An introduction to the study of critical points of solutions of elliptic and parabolic equations

We give a survey at an introductory level of old and recent results in the study of critical points of solutions of elliptic and parabolic partial differential equations. To keep the presentation simple, we mainly consider four exemplary boundary value problems: the Dirichlet problem for the Laplace's equation; the torsional creep problem; the case of Dirichlet eigenfunctions for the Laplace's equation; the initial-boundary value problem for the heat equation. We shall mostly address three issues: the estimation of the local size of the critical set; the dependence of the number of critical points on the boundary values and the geometry of the domain; the location of critical points in the domain.Comment: 34 pages, 13 figures; a few slight changes and some references added; to appear in the special issue, in honor of G. Alessandrini's 60th birthday, of the Rendiconti dell'Istituto Matematico dell'Universit\`a di Triest

Exact and Asymptotic Degeneracies of Small Black Holes

We examine the recently proposed relations between black hole entropy and the topological string in the context of type II/heterotic string dual models. We consider the degeneracies of perturbative heterotic BPS states. In several examples with N=4 and N=2 supersymmetry, we show that the macroscopic degeneracy of small black holes agrees to all orders with the microscopic degeneracy, but misses non-perturbative corrections which are computable on the heterotic side. Using these examples we refine the previous proposals and comment on their domain of validity as well as on the relevance of helicity supertraces.Comment: 35pp. harvmac b-mode; v2 is substantially rewritten and includes new results; v3 contains further clarifications, and some new results; v3: final version to match published versio