These are the informal notes of two seminars held at the Universita di Roma -La Sapienza-, and at the Scuola Normale Superiore in Pisa in spring and autumn 1997. We discuss in detail the content of the parts of a paper by Givental dealing with mirror symmetry for projective complete intersections.

These are transcripts of notes taken at (some of) the lectures given at the Mittag-Leffler Institute during the first semester of the year 1996/97 on Enumerative geometry and its interaction with theoretical physics. The first part of this collection consists of notes from talks on the basics of quantum cohomology; the second part treats more advanced topics in quantum cohomology; the third part consists of background material and related topics; an appendix gives a description of Kresch's C-program Farsta for quantum cohomology computations.These notes are meant as a series of snapshots of...

The main goal of these lectures is to get students acquainted with a glossary of nonlinear physics. I try to present the material of the lectures (delivered at SNS Pisa in April -- June 1999) in such a way that the students do not need to use additional literature during the first reading. The lectures focus on the following arguments: discussion of nonlinear problems in classical physics; the use of the multiple time scale expansion method to solve problems of nonlinear oscillations; description of nonlinear wave steepening and shock wave formation using the Lagrange variables; solution of...

In this thesis we deal with the models of subspace arrangements introduced by De Concini and Procesi. In particular we study their integer cohomology rings, which are torsion free Z-modules of which we find Z-bases. When the considered arrangement is the braid hyperplane arrangement, this leads to the study of the integer cohomology rings of the moduli spaces of n-pointed curves of genus 0 and of their Mumford-Deligne compactifications. We deal with the action of the symmetric group on the cohomology rings: we give explicit formulas for the associated generalized Poincare series, and provide...

Features proceedings that contain the contributions of some of the participants in the "intensive research period" held at the De Giorgi Research Center in Pisa, during the period May-June 2010.

Since 2001 the Scuola Normale Superiore di Pisa has organized the "Colloquio De Giorgi", a series of colloquium talks named after Ennio De Giorgi. The Colloquio is addressed to a general mathematical audience, and especially meant to attract graduate students and advanced undergraduate students. The lectures are intended to be not too technical, in fields of wide interest. They must provide an overview of the general topic, possibly in a historical perspective, together with a description of more recent progress. The idea of collecting the materials from these lectures and publishing them in...

In this thesis, we study the regularity of optimal transport maps and its applications to the semi-geostrophic system. The first two chapters survey the known theory, in particular there is a self-contained proof of Brenier' theorem on existence of optimal transport maps and of Caffarelli's Theorem on Holder continuity of optimal maps. In the third and fourth chapter we start investigating Sobolev regularity of optimal transport maps, while in Chapter 5 we show how the above mentioned results allows to prove the existence of Eulerian solution to the semi-geostrophic equation. In Chapter 6 we...

The aim of the book is to study some aspects of geometric evolutions, such as mean curvature flow and anisotropic mean curvature flow of hypersurfaces. We analyze the origin of such flows and their geometric and variational nature. Some of the most important aspects of mean curvature flow are described, such as the comparison principle and its use in the definition of suitable weak solutions. The anisotropic evolutions, which can be considered as a generalization of mean curvature flow, are studied from the view point of Finsler geometry. Concerning singular perturbations, we discuss the...