k denotes a variable synthetic lower bound for the Ricci curvature of X and l denotes a lower bound for the “curvature of the boundary” of Y, defined in purely metric terms.

15h10: Geometric Measure Theory on non smooth spaces with lower Ricci bounds

Daniele Semola (Oxford)

Abstract: There is a celebrated connection between Geometric Measure Theory and Ricci curvature in Geometric Analysis, often boiling down to the appearance of a Ricci term in the second variation formula for the area of codimension one hypersurfaces. In the seminar I will discuss some recent joint works with Andrea Mondino, Gioacchino Antonelli, Enrico Pasqualetto and Marco Pozzetta aimed at investigating the analogous connection on RCD spaces, i.e. infinitesimally hilbertian metric measure spaces verifying synthetic lower Ricci bounds and dimension upper bounds. The main focus will be on the ability of estimating first and second variation of the area for solutions of geometric variational problems as in the smooth case, without appealing to any classical regularity theory.

In the end of the talk I will describe how these new tools can be employed to study some questions of Geometric Analysis on smooth open manifolds with lower Ricci curvature bounds.

September 12-16 Workshop

September 21 Eric Woolgar and Sharmila Gunasekaran

14h00 On the topology of general cosmological models

Eric Woolgar (U Alberta)

Abstract: Is the Universe finite or infinite in spatial extent, and what `shape' does it have? These fundamental questions are typically studied within the context of the standard model of cosmology where the Universe is assumed to be homogeneous and isotropic and the spacetime metric is of FLRW form. But these questions can be addressed in much more general cosmological models, with the only assumption being that the average flow of matter is irrotational. To do so in general, without a precise form of the spacetime metric, we use (smooth) Bakry-\'Emery generalizations of the Bonnet-Myers theorem and the almost splitting theorem. Time permitting, I will also discuss the application of these tools to the topology of black hole horizons.

15h10 Decay of waves in gravitational solitons

Sharmila Gunasekaran (Fields Institute)

Abstract: Gravitational solitons are globally stationary, horizonless asymptotically flat spacetimes with positive energy. I will address the stability of a particular family of solitons at the simplest level by investigating solutions to the linear wave equation. I will describe a methodology to prove that massless scalar waves in this family of soliton spacetimes cannot decay faster than inverse logarithmically in time. This slow decay can be attributed to the stable trapping of null geodesics and is suggestive of instability at the nonlinear level. This is joint work with Hari Kunduri.

September 28 John Lott and Jan Sbierski

14h00 Some Rigorous Results about the Past and Future Behavior of Expanding Vacuum Spacetimes

John Lott (UC Berkeley)

Abstract: It's an old problem to understand the asymptotic geometry of an expanding vacuum spacetime as one approaches an initial singularity, or as one goes to the future. The best results assume some continuous symmetries. I will discuss recent work that does not involve any symmetry assumptions, but does have a reasonable scale-invariant curvature assumption. In particular, one goal is to characterize the existence of Kasner-like regions near an initial crushing singularity.

15h10 On the uniqueness problem of spacetime extensions

Jan Sbierski (U Edinburgh)

Abstract: This talk discusses the problem under what conditions two extensions of a Lorentzian manifold have to be the same at the boundary. After making precise the notion of two extensions agreeing at the boundary, we recall a classical example that shows that even under the assumption of analyticity of the extensions, uniqueness at the boundary is in general false — in stark contrast to the extension problem for functions on Euclidean space. We proceed by presenting a recent result that gives a necessary condition for two extensions with at least Lipschitz continuous metrics to agree at the boundary. Furthermore, we discuss the relation to a previous result by Chruściel and demonstrate a new non-uniqueness mechanism for extensions below Lipschitz regularity.

October 3-7 Workshop

October 12 Ruth Gregory and Melanie Graf

14h00 The Maths and Physics of Cosmic Strings

Ruth Gregory (King’s College London)

Abstract: I will review the basics about "Cosmic Strings" -- topological defects in the vacuum of some particle physics models. I will explain how the geometry around a string is either a conical deficit, or a highly nontrivial self-compactified manifold, depending on the nature of the broken symmetry of the vacuum. Time permitting, I will discuss how strings and black holes interact, and how the simple "cut a deficit out" prescription is not quite the whole story.

15h10 Low-regularity Lorentzian geometry (from an analytic perspective)

Melanie Graf

Abstract: An important mathematical subtlety – that is also highly physically relevant – permeating Mathematical General Relativity is the regularity one imposes on – or, from a perhaps more physical point of view, expects of – the metric, the manifold and/or chosen coordinates. In recent years, low regularity analytic methods have become increasingly important in Lorentzian geometry, allowing one to address several longstanding questions. In my talk I will present some of these developments and in particular focus on recent advances regarding the extension of the classical singularity theorems of General Relativity to non-C^2 Lorentzian metrics.

October 19 Nicola Gigli and Emanuele Caputo

14h00 Differentiating in a non-differentiable environment

Nicola Gigli (SISSA)

Abstract: We all know what the differential of a smooth map from R to R is. By looking at coordinates and then at charts, we also know what it is the differential of a smooth map between differentiable manifolds. With a little bit of work, we can also define a (weak) differential for Sobolev/BV maps in this setting (but the case of manifold-valued maps presents challenges already at this level). In this talk I will discuss how it is possible to differentiate maps between spaces that have no underlying differentiable structure at all. The concepts of Sobolev/BV maps in this setting will also be discussed.

15h10 Parallel transport on non-collapsed RCD(K,N) spaces

Emanuele Caputo (University of Jyväskylä)

Abstract: In this seminar, I address the construction of parallel transport in the setting of non-collapsed RCD(K,N) spaces, i.e. infinitesimally Hilbertian mms verifying a synthetic notion of lower bound on the Ricci curvature and upper bound on the dimension, equipped with the N-dimensional Hausdorff measure. In this generality, we cannot study parallel transport along a single curve, but along a well distributed family of flow lines, in a sense related to the notion of regular Lagrangian flows in the non smooth setting. After a brief introduction to the theory of RLF in the non smooth setting, I address how, in a joint work with N. Gigli and E. Pasqualetto, we obtained existence and uniqueness of parallel transport in this setting. To be more specific, we established a Leibniz formula granting uniqueness and I would like to emphasize the functional analytic tools developed for the proof of this formula.

October 26 Catherine Searle and Nicola Cavallucci

14h00 When is an Alexandrov space smoothable?

Catherine Searle (Wichita State University)

Abstract: In this talk, I will discuss the problem of when an Alexandrov space is smoothable. We will review the history of this question and discuss a new result that partially answers it. This is joint work in progress with Pedro Solorzano and Fred Wilhelm.

15h10 The L^2-completion of the space of Riemannian metrics is CAT(0)

Nicola Cavallucci

Abstract: We reprove in an easier way a result of Brian Clarke: the completion of the space of Riemannian metrics of a compact, orientable smooth manifold with respect to the L^2-distance is CAT(0). In particular we show that this completion is isometric to the space of L^2-maps from a standard probability space to a fixed CAT(0) space.

November 2 Hari Kunduri and Stefano Borghini

14h00 Constructing Ricci flat gravitational instantons

Hari Kunduri (Memorial University of Newfoundland)

Abstract: I will discuss uniqueness and existence theorems for four-dimensional, non-compact complete Ricci-flat manifolds with a torus symmetry. Natural asymptotic conditions for these spaces (referred to as `gravitational instantons' are asymptotically flat (S^1 X R^3 with the flat metric), asymptotically locally Euclidean (ALE) and asymptotically Taub-NUT. Solutions are characterised by data (rod structure) that encodes the fixed point sets of the torus action. Furthermore, we establish that for every admissible rod structure there exists an instanton that is smooth up to possible conical singularities at the axes of symmetry. This is in sharp contrast to the analogous problem in the Lorentzian setting (stationary and axisymmetric black hole solutions). I will also discuss generalisations to higher dimensions.

15h10 Comparison geometry for substatic manifolds

Stefano Borghini (Università Milano Bicocca)

Substatic manifolds arise naturally in General Relativity as spatial slices of static spacetimes

satisfying the Null Energy Condition. We will show that the substatic condition captures a large

class of interesting model solutions. We will then discuss the strong connection between the

substatic condition and the Bakry-Emery Ricci tensor. This will allow us to perform comparison

arguments leading to a Bishop-Gromov monotonicity and a splitting theorem for this class of

manifolds. This is a work in progress with Mattia Fogagnolo.

November 9 Iva Stavrov and Elefterios Soultanis

14h00 Representing relativistic objects as aggregates of point-sources

Iva Stavrov (Lewis and Clark)

In some ways calculus and Newtonian physics are reductive theories: they explain phenomena in terms of aggregates of (infinitesimal) units. On the other hand, even if one were to believe that Schwarzschild metrics somehow correspond to point-sources in General Relativity, the non-linearity of the Einstein's equations makes it hard to see that a concept of a point-source could lend itself to superposition. In this talk I present some surprising results regarding ``macroscopic” relativistic objects seen as aggregates of idealized relativistic point-sources.

15h10 Volume rigidity and filling minimality of convex bodies

Elefterios Soultanis (Radboud U)

Given a metric manifold Y, its minimal filling volume is defined as the infimum of volumes of manifolds whose boundary is Y. In general, the infimum need not be attained by a manifold but rather by an integral current space. In this talk I describe how convex bodies are the unique minimal fillings of their boundaries and how this relates to volume a property called "Lipschitz-volume rigidity".

November 16 Workshop

November 23 Vitali Kapovitch and Christian Ketterer

14h00 Submetries of manifolds

Vitali Kapovitch (U Toronto)

Abstract: Submetries are generalizations of Riemannian submersions and quotients by proper isometric group actions. They naturally arise in a number of areas such as collapsing with various curvature bounds, singular Riemannian foliations and various rigidity problems in geometry. We study geometric and topological structure of submetries of manifolds. I will also discuss possible applications to collapsing with lower curvature bounds. This is joint work with Alexander Lytchak.

15h10 Rigidity and stability results for mean convex subsets in $RCD$ spaces

Christian Ketterer

Abstract: I present splitting theorems for mean convex subsets in RCD spaces. This extends results for Riemannian manifolds with boundary by Kasue, Croke and Kleiner to a non-smooth setting. A corollary is a Frankel-type theorem. I also show that the notion of mean curvature bounded from below for the boundary of an open subset is stable w.r.t. to uniform convergence of the corresponding boundary distance function.

November 30 Shouhei Honda and Therese-Marie Landry

14h00 Open problems for RCD spaces

Shouhei Honda

Abstract: In this talk we provide several open problems for RCD spaces. Mainly we focus on (pointed) measured Gromov-Hausdorff convergence of RCD spaces, with potential applications.

15h10 Towards Analysis on Fractals: Piecewise $C^1$-Fractal Curves, Spectral Triples, and the Gromov-Hausdorff Propinquity

Therese-Marie Landry (Fields Institute)

Abstract: Many important physical processes can be described by differential equations. The solutions of such equations are often formulated in terms of operators on smooth manifolds. A natural question is to determine whether differential structures defined on fractals can be realized as a metric limit of differential structures on their approximating finite graphs. One of the fundamental tools of noncommutative geometry is Alain Connes’ spectral triple. Because spectral triples generalize differential structure, they open up promising avenues for extending analytic methods from mathematical physics to fractal spaces. The Gromov-Hausdorff distance is an important tool of Riemannian geometry and building on the earlier work of Marc Rieffel, Frederic Latremoliere introduced a generalization of the Gromov-Hausdorff distance that was recently extended to spectral triples. The class of piecewise $C^1$-fractal curves was first characterized by Michel Lapidus and Jonathan Sarhad as a generalized setting for the spectral triple construction developed by Christensen, Ivan, and Lapidus in the context of the Sierpinski gasket. We provide an analytic framework for the metric approximation of the Lapidus-Sarhad spectral triple on a piecewise $C^1$-fractal curve by spectral triples defined on an approximating sequence of finite graphs which exhibit properties motivated by the setting of the Sierpinski gasket.

December 7 Philippe LeFloch and Xingyu Zhu

14h00 Mathematical methods for self-gravitating fields: dispersion or collapse ?

Philippe G. LeFloch (Sorbonne University and CNRS )

The nonlinear stability of spacetimes satisfying Einstein equations is understood only under very special circumstances: small perturbations of ``known'' solutions or solutions enjoying certain symmetry restrictions. Gravitational attraction may lead to collapse or dispersion, and it is extremely challenging to cope with these phenomena mathematically. In this lecture, I will present a selection of recent advances on the global evolution of self-gravitating fields and on the theory of f(R) gravity. Blog: philippelefloch.org

15h10 Boundary of RCD spaces and RCD subspaces

Xingyu Zhu (Fields Institute)

Abstract: There are two Intrinsic notions of boundary for non-collapsed RCD (ncRCD in short) spaces and the regularity and topology of both notions are relatively well understood. On the other hand, there is also a notion of topological boundary when a ncRCD(K,N) space contains a ncRCD(K,N) subspace, which is extrinsic. In this talk, I will present a recent study on the relation between the extrinsic and intrinsic notions of boundary. For Alexandrov spaces, an invariance of domain theorem is employed to get some stronger results. This is joint work with Vitali Kapovitch.

December 14 Christina Sormani and Brian Allen

14h00 Intrinsic Flat Convergence of Spacetimes

Christina Sormani (Lehman College and CUNY Graduate Center)

Abstract: Intrinsic Flat Convergence is a notion of convergence of Riemannian manifolds which has been applied to study sequences of time symmetric asymptotically flat initial data sets whose ADM mass is converging to zero. It is useful in settings where there is no smooth convergence due to the existence of gravity wells and/or black holes. After quickly reviewing the above results, I will present joint work with Sakovich developing Spacetime Intrinsic Flat Convergence of Lorentzian manifolds using the null distance defined jointly with Vega and the Cosmological Time function of Andersson-Galloway-Howard.

15h10 Estimating the Intrinsic Flat Distance between Time Symmetric Spacetime Slices

Brian Allen (Lehman College, CUNY)

Abstract: What is the distance between two time symmetric, asymptotically flat spacetime slices satisfying the dominant energy condition? In the case where the spacetime slices have small mass it has been conjectured by Lee and Sormani that the distance should be small in the Intrinsic Flat sense. In this talk we will review progress on this conjecture and present practical tools which allow one to address the initial question in the case of Intrinsic Flat distance.

SUMMER 2022

We had the same format but different meeting times and longer talks:

2:00-3:00 pm: a senior speaker presents an introductory talk

3:30-4:30 pm: a junior speaker presents a related talk

SUMMER SCHEDULE:

July 6 Christina Sormani and Jikang Wang

2:00pm Welcome to Nonsmooth Riemannian and Lorentzian Geometry

Christina Sormani (CUNYGC and Lehman College)

Abstract: This talk will review some of the key ideas that went into the initiation of the Thematic Program on Nonsmooth Riemannian and Lorentzian Geometry. It will itemize reasons why physicists and geometers studying general relativity need to explore an approach which includes Riemannian and Lorentzian manifolds of lower regularity. To facilitate collaboration, the talk will survey various approaches to this challenging problem, emphasizing the work of participants in our program. See this page for links to more lectures on these topics.

3:30pm An Introduction to Gromov-Hausdorff Convergence and Low Regularity Riemannian Geometry

Jikang Wang (Fields Institute)

Abstract: In this talk, I will describe the definition of Gromov-Hausdorff (GH) convergence for metric spaces. Gromov's precompactness Theorem guarantees that any sequence of $n$-dim Riemannian manifolds with a Ricci curvature lower bound has a subsequence GH converging to a metric space, which we call a Ricci limit space. Then I will discuss Cheeger-Colding-Naber Theory about geometric structure of a Ricci limit space. Finally, I will show some topological results about Ricci limit spaces.

July 13 Ghazal Geshnizjani and Jerome Quintin

2:00pm Introduction to Cosmological Perturbation Theory

Ghazal Geshnizjani (Waterloo and Perimeter Institute)

Abstract: One of the first principles of standard cosmology is that the universe is to a very good approximation homogeneous and isotropic on large scales. This justifies the use of Friedmann–Lemaître–Robertson–Walker (FLRW) metric to describe the geometry of spacetime and different energy and matter contents of the universe as uniformly distributed perfect fluids. Using these ansatz the dynamics of background space-time can be obtained by Einstein theory of gravity. However, clearly the universe is not perfectly isotropic and homogeneous. There are large scale structures (LSS) such as galaxies and clusters of galaxies. In fact, maybe one the most significant accomplishments of the early universe cosmologists is the development of a semi-classical framework for early universe such as inflationary scenarios that can provide the mechanisms in which the initial conditions for LSS are generated from vacuum quantum fluctuations and through linear perturbations theory around FLRW metric. In this talk, I will review the cosmological perturbation theory and how the quantum fluctuations undergoing these scenarios can seed the generation of the large scale structures and provide connections to present observables.

3:30pm Stability of singularity-resolving effective theories of gravity

Jerome Quintin (Fields Institute)

Abstract: A fundamental question in physics is whether spacetime singularities really exist or whether they are mere limitations of our incomplete theories. To answer the question without a full understanding of quantum gravity, a modest approach is to try finding healthy semi-classical theories that could represent effective bottom-up descriptions of quantum gravity and which admit non-singular spacetime solutions. I will present a few proposals, but the focus will be on the Cuscuton, an incompressible scalar field that modifies general relativity dynamics without introducing new degrees of freedom. To connect with the previous talk by Prof. Ghazal Geshnizjani, I will then discuss how linear perturbation theory can be used to assess the stability and soundness of the theory. I will end with a few words about non-linear stability.

July 20 Robert McCann and Mathias Braun

2:00pm A Nonsmooth Approach to Einstein's Theory of Gravity

Robert McCann (U Toronto)

Abstract: The theory of metric measure spaces with upper dimension and lower Ricci curvature bounds has provided an extremely fruitful approach to nonsmooth Riemannian geometry in recent years. In this talk we explore an analogous framework for nonsmooth Lorentzian geometry, based on entropic convexity conditions along timelike geodesics of probability measures on spacetime.

Based in part on work with Clemens S\"amann (University of Vienna) [68][79] at http://www.math.toronto.edu/mccann/publications

3:30pm Timelike Curvature-Dimension Conditions for Lorentzian Spaces via Rényi's Entropy

Mathias Braun (U Toronto)

Abstract: We survey recently introduced timelike curvature-dimension conditions for zzmeasured Lorentzian spaces which, using optimal transport means, are formulated by convexity properties of the Rényi entropy along chronological geodesics of probability measures. These constitute Lorentzian analogues of the (reduced) CD condition of Sturm and Bacher-Sturm for metric measure spaces, and complement the recent entropic approach by Cavalletti-Mondino, who use the Boltzmann entropy, after Erbar-Kuwada-Sturm. We discuss basic properties of (and relations between) these conditions, such as compatibility with the smooth case, stability, equivalences, sharp geometric inequalities, etc. Finally, we outline various possible directions of future research about spaces with synthetic timelike Ricci curvature bounds.

July 27 Bo'az Klartag and Krzysztof Ciosmak

2:00pm Rigidity of Riemannian embeddings of discrete metric spaces

Bo'az Klartag (Weizmann Institute of Science)

Abstract: Let M be a complete, connected Riemannian surface and

suppose that S is a discrete subset of M. What can we learn about M

from the knowledge of all distances in the surface between pairs of

points of S? We prove that if the distances in S correspond to the

distances in a 2-dimensional lattice, or more generally in an

arbitrary net in R^2, then M is isometric to the Euclidean plane. We

thus find that Riemannian embeddings of certain discrete metric spaces

are rather rigid. A corollary is that a subset of Z^3 that strictly

contains a two-dimensional lattice cannot be isometrically embedded in

any complete Riemannian surface. This is a joint work with M. Eilat.

3:30pm Towards multi-dimensional localisation

Krzysztof Ciosmak (University of Oxford)

Abstract: Localisation is a powerful tool in proving and analysing various geometric inequalities, including isoperimertic inequality in the context of metric measure spaces. Its multi-dimensional generalisation is linked to optimal transport of vector measures and vector-valued Lipschitz maps. I shall present recent developments in this area: a partial affirmative answer to a conjecture of Klartag concerning partitions associated to Lipschitz maps on Euclidean space, and a negative answer to another conjecture of his concerning the mass-balance condition for absolutely continuous vector measures. During the course of the talk I shall also discuss an intriguing notion of ghost subspaces related to the above mentioned partitions.

August 3 Siyuan Lu and Cale Rankin

2:00pm Rigidity of Riemannian Penrose inequality with corners

Siyuan Lu (Rutgers University)

Abstract: In this talk, I will discuss rigidity of Riemannian Penrose inequality with corners. I will also discuss its applications in quasi-local mass and isometric embedding. This is based on joint works with P. Miao.

3:30pm Regularity and the A3w condition for Monge–Ampère type equations

Cale Rankin (Fields Institute and U Toronto)

Abstract: The potential functions in optimal transport are, in a suitably weak sense, solutions of a Monge–Ampère type equation. In this talk we discuss the regularity for these equations. We pay particular attention to a necessary condition for regularity, known as A3w, and its geometric interpretation. Then we discuss a more general class of Monge–Ampère type equations which arise from problems in geometric optics. We explain how the highly successful regularity theory for optimal transport has been extended to this new class of equations.

August 10 Latham Boyle and Clemens Sämann

2:00pm The Penrose tiling, self-similar quasicrystals, and fundamental physics

Latham Boyle (Perimeter Institute)

Abstract: I will begin with the Penrose tiling -- the most famous example of a self-similar quasi-periodic pattern. In addition to its beauty and mathematical interest, this pattern has a famous physical application to exotic materials called quasicrystals. But, in this talk, I will explain two new physical contexts in which such patterns appear:

(1) First, I will show that a regular tiling of hyperbolic space naturally decomposes into a sequence of self-similar quasicrystalline slices, with each slice related to the next by an invertible local "inflation/deflation" rule, so that the whole tiling may be reconstructed from a single slice. (In particular, the self-dual tiling of hyperbolic space by icosahedra essentially breaks into a sequence of Penrose tilings, as conjectured by Thurston.) I will discuss how this relates to recent efforts to formulate discrete versions of the holographic principle.

(2) Second, I will show how the symmetries of the remarkable lattice II_{9,1} (the even self-dual lattice in 9+1 dimensional Minkowski space) has several natural 3+1 dimensional quasicrystals living inside it. I will explain some (speculative) reasons to wonder if such a quasicrystal might provide a relevant model for our own 3+1 dimensional spacetime.

3:30pm Lorentzian length spaces - a synthetic approach to Lorentzian geometry

Clemens Sämann (Fields Institute and U Vienna)

Abstract: In this talk I will review a synthetic approach to Lorentzian geometry akin to Alexandrov- or CAT(k)-spaces in metric geometry. The goal is to have a notion of curvature (bounds) even if the underlying space is non-smooth or not a manifold. The talk will give some background for the talks of Robert McCann and Mathias Braun three weeks ago. Moreover, applications to General Relativity are given, in particular singularity theorems and (in)extendibility questions.

August 17 Erik Schnetter and Maxence Corman

2:00 pm Numerical General Relativity

Erik Schnetter (Perimeter Institute)

Abstract: I will describe general relativity from a numerical perspective. This will include formulations for an initial value problem, gauge conditions, constraints, boundary conditions, singularities, horizons, discrete stability, and related topics. The astrophysics and cosmology community which is using numerical methods to solve the Einstein equations has assembled a host of techniques that deserve to be presented to others (and their criticism and ideas).

3:30pm Black holes as a tool to study the dynamics of spacetime

Maxence Corman (Perimeter Institute)

Abstract: Black holes, through theoretical predictions or more recently observations of gravitational waves, have allowed us to gain an understanding of the strongly curved spacetime originally predicted by Einstein. Going hand in hand with this, there has been rapid development in both the theoretical and computational tools we use to study the dynamics of gravity in this strong field regime. In this talk, I will demonstrate that black holes are a unique probe of fundamental physics. I will illustrate this, by numerically solving for the dynamics of black holes in the early universe within the context of bouncing and inflationary models. I will also discuss some recent progress in modeling binary black hole mergers in modified theories of gravity, in the hope to perform model dependent tests of General Relativity.

August 24 David Garfinkle and Eric Ling

2:00pm Cosmic strings and distributions

David Garfinkle (Oakland)

Abstract: Cosmic strings are objects that occur in theories where the space of vacua is not simply connected. The thickness of these objects is typically tiny compared to their length. Thus one would like to treat these objects as distributions. This sort of treatment is straightforward in weak field (linearized) gravity. However (as this talk will explore) treating cosmic strings as distributions is problematic in general relativity.

3:30pm: Remarks on the cosmological constant appearing as an initial condition for Milne-like spacetimes

Eric Ling (Rutgers)

Abstract: Milne-like spacetimes are a class of k = -1 FLRW spacetimes which admit continuous extensions through the big bang. Under suitable assumptions on the scale factor, it was previously shown that the cosmological constant appears as an initial condition for Milne-like spacetimes; this yields a "quasi de Sitter" expansion for the early universe which could have applications to inflationary scenarios. In this talk, we generalize these results to spacetimes which share similar geometrical properties with Milne-like spacetimes but without any spatially isotropic or homogeneous assumptions.