Linear fields on anti-de Sitter spacetimes.
Claude Warnick, Imperial College London.
Spacetimes with negative cosmological constant are of interest both from a mathematical point of view, but also from a physical perspective in view of the conjectured AdS/CFT correspondence. A crucial feature of these spacetimes is their timelike null infinity, on which boundary conditions must be imposed. I will discuss several results in the theory of linear fields on anti-de Sitter backgrounds, including renormalisation and well-posedness, quasinormal modes and black hole stability.
Exact entropy of $1/4$-BPS black holes in $N=4$ supergravity and the mixed Rademacher expansion.
Valentin Reys, University of Milano-Bicocca.
In this talk, I will present some recent developments in computing the exact entropy of dyonic $1/4$-BPS black holes in four-dimensional $N=4$ supergravity theories originating from Type IIB string theory compactified on $K3 \times T_2$. The exact entropy is obtained in the Quantum Entropy Function formalism by means of supersymmetric localization techniques. The result can then be compared to the degeneracy of the brane/momentum system making up the black hole in the string theory picture. Such degeneracies are given by the Fourier coefficients of so-called mock Jacobi forms, a concept I will review. An exact formula for the coefficients can be obtained via a suitable generalization of the Hardy-Ramanujan-Rademacher circle method which takes into account the mock character of the counting functions. After presenting these results, I will outline some discrepancies (at sub-leading order in the charges) between the supergravity result for the exact entropy and the degeneracies of the brane/momentum system, and point to some aspects of the supergravity calculations which should be examined in more detail if one hopes to get a complete matching.
A geometria simpléctica e a matemática discreta estão fortemente interligadas devido à existência de acções hamiltonianas de toros. Estas acções estão associadas a uma aplicação (denominada aplicação momento) que transforma uma variedade simpléctica compacta num polítopo convexo. Nesta palestra vamos concentrar-nos numa classe de polítopos, definida por Batyrev no contexto de simetria-espelho e que tem atraído muita atenção recentemente: os polítopos reflexivos. Em particular, vamos ver como as famosas propriedades "12 e 24" em dimensão 2 e 3 podem ser generalizadas com a ajuda da geometria simpléctica.