In this talk we study the metastability phenomena for a class of "long-range" Ising models. We will start with the one-dimensional case, where, under suitable general conditions, the configuration -1 is the only metastable state. Moreover, we illustrate the theory with two examples (exponentially and polynomially decaying interaction) and we show that the critical droplet can be macroscopic or mesoscopic, according to the value of the external magnetic field. Heuristics and preliminary results for the two-dimensional case will be briefly discussed as well.
We will describe also polarizations and groups of automophisms. Under certain mild assumption on the order R, we will be able to effectively compute the abelian varieties up to isomorphism and in the case r=1 we can also list all polarizations of a fixed degree (up to polarized isomorphisms) with automorphism groups and period matrix of the canonical lift.
Abstract: "Mirror symmetry is a powerful dualitybetween type II string compactifications on a pair of mirror dual Calabi-Yaumanifolds. Apart from its relevance in determining non-perturbative correctionsto the low energy effective action and in computing enumerative invariants ofCalabi-Yau manifolds, mirror symmetry also predicts an equivalence between theset of D-brane states appearing in such a pair of mirror dual Calabi-Yaucompactifications. In this talk we review some general aspects of mirrorsymmetry on the level of D-branes. We give evidence and illustrate with anexample that for particular classes of D-branes in Calabi-Yau 3-folds thecorresponding mirror branes in the mirror geometry are associated withhyperbolic 3-manifolds.”
two conical singularities of a flat surface. The proof of this formula involves
both ingredients from representation theory via the Bloch-Okounkov formalism
and from intersection theory on moduli spaces.