Sep 09, 2013
Monday
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09:15 AM - 09:30 AM
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Welcome
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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09:30 AM - 10:30 AM
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Spacetime Geometry: A Setting for General Relativity
Daniel Pollack (University of Washington)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- --
- Supplements
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10:30 AM - 11:00 AM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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The Einstein Field Equations: A PDE Perspective
Daniel Pollack (University of Washington)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- --
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
- --
- Supplements
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02:00 PM - 03:00 PM
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The conformal method of constructing Cauchy data for the Einstein equations
David Maxwell (University of Alaska)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- Initial data for the Cauchy problem in general relativity cannot be freely specified, but must solve an underdetermined system of PDEs known as the Einstein constraint equations. This two-part series of lectures discusses the conformal method and its cousins, which are the most prevalent techniques used for constructing solutions of the constraint equations. The talks will include a development of the conformal method, as well as a discussion of more recent results in the field and remaining open problems.
- Supplements
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Notes
334 KB application/pdf
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Notes
238 KB application/pdf
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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04:00 PM - 05:00 PM
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MSRI Evans Talk: On the topology of black holes and beyond.
Gregory Galloway (University of Miami)
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- Location
- Evans Hall
- Video
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- Abstract
- There is a widely held belief in physics that a true astrophysical black hole, formed from the gravitational collapse of some stellar object, can be described by a certain exact solution to the Einstein equations discovered by Kerr in the 60's. This belief is based largely on a series of powerful results which shows that the Kerr solution is the unique solution to the vacuum (source-free) Einstein equations with certain prescribed properties. A basic step in the proof is Hawking's theorem on the topology of black holes which asserts that, under physically natural conditions, the surface of a black hole (cross-section of the event horizon) must be topologically a 2-sphere.
Various developments in string theory have generated a great deal of interest in gravity in higher dimensions and, in particular, in higher dimensional black holes. The remarkable discovery of Emparan and Reall of a 4+1 dimensional vacuum black hole solution to the Einstein equations with nonspherical horizon topology raised the question as to what horizon topologies are allowable in higher dimensions.
In this talk we review Hawking's theorem on the topology of black holes in 3+1 dimensions and present a generalization of it to higher dimensions. The latter is a geometric result which places restrictions on the topology of black holes in higher dimensions. We shall also discuss recent work on the topology of space exterior to a black hole. This is closely connected to the Principle of Topological Censorship, which roughly asserts that the topology of the region outside of all black holes (and white holes) should be simple. All of the results to be discussed rely on the recently developed theory of marginally outer trapped surfaces, which are natural spacetime analogues of minimal surfaces in Riemannian geometry.
This talk is based primarily on joint work with Rick Schoen and with Michael Eichmair and Dan Pollack
- Supplements
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Sep 10, 2013
Tuesday
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09:30 AM - 10:30 AM
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Introduction to decay of fields outside black holes.
Pieter Blue (University of Edinburgh)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- --
- Supplements
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Notes
258 KB application/pdf
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10:30 AM - 11:00 AM
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Tea
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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--
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11:00 AM - 12:00 PM
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The conformal method of constructing Cauchy data for the Einstein equations
David Maxwell (University of Alaska)
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- Location
- SLMath: Eisenbud Auditorium
- Video
-
- Abstract
- nitial data for the Cauchy problem in general relativity cannot be freely specified, but must solve an underdetermined system of PDEs known as the Einstein constraint equations. This two-part series of lectures discusses the conformal method and its cousins, which are the most prevalent techniques used for constructing solutions of the constraint equations. The talks will include a development of the conformal method, as well as a discussion of more recent results in the field and remaining open problems.
- Supplements
-
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Notes
238 KB application/pdf
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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--
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02:00 PM - 03:00 PM
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The cosmic censorship conjectures
Mihalis Dafermos (Princeton University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- --
- Supplements
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Notes
179 KB application/pdf
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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03:30 PM - 04:30 PM
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Constructing localized solutions of the Einstein constraint equations
Richard Schoen (Stanford University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- There are obstructions, such as the postive mass theorem, which require that general solutions of the Einstein equations must have nonvanishing asymptotic terms at infinity, and so cannot be localized. The extent to which they can be localized in light of these obstructions leads to constructions of interesting solutions with very special asymptotic behavior. We will describe such results and how they are obtained in this talk
- Supplements
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Notes
177 KB application/pdf
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04:30 PM - 06:20 PM
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Reception
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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Sep 11, 2013
Wednesday
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09:30 AM - 10:30 AM
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Further topics in decay of fields outside black holes.
Pieter Blue (University of Edinburgh)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- --
- Supplements
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Notes
189 KB application/pdf
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10:30 AM - 11:00 AM
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Tea
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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--
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11:00 AM - 12:00 PM
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The cosmic censorship conjectures
Mihalis Dafermos (Princeton University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- --
- Supplements
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Notes
142 KB application/pdf
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Sep 12, 2013
Thursday
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09:30 AM - 10:30 AM
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Quasi-local mass in general relativity
Mu-Tao Wang (Columbia University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- One of the greatest accomplishments of the theory of general relativity in the past century is the proof of the positive mass/energy theorem for asymptotically flat space-times. However, the concept of mass/energy remains a challenging problem because of the lack of a quasi-local description. Most observable physical models are finitely extended spatial regions and measurement of mass/energy on such a region is essential in many fundamental issues. In my talks, I shall first survey several notions of quasi-local mass including the Hawking mass, the Bartnik mass, the Brown-York mass, and their applications. At the end, I shall describe a new proposal of quasi-local mass/energy and discuss its application in the invariant mass conjecture of general relativity.
- Supplements
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Notes
168 KB application/pdf
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Notes
168 KB application/pdf
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10:30 AM - 11:00 AM
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Tea
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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11:00 AM - 12:00 PM
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Null hypersurfaces in Lorentzian spacetimes
Lydia Bieri (University of Michigan)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- In the mathematical theory of general relativity, null hypersurfaces in Lorentzian space-times play a crucial role. In this talk, I introduce null geodesic vector fields, which are used to construct null hypersurfaces. Then the most important geometric features of such hypersurfaces are discussed, including the definitions of shear and torsion. We give an overview of the analysis of the latter. This will be used to explain gravitational radiation in the second talk. Results on structure and asymptotic behavior of null hypersurfaces yield insight into gravitational waves.
- Supplements
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Notes
179 KB application/pdf
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Density Theorems for the Einstein Constraint Equations
Lan-Hsuan Huang (University of Connecticut)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- We will start with a beautiful theorem of Corvino and Schoen which says that the asymptotically flat initial data sets with harmonic asymptotics are dense among general initial data sets. Then we will discuss several generalizations of the density theorem and their applications in the positive mass theorem and the constructions relating to the center of mass and angular momentum.
- Supplements
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Notes
155 KB application/pdf
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
- --
- Supplements
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03:30 PM - 04:30 PM
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An introduction to the Penrose inequality conjecture
Marc Mars (University of Salamanca)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- The Penrose inequality is a conjecture relating the total mass of asymptotically flat space-times satisfying appropriate curvature conditions and the area of certain closed, space-like and co-dimension-two surfaces describing black holes from a quasi-local perspective. This conjecture is motivated by physical properties of black holes and has been proven in a number of particular but very interesting cases, mainly concerning the so-called Riemannian Penrose inequality.
In this talk I will present the general inequality, explain in which sense it is supported by black hole physics and describe several known results in the Riemannian case, including recent results for graphs in Euclidean space where the proof becomes particularly simple. If time permits, an approach to the inequality in the general case will also be mentioned.
- Supplements
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Notes
359 KB application/pdf
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Sep 13, 2013
Friday
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09:30 AM - 10:30 AM
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Quasi-local mass in general relativity
Mu-Tao Wang (Columbia University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
-
- Abstract
- One of the greatest accomplishments of the theory of general relativity in the past century is the proof of the positive mass/energy theorem for asymptotically flat space-times. However, the concept of mass/energy remains a challenging problem because of the lack of a quasi-local description. Most observable physical models are finitely extended spatial regions and measurement of mass/energy on such a region is essential in many fundamental issues. In my talks, I shall first survey several notions of quasi-local mass including the Hawking mass, the Bartnik mass, the Brown-York mass, and their applications. At the end, I shall describe a new proposal of quasi-local mass/energy and discuss its application in the invariant mass conjecture of general relativity.
- Supplements
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Notes
170 KB application/pdf
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10:30 AM - 11:00 AM
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Tea
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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11:00 AM - 12:00 PM
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Gravitational radiation - a geometric-analytic approach
Lydia Bieri (University of Michigan)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- Gravitational waves are predicted by the general theory of relativity. Several experiments aim at detecting them in the near future. We can think of gravitational waves as fluctuations of the curvature of the space-time manifold. They propagate at the speed of light along null hypersurfaces. In this talk, I show how by geometric-analytic methods we can lay open the structures of the relevant manifolds, and in particular of the asymptotics of the null hypersurfaces. Thereby, I explain the geometric picture of gravitational radiation and discuss also a specific nonlinear phenomenon. Finally, we relate the mathematical findings to experiments.
- Supplements
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Notes
275 KB application/pdf
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Cosmology
Hans Ringström (Royal Institute of Technology (KTH))
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- The talk will contain an overview of mathematical problems arising in the study of the universe. Moreover, it will provide a description of various methods that have been used to address these problems, as well as results that have been obtained.
- Supplements
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Notes
138 KB application/pdf
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
- --
- Supplements
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