K3 Projective Models In Scrolls

The exposition studies projective models of K3 surfaces whose hyperplane sections are non-Clifford general curves. These models are contained in rational normal scrolls. The exposition supplements standard descriptions of models of general K3 surfaces in projective spaces of low dimension, and leads to a classification of K3 surfaces in projective spaces of dimension at most 10. The authors bring further the ideas in Saint-Donat's classical article from 1974, lifting results from canonical curves to K3 surfaces and incorporating much of the Brill-Noether theory of curves and theory of syzygies developed in the mean time.

Written to honor the enduring influence of William Fulton, these articles present substantial contributions to algebraic geometry.

The book deals with the random perturbation of PDEs which lack well-posedness, mainly because of their non-uniqueness, in some cases because of blow-up. The aim is to show that noise may restore uniqueness or prevent blow-up. This is not a general or easy-to-apply rule, and the theory presented in the book is in fact a series of examples with a few unifying ideas. The role of additive and bilinear multiplicative noise is described and a variety of examples are included, from abstract parabolic evolution equations with non-Lipschitz nonlinearities to particular fluid dynamic models, like the dyadic model, linear transport equations and motion of point vortices.

This work reflects sixteen hours of lectures delivered by the author at the 2009 St Flour summer school in probability. It provides a rapid introduction to a range of mathematical models that have their origins in theoretical population genetics. The models fall into two classes: forwards in time models for the evolution of frequencies of different genetic types in a population; and backwards in time (coalescent) models that trace out the genealogical relationships between individuals in a sample from the population. Some, like the classical Wright-Fisher model, date right back to the origins of the subject. Others, like the multiple merger coalescents or the spatial Lambda-Fleming-Viot process are much more recent. All share a rich mathematical structure. Biological terms are explained, the models are carefully motivated and tools for their study are presented systematically.

Granular matter displays a variety of peculiarities that distinguish it from other appearances studied in condensed matter physics and renders its overall mathematical modelling somewhat arduous. Prominent directions in the modelling granular flows are analyzed from various points of view. Foundational issues, numerical schemes and experimental results are discussed. The volume furnishes a rather complete overview of the current research trends in the mechanics of granular matter. Various chapters introduce the reader to different points of view and related techniques. New models describing granular bodies as complex bodies are presented. Results on the analysis of the inelastic Boltzmann equations are collected in different chapters. Gallavotti-Cohen symmetry is also discussed.

The articles in this volume are the outcome of the Impanga Conference on Algebraic Geometry in 2010 at the Banach Center in Bedlewo. The following spectrum of topics is covered: K3 surfaces and Enriques surfaces Prym varieties and their moduli invariants of singularities in birational geometry differential forms on singular spaces Minimal Model Program linear systems toric varieties Seshadri and packing constants equivariant cohomology Thom polynomials arithmetic questions The main purpose of the volume is to give comprehensive introductions to the above topics, starting from an elementary level and ending with a discussion of current research. The first four topics are represented by the notes from the mini courses held during the conference. In the articles, the reader will find classical results and methods, as well as modern ones. This book is addressed to researchers and graduate students in algebraic geometry, singularity theory, and algebraic topology. Most of the material in this volume has not yet appeared in book form.

The exposition studies projective models of K3 surfaces whose hyperplane sections are non-Clifford general curves. These models are contained in rational normal scrolls. The exposition supplements standard descriptions of models of general K3 surfaces in projective spaces of low dimension, and leads to a classification of K3 surfaces in projective spaces of dimension at most 10. The authors bring further the ideas in Saint-Donat's classical article from 1974, lifting results from canonical curves to K3 surfaces and incorporating much of the Brill-Noether theory of curves and theory of syzygies developed in the mean time.