Second Minho Meeting on Mathematical Physics

Michael Bradley (Department of Physics, Umea University, Sweden)

The growth of density perturbations in Kantowski-Sachs cosmologies with cosmological constant

The growth of density perturbations in Kantowski-Sachs cosmologies with a cosmological constant is studied, using the 1+3 and 1+1+2 covariant formalisms. For each wave number we obtain a closed system for scalars formed from quantities that are zero on the background and hence are gauge invariant. The solutions to this system are then analyzed both analytically and numerically. In particular the effects of anisotropy and the behaviour close to bounces is considered. 

Simone Calogero (Department of Applied Mathematics, University of Granada, Spain)

Exponential Convergence to Equilibrium for Kinetic Fokker-Planck Equations on Riemannian Manifolds

In this talk I consider a class of linear kinetic Fokker-Planck equations with a non-trivial diffusion matrix and with periodic boundary conditions in the spatial variable. After formulating the problem in a geometric setting, I will prove that global regular solutions convergein time to equilibrium with exponential rate, provided the velocity field, the energy function and the diffusion matrix satisfy certain geometric inequalities. The result is proved by estimating the time derivative of a modified entropy functional, as recently proposed by Villani. For spatially homogeneous solutions the assumptions of the main theorem reduce to the curvature bound condition for the validity of logarithmic Sobolev inequalities discovered by Bakry and Emery.  

Jaume Carot (Department de Física, Universitat de les Illes Balears, Spain )

Some developments on the relativistic theory of elasticity  

The theory of elasticity is reviewed in the general relativistic setup and its potential physical applications are discussed, in particular the modeling of star interiors possessing elastic properties such as the ones expected in neutron stars. In this context, two geometrical configurations turn out to be of special relevance: spherically symmetric spacetimes and stationary and axially symmetric spacetimes. Both cases are discussed and specific examples are presented, including a static two-layer star ‘toy model’ consisting of an elastic core surrounded by a perfect fluid corresponding to the interior Schwarzschild solution matched to the vacuum Schwarzschild solution. The work here presented extends and generalizes the pioneering work by Magli and Kijowski, and complements, in a sense, that by Karlovini and Samuelsson in their interesting series of papers.  

Michele Cirafici  (Department of Mathematics, Instituto Superior Técnico, Universidade Técnica de Lisboa, Portugal)

Non-perturbative effects in Calabi-Yau compactifications

String theory compactified on Calabi-Yau manifolds is approximated by supergravity at low energies. I will discuss a general formalism to compute non-perturbative corrections, in the form of worldsheet or D brane instantons, to the low energy supergravity action. This formalism supports some recent conjectures on the quantum structure of the hypermultiplet moduli space.

Filipe Paccetti  (Department of Physics, University of Porto, Portugal)

Hermitian YM instantons on resolved Calabi-Yau cones

I shall discuss the construction of Hermitian Yang-Mills instantons over Calabi-Yau cones, as well as over their resolutions. In particular, in d complex dimensions, I shall present an infinite family of SU(d) instantons, parametrised by an integer k and a continuous modulus. A detailed study of their properties, including the computation of the instanton numbers is provided.

Then, I shall also explain how they can be used in the construction of heterotic non-Kahler solutions, including heterotic string duals of 4d N=1 SYM theories.