mercoledi' 16-02-2005 (16:00) - Aula Magna Renling Jin (Charleston, U.S.A) : Colloquium Dipartimento di Matematica :Magic of Infinitely Large Integers

In a nonstandard model we have many integers, which are greater than all known positive integers in our standard world. Are these integers purely for a mind game or useful tools for understanding our standard world of numbers? In the talk, the speaker will present some of his results that illustrate how these infinitely large integers are used for obtaining new standard theorems in combinatorial number theory. For example, he recently derived a theorem on Freiman's inverse problems that if A+A is small, then A must have some arithmetic structure. Freiman's inverse problems have been popular research topics in combinatorial number theory since 1960s. The theorem characterizes the arithmetic structure of A when the size of A+A is 3 times the size of A plus b, where b can be, for example, any constant integer independent of the size of A. The best results before the speaker's work had been for b < -1. The audience will not be assumed to have prior knowledge of nonstandard models. Any people above undergraduate math level should be able to understand the questions and theorems without difficulty.

mercoledi' 16-02-2005 (18:00) - Sala delle Riunioni Antonio Siconolfi (Roma I) : Seminari di calcolo delle variazioni -Aubry set and applications

For a given Hamiltonian H(x,p) continuous and quasiconvex in the second argument, defined in RN *RN or on the cotangent bundle of a compact boundaryless manifold, we consider the equation H = c with c critical value, i.e. for which the equation admits locally Lipschitz-continuous a.e. subsolutions, but not strict subsolutions. We show that there is a subset of the state variable space, called Aubry set and denoted by \A, where the obstruction to the existence of such subsolution is concentrated. We give a metric characterization of \A, and we discuss its main properties. They are applied to a projection problem in a Banach space, to the study of the large-time behavior of subsolutions to a time-dependent Hamilton-Jacobi equation,and to construct a Lyapunov function for a perturbed dynamics, under suitable stability assumptions.

giovedi' 17-02-2005 (15:00) - Sala Seminari V. Kharlamov (Strasbourg)) : Logarithmic asymptotics of Welschinger and Gromov-Witten invariants

Argomento: Geometria

For toric Del-Pezzo surfaces, one shows the existence of real solutions for the problem of interpolating real points by real rational curves, and we establish the logarithmic equivalence between the number of real solutions and the number of complex ones. We will discuss also the logarithmic asymptotics of genus-0 Gromov-Witten invariants

Giulia Curciarello Segreteria Didattica tel: 050-2213219 e-mail curciare@dm.unipi.it

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