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Mathematics Department Técnico Técnico

Mathematical Physics Seminar  RSS

Sessions

Past

02/04/2009, 16:00 — 17:00 — Room P3.10, Mathematics Building
, University of Oxford

Mason’s Theorem on solutions of the Einstein Field Equations

It is possible to integrate the vacuum Einstein equations in the particular case of algebraically-special metrics far enough to obtain a good picture of the resulting space-times. For this reason, there was for a long time a hope that explicit, asymptotically-flat vacuum solutions with gravitational radiation could be found among these solutions. Mason published an argument in 1998 to show that this hope was misplaced. I shall describe some joint work with Piotr Chrusciel which gives a rigorous proof of Mason’s Theorem, under some restrictions. Based on arXiv:0712.3846.

18/02/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building
Andreas Döring, Imperial College

Quantum states as measures on the spectral presheaf

States of a classical physical system are probability measures on the state space of the system, while in standard quantum theory, a state is a positive linear functional of norm 1 on the algebra of physical quantities. In the topos approach to quantum theory, the spectral presheaf serves as an analogue of the state space. I will show that states of a quantum system correspond bijectively to certain measures on the spectral presheaf. In the second half of the talk, which is meant to be more informal, I will present some further aspects that lead to open problems: the relation between pure states and measures and the question if all probabilistic aspects of quantum theory could be subsumed by the logic of a suitable topos, and the potential characterisation of entangled states via the corresponding measures.

29/02/2008, 11:30 — 12:30 — Room P3.10, Mathematics Building
, Instituto Superior Técnico

Spherically symmetric self-gravitating scalar field

We will give an overview of the paper by D. Christodoulou "The problem of a self-gravitating scalar field", Commun. Math. Phys., 105 (1986), 337-361.

29/02/2008, 10:00 — 11:00 — Room P3.10, Mathematics Building
, Universidade do Minho

Dynamics of Bianchi I elastic spacetimes

Understanding the asymptotic dynamics of the Universe is one of the main goals of theoretical cosmology. In the context of homogeneous cosmological models, where the Einstein equations of general relativity reduce to a system of ordinary differential equations, our knowledge on this problem has increased substantially over the years, thanks to the application of methods from the theory of (finite dimensional) dynamical systems. After reviewing some results on the dynamics of Bianchi I perfect fluid cosmological models, this talk will focus on the more complicated case of a space-time filled with an elastic material. In the latter case a new behavior for Bianchi type I solutions is found in the limit toward the initial (big-bang) singularity, namely the existence of oscillations between different type of vacuum solutions. This behavior is associated to energy condition violation as the singularity is approached.

22/11/2007, 15:00 — 16:00 — Room P4.35, Mathematics Building
, Instituto Superior Técnico

General Relativity VII

This will be the tenth of a series of seminars devoted to hyperbolic partial differential equations and general relativity. The objective is to understand the well-known papers by D. Christodoulou on the spherically symmetric gravitational collapse of a massless scalar field. No prior knowledge of hyperbolic partial differential equations or general relativity is assumed.

19/11/2007, 14:30 — 15:30 — Room P3.10, Mathematics Building
, Instituto Superior Técnico

General Relativity VI

This will be the ninth of a series of seminars devoted to hyperbolic partial differential equations and general relativity. The objective is to understand the well-known papers by D. Christodoulou on the spherically symmetric gravitational collapse of a massless scalar field. No prior knowledge of hyperbolic partial differential equations or general relativity is assumed.

15/11/2007, 15:00 — 16:00 — Room P4.35, Mathematics Building
, Instituto Superior Técnico

General Relativity V

This will be the eighth of a series of seminars devoted to hyperbolic partial differential equations and general relativity. The objective is to understand the well-known papers by D. Christodoulou on the spherically symmetric gravitational collapse of a massless scalar field. No prior knowledge of hyperbolic partial differential equations or general relativity is assumed.

12/11/2007, 15:30 — 16:30 — Room P3.10, Mathematics Building
, Instituto Superior Técnico

General Relativity IV

This will be the seventh of a series of seminars devoted to hyperbolic partial differential equations and general relativity. The objective is to understand the well-known papers by D. Christodoulou on the spherically symmetric gravitational collapse of a massless scalar field. No prior knowledge of hyperbolic partial differential equations or general relativity is assumed.

08/11/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
, Instituto Superior Técnico

General Relativity III

This will be the sixth of a series of seminars devoted to hyperbolic partial differential equations and general relativity. The objective is to understand the well-known papers by D. Christodoulou on the spherically symmetric gravitational collapse of a massless scalar field. No prior knowledge of hyperbolic partial differential equations or general relativity is assumed.

05/11/2007, 14:30 — 15:30 — Room P3.10, Mathematics Building
Jorge Drumond Silva, Instituto Superior Técnico

Hyperbolic Partial Differential Equations III

This will be the fifth of a series of seminars devoted to hyperbolic partial differential equations and general relativity. The objective is to understand the well-known papers by D. Christodoulou on the spherically symmetric gravitational collapse of a massless scalar field. No prior knowledge of hyperbolic partial differential equations or general relativity is assumed.

31/10/2007, 15:00 — 16:00 — Room P4.35, Mathematics Building
, Instituto Superior Técnico

General Relativity II

This will be the fourth of a series of seminars devoted to hyperbolic partial differential equations and general relativity. The objective is to understand the well-known papers by D. Christodoulou on the spherically symmetric gravitational collapse of a massless scalar field. No prior knowledge of hyperbolic partial differential equations or general relativity is assumed.

29/10/2007, 14:30 — 15:30 — Room P3.10, Mathematics Building
Jorge Drumond Silva, Instituto Superior Técnico

Hyperbolic Partial Differential Equations II

This will be the third of a series of seminars devoted to hyperbolic partial differential equations and general relativity. The objective is to understand the well-known papers by D. Christodoulou on the spherically symmetric gravitational collapse of a massless scalar field. No prior knowledge of hyperbolic partial differential equations or general relativity is assumed.

25/10/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
, Instituto Superior Técnico

General Relativity I

This will be the second of a series of seminars devoted to hyperbolic partial differential equations and general relativity. The objective is to understand the well-known papers by D. Christodoulou on the spherically symmetric gravitational collapse of a massless scalar field. No prior knowledge of hyperbolic partial differential equations or general relativity is assumed.

22/10/2007, 14:30 — 15:30 — Room P3.10, Mathematics Building
Jorge Drumond Silva, Instituto Superior Técnico

Hyperbolic Partial Differential Equations I

This will be the first of a series of seminars devoted to hyperbolic partial differential equations and general relativity. The objective is to understand the well-known papers by D. Christodoulou on the spherically symmetric gravitational collapse of a massless scalar field. No prior knowledge of hyperbolic partial differential equations or general relativity is assumed.

28/03/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
, University of Oxford

New results in black-hole uniqueness

I shall review the proofs of black-hole uniqueness for four-dimensional black holes, and describe how some recent work has closed off some of the remaining open questions (based on gr-qc/0512041, gr-qc/0512043, gr-qc/0611035).

21/03/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
, University of Oxford

Lanczos potentials and a definition of gravitational entropy for perturbed FLRW space-times

We give a prescription for constructing a Lanczos potential for a cosmological model which is a purely gravitational perturbation of a Friedmann-Lemaitre-Robertson- Walker (FLRW) space-time. For the radiation equation of state, we find the Lanczos potential explicitly via Fourier transform. As an application, we follow up a suggestion of Penrose and propose a definition of gravitational entropy for these cosmologies. With this definition, the gravitational entropy initially is finite if and only if the initial Weyl tensor is finite. (Based on gr-qc/0702057.)

22/06/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building
Paul Lescot, Universidade de Lisboa

Preliminaries to a possible construction of a unitarizing measure for the Virasoro algebra (after Kirillov and Malliavin)

The Virasoro algebra Vir is the (essentially unique) nontrivial central extension of the Lie algebra diff (S 1 ) of C --vector fields on the circle. We shall present KirillovŽs work, identifying the space Diff (S 1 )/S 1 with a space of univalent functions and constructing a family of representations of Vir . The existence of a unitarizing measure on for these representations is a difficult open problem ; we shall establish certain algebraic conditions that such a measure would have to satisfy.

09/03/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building
, Instituto Superior Técnico

Subfactor or RCFT realisation of modular invariant partition functions

Subfactor theory initiated by V. Jones is a natural framework to systematically classify modular invariant partition functions in RCFT, e.g. WZW $SU(n)$ level $k$ and holomorphic orbifolds models. With this formalism, the highest weights reps or primary fields are replaced by braided endomorphisms on a fixed von Neumann factor $N$, and recaptures the underlying unitary representation of the modular group $SL(2,Z)$ in RCFT. Then every subfactor $N \subset M$ yields a modular invariant (MI) with nice representations of the Verlinde fusion rules and graphs empirically obtained before. One major problem is to decide if we can construct a subfactor producing a given MI. Following joint work with D. Evans we plan to provide the basics of this framework together with recent results illustrated in the (possibly $H^3$-twisted) quantum $S_3$ and $Z_p$ finite groups as well as in simple currents. Finally we will yield a model for which no physical realisation has been found yet.

References

  1. A.Coste, T.Gannon, P.Ruelle, Finite group modular data. Nuc. Phys.B 581 (2000), 679--717, hep-th/0001158.
  2.  D.E.Evans, P.R.Pinto, Subfactor realisation of modular invariants. Commun. Math. Phys. 237 (2003), 309-363.

14/10/2004, 14:00 — 15:00 — Room P3.10, Mathematics Building
, Instituto Superior Técnico

Geometric Quantization, Complex Structures and the Coherent State Transform

We establish the equivalence of geometric quantizations for a natural one-parameter family of complex structures on the cotangent bundle T* K of a compact Lie group K. This equivalence arises as the unitary parallel transport of a Hermitian connection which is shown to be intimately related to the heat operator on K and to be equivalent to the coherent state transform for this group.

29/04/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building
, Instituto Superior Técnico

Soliton equations and the infinite Grassmannian

Following the book "Solitons" by Miwa, Jimbo and Date, this talk will be an introduction to the study of some partial differential equations such as the KdV and the KP hierarchies from a geometric point of view. We will describe the Lax form of these equations, their infinitely many commuting symmetries, the tau functions and their parametrization in terms of Sato's infinite Grassmannian.

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