This book provides a quick access to computational tools for algebraic geometry, the mathematical discipline which handles solution sets of polynomial equations. Originating from a number of intense one week schools taught by the authors, the text is designed so as to provide a step by step introduction which enables the reader to get started with his own computational experiments right away. The authors present the basic concepts and ideas in a compact way.

The relation between mathematics and physics has a long history, in which the role of number theory and of other more abstract parts of mathematics has recently become more prominent.

More than ten years after a first meeting in 1989 between number theorists and physicists at the Centre de Physique des Houches, a second 2-week event focused on the broader interface of number theory, geometry, and physics.

This book is the result of that exciting meeting, and collects, in 2 volumes, extended versions of the lecture courses, followed by shorter texts on special topics, of eminent mathematicians and physicists.

The present volume has three parts: Random matrices, Zeta functions, Dynamical systems.

The companion volume is subtitled: On Conformal Field Theories, Discrete Groups and Renormalization and will be published in 2006 (Springer, 3-540-30307-3).

Vector?eldsonmanifoldsplaymajorrolesinmathematicsandothersciences. In particular, the Poincar' e-Hopf index theorem and its geometric count- part,the Gauss-Bonnettheorem, giveriseto the theoryof Chernclasses,key invariants of manifolds in geometry and topology. One has often to face problems where the underlying space is no more a manifold but a singular variety. Thus it is natural to ask what is the "good" notionofindexofavector?eld,andofChernclasses,ifthespaceacquiress- gularities.Thequestionwasexploredbyseveralauthorswithvariousanswers, starting with the pioneering work of M.-H. Schwartz and R. MacPherson. We present these notions in the framework of the obstruction theory and the Chern-Weil theory. The interplay between these two methods is one of the main features of the monograph. Marseille Jean-Paul Brasselet Cuernavaca Jos' e Seade Tokyo Tatsuo Suwa September 2009 v Acknowledgements Parts of this monograph were written while the authors were staying at various institutions, such as Hokkaido University and Niigata University in Japan, CIRM, Universit' e de la Mediterran' ee and IML at Marseille, France, the Instituto de Matem' aticas of UNAM at Cuernavaca, Mexico, ICTP at Trieste, Italia, IMPA at Rio de Janeiro, and USP at S" ao Carlos in Brasil, to name a few, and we would like to thank them for their generous hospitality and support. Thanks are also due to people who helped us in many ways, in particular our co-authors of results quoted in the book: Marcelo Aguilar, Wolfgang Ebeling, Xavier G' omez-Mont, Sabir Gusein-Zade, LeDung " Tran ' g, Daniel Lehmann, David Massey, A.J. Parameswaran, Marcio Soares, Mihai Tibar, Alberto Verjovsky,andmanyother colleagueswho helped usin variousways.

A group of Gerry Schwarz's colleagues and collaborators gathered at the Fields Institute in Toronto for a mathematical festschrift in honor of his 60th birthday. This volume is an outgrowth of that event, covering the wide range of mathematics to which Gerry Schwarz has either made fundamental contributions or stimulated others to pursue. The articles are a sampling of modern day algebraic geometry with associated group actions from its leading experts, with a particular focus on characteristic 0 and modular invariant theory.

Contributors:

M. Brion

A. Broer

D. Daigle

J. Elmer

P. Fleischmann

G. Freudenberg

D. Greb

P. Heinzner

A. Helminck

B. Kostant

H. Kraft

R. J. Shank

W. Traves

N. R. Wallach

D. Wehlau

An original motivation for algebraic geometry was to understand curves and surfaces in three dimensions. Recent theoretical and technological advances in areas such as robotics, computer vision, computer-aided geometric design and molecular biology, together with the increased availability of computational resources, have brought these original questions once more into the forefront of research. One particular challenge is to combine applicable methods from algebraic geometry with proven techniques from piecewise-linear computational geometry (such as Voronoi diagrams and hyperplane arrangements) to develop tools for treating curved objects. These research efforts may be summarized under the term nonlinear computational geometry. This volume grew out of an IMA workshop on Nonlinear Computational Geometry in May/June 2007 (organized by I.Z. Emiris, R. Goldman, F. Sottile, T. Theobald) which gathered leading experts in this emerging field. The research and expository articles in the volume are intended to provide an overview of nonlinear computational geometry. Since the topic involves computational geometry, algebraic geometry, and geometric modeling, the volume has contributions from all of these areas. By addressing a broad range of issues from purely theoretical and algorithmic problems, to implementation and practical applications this volume conveys the spirit of the IMA workshop.

"The book is largely self-contained...There is a nice introduction to symplectic geometry and a charming exposition of equivariant K-theory. Both are enlivened by examples related to groups...An attractive feature is the attempt to convey some informal wisdom rather than only the precise definitions. As a number of results are] due to the authors, one finds some of the original excitement. This is the only available introduction to geometric representation theory...it has already proved successful in introducing a new generation to the subject." (Bulletin of the AMS)

Calabi-Yau manifolds have been an object of extensive research during the last two decades. One of the reasons is the importance of Calabi-Yau 3-manifolds in modern physics - notably string theory. An interesting class of Calabi-Yau manifolds is given by those with complex multiplication (CM). Calabi-Yau manifolds with CM are also of interest in theoretical physics, e. g. in connection with mirror symmetry and black hole attractors. It is the main aim of this book to construct families of Calabi-Yau 3-manifolds with dense sets of ?bers with complex multiplication. Most - amples in this book are constructed using families of curves with dense sets of ?bers with CM. The contents of this book can roughly be divided into two parts. The ?rst six chapters deal with families of curves with dense sets of CM ?bers and introduce the necessary theoretical background. This includes among other things several aspects of Hodge theory and Shimura varieties. Using the ?rst part, families of Calabi-Yau 3-manifolds with dense sets of ?bers withCM are constructed in the remaining ?ve chapters. In the appendix one ?nds examples of Calabi-Yau 3-manifolds with complex mul- plication which are not necessarily ?bers of a family with a dense set ofCM ?bers. The author hopes to have succeeded in writing a readable book that can also be used by non-specialists.

These notes present a polished introduction to tropical geometry and contain some applications of this rapidly developing and attractive subject. It consists of three chapters which complete each other and give a possibility for non-specialists to make the first steps in the subject which is not yet well represented in the literature. The notes are based on a seminar at the Mathematical Research Center in Oberwolfach in October 2004. The intended audience is graduate, post-graduate, and Ph.D. students as well as established researchers in mathematics.

The aim of this monograph is to give an overview of various classes of in?ni- dimensional Lie groups and their applications, mostly in Hamiltonian - chanics, ?uid dynamics, integrable systems, and complex geometry. We have chosen to present the unifying ideas of the theory by concentrating on speci?c typesandexamplesofin?nite-dimensionalLiegroups. Ofcourse, theselection of the topics is largely in?uenced by the taste of the authors, but we hope thatthisselectioniswideenoughtodescribevariousphenomenaarisinginthe geometry of in?nite-dimensional Lie groups and to convince the reader that they are appealing objects to study from both purely mathematical and more applied points of view. This book can be thought of as complementary to the existing more algebraic treatments, in particular, those covering the str- ture and representation theory of in?nite-dimensional Lie algebras, as well as to more analytic ones developing calculus on in?nite-dimensional manifolds. This monograph originated from advanced graduate courses and mi- courses on in?nite-dimensional groups and gauge theory given by the ?rst author at the University of Toronto, at the CIRM in Marseille, and at the Ecole Polytechnique in Paris in 2001-2004. It is based on various classical and recentresultsthathaveshapedthisnewlyemergedpartofin?nite-dimensional geometry and group theory. Our intention was to make the book concise, relatively self-contained, and useful in a graduate course. For this reason, throughout the text, we have included a large number of problems, ranging from simple exercises to open questions

In this monograph, we de?ne and investigate an algebro-geometric analogue of Donaldson invariants by using moduli spaces of semistable sheaves with arbitrary ranks on a polarized projective surface. We may expect the existence of interesting "universal relations among invariants", which would be a natural generalization of the "wall-crossing formula" and the "Witten conjecture" for classical Donaldson invariants. Our goal is to obtain a weaker version of such relations, in other brief words, to describe a relation as the sum of integrals over the products of m- uli spaces of objects with lower ranks. Fortunately, according to a recent excellent work of L. Gottsche, H. Nakajima and K. Yoshioka, [53], a wall-crossing formula for Donaldson invariants of projective surfaces can be deduced from such a weaker result in the rank two case. We hope that our work in this monograph would, at least tentatively, provides a part of foundation for the further study on such universal relations. In the rest of this preface, we would like to explain our motivation and some of important ingredients of this study. See Introduction for our actual problems and results. Donaldson Invariants Let us brie?y recall Donaldson invariants. We refer to [22] for more details and precise. We also refer to [37], [39], [51] and [53]. LetX be a compact simply con- ? nected oriented real 4-dimensional C -manifold with a Riemannian metric g. Let P be a principalSO(3)-bundle on X.

The characterization of combinatorial or geometric structures in terms of their groups of automorphisms has attracted considerable interest in the last decades and is now commonly viewed as a natural generalization of Felix Klein's Erlangen program(1872).Inaddition,especiallyfor?nitestructures,importantapplications to practical topics such as design theory, coding theory and cryptography have made the ?eld even more attractive. The subject matter of this research monograph is the study and class- cation of ?ag-transitive Steiner designs, that is, combinatorial t-(v,k,1) designs which admit a group of automorphisms acting transitively on incident point-block pairs. As a consequence of the classi?cation of the ?nite simple groups, it has been possible in recent years to characterize Steiner t-designs, mainly for t=2,adm- ting groups of automorphisms with su?ciently strong symmetry properties. For Steiner 2-designs, arguably the most general results have been the classi?cation of all point 2-transitive Steiner 2-designs in 1985 by W. M. Kantor, and the almost complete determination of all ?ag-transitive Steiner 2-designs announced in 1990 byF.Buekenhout,A.Delandtsheer,J.Doyen,P.B.Kleidman,M.W.Liebeck, and J. Saxl. However, despite the classi?cation of the ?nite simple groups, for Steiner t-designs witht> 2 most of the characterizations of these types have remained long-standing challenging problems. Speci?cally, the determination of all ?- transitive Steiner t-designs with 3? t? 6 has been of particular interest and object of research for more than 40 years.

How does an algebraic geometer studying secant varieties further the understanding of hypothesis tests in statistics? Why would a statistician working on factor analysis raise open problems about determinantal varieties? Connections of this type are at the heart of the new field of "algebraic statistics." In this field, mathematicians and statisticians come together to solve statistical inference problems using concepts from algebraic geometry as well as related computational and combinatorial techniques. The goal of these lectures is to introduce newcomers from the different camps to algebraic statistics. The introduction will be centered around the following three observations: many important statistical models correspond to algebraic or semi-algebraic sets of parameters; the geometry of these parameter spaces determines the behaviour of widely used statistical inference procedures; computational algebraic geometry can be used to study parameter spaces and other features of statistical models.

This commemorative book contains the 28 major articles that appeared in the 2008 Twentieth Anniversary Issue of the journal Discrete & Computational Geometry, and presents a comprehensive picture of the current state of the field. Formed during the past few decades by the merger of the classical discipline of combinatorial and discrete geometry with the new field of computational geometry that sprang up in the 1970s, discrete and computational geometry now claims the allegiance of a sizeable number of mathematicians and computer scientists all over the world, whose most important work has been appearing since 1986 in the pages of the journal.

The articles in this volume, a number of which solve long-outstanding problems in the field, were chosen by the editors of DCG for the importance of their results, for the breadth of their scope, and to show the intimate connections that have arisen between discrete and computational geometry and other areas of both computer science and mathematics. Apart from the articles, the editors present an expanded preface, along with a set of photographs of groups and individuals who have played a major role in the history of the field during the past twenty years.

Contributors include:

E. Ackerman

P.K. Agarwal

I. Aliev

I. BArAny

A. Barvinok

S. Basu

L.J. Billera

J.-D. Boissonnat

C. Borcea

E. Boros

K. Borys

B. Braun

K. Buchin

O. Cheong

D. Cohen-Steiner

M. Damian

K. Elbassioni

R. Flatland

T. Gerken

J.E. Goodman

X. Goaoc

P. Gronchi

V. Gurvich

S. Har-Peled

J. Hershberger

A. Holmsen

S.K. Hsiao

A. Hubard

J. JerA3nimo

L. Khachiyan

R. Klein

C. Knauer

S.Langerman

J.-Y. Lee

M. Longinetti

E. Miller

P. Morin

U. Nagel

E. Nevo

P. Niyogi

I. Novik

J. Oa (TM)Rourke

J. Pach

I. Pak

M.J. Pelsmajer

S. Petitjean

F. Pfender

R. Pinchasi

R. Pollack

J.S. Provan

K. Przeslawski

R.M. Richardson

G. Rote

M. Schaefer

Y. Schreiber

M. Sharir

J.R. Shewchuk

S. Smale

B. Solomyak

M. Soss

D. A tefankovic

G. Vegter

V.H. Vu

S. Weinberger

L. Wu

D. Yost

H. Yu

T. Zell

This book constitutes the thoroughly refereed post-conference proceedings of the Kyoto Conference on Computational Geometry and Graph Theory, KyotoCGGT 2007, held in Kyoto, Japan, in June 2007, in honor of Jin Akiyama and Vasek Chvatal, on the occasion of their 60th birthdays.

The 19 revised full papers, presented together with 5 invited papers, were carefully selected during two rounds of reviewing and improvement from more than 60 talks at the conference. All aspects of Computational Geometry and Graph Theory are covered, including tilings, polygons, impossible objects, coloring of graphs, Hamilton cycles, and factors of graphs.

Recent advances in both the theory and implementation of computational algebraic geometry have led to new, striking applications to a variety of fields of research. The articles in this volume highlight a range of these applications and provide introductory material for topics covered in the IMA workshops on "Optimization and Control" and "Applications in Biology, Dynamics, and Statistics" held during the IMA year on Applications of Algebraic Geometry. The articles related to optimization and control focus on burgeoning use of semidefinite programming and moment matrix techniques in computational real algebraic geometry. The new direction towards a systematic study of non-commutative real algebraic geometry is well represented in the volume. Other articles provide an overview of the way computational algebra is useful for analysis of contingency tables, reconstruction of phylogenetic trees, and in systems biology. The contributions collected in this volume are accessible to non-experts, self-contained and informative; they quickly move towards cutting edge research in these areas, and provide a wealth of open problems for future research.

The aim of this monograph is to give an overview of various classes of in?ni- dimensional Lie groups and their applications, mostly in Hamiltonian - chanics, ?uid dynamics, integrable systems, and complex geometry. We have chosen to present the unifying ideas of the theory by concentrating on speci?c typesandexamplesofin?nite-dimensionalLiegroups. Ofcourse, theselection of the topics is largely in?uenced by the taste of the authors, but we hope thatthisselectioniswideenoughtodescribevariousphenomenaarisinginthe geometry of in?nite-dimensional Lie groups and to convince the reader that they are appealing objects to study from both purely mathematical and more applied points of view. This book can be thought of as complementary to the existing more algebraic treatments, in particular, those covering the str- ture and representation theory of in?nite-dimensional Lie algebras, as well as to more analytic ones developing calculus on in?nite-dimensional manifolds. This monograph originated from advanced graduate courses and mi- courses on in?nite-dimensional groups and gauge theory given by the ?rst author at the University of Toronto, at the CIRM in Marseille, and at the Ecole Polytechnique in Paris in 2001-2004. It is based on various classical and recentresultsthathaveshapedthisnewlyemergedpartofin?nite-dimensional geometry and group theory. Our intention was to make the book concise, relatively self-contained, and useful in a graduate course. For this reason, throughout the text, we have included a large number of problems, ranging from simple exercises to open questions

In this volume, the authors construct a theory of weights on the log crystalline cohomologies of families of open smooth varieties in characteristic p>0, by defining and constructing four filtered complexes. Fundamental properties of these filtered complexes are proved, in particular the p-adic purity, the functionality of three filtered complexes, the weight-filtered base change formula, the weight-filtered Kunneth formula, the weight-filtered Poincare duality, and the E2-degeneration of p-adic weight spectral sequences. In addition, the authors state some theorems on the weight filtration and the slope filtration on the rigid cohomology of a separated scheme of finite type over a perfect field of characteristic p>0.

It is a pleasure to welcome you to the proceedings of the second International Castle Meeting on Coding Theory and its Applications, held at La Mota Castle in Medina del Campo. The event provided a forum for the exchange of results and ideas, which we hope will foster future collaboration. The ?rst meeting was held in 1999, and, encouraged by that experience, we now intend to hold the meeting every three years. Springer kindly accepted to publish the proceedings volume you have in your hands in their LNCS series. The topics were selected to cover some of the areas of research in Coding Theory that are currently receiving the most attention. The program consisted of a mixture of invited and submitted talks, with the focus on quality rather than quantity. A total of 34 papers were submitted to themeeting.Afteracarefulreviewprocessconductedbythescienti?ccommittee aided by external reviewers, we selected 14 of these for inclusion in the current volume, along with 5 invited papers. The program was further augmented by the remaining invited papers in addition to papers on recent results, printed in a separate volume. We would like to thank everyone who made this meeting possible by helping with the practical and scienti?c preparations: the organization committee, the scienti?c committee, the invited speakers, and the many external reviewers who shall remain anonymous. I would especially like to mention the General Advisor ofthe meeting, OyvindYtrehus.Finally Iextend mygratitudeto allthe authors and participants who contributed to this meeting."

Stochastic geometry deals with models for random geometric structures. Its early beginnings are found in playful geometric probability questions, and it has vigorously developed during recent decades, when an increasing number of real-world applications in various sciences required solid mathematical foundations. Integral geometry studies geometric mean values with respect to invariant measures and is, therefore, the appropriate tool for the investigation of random geometric structures that exhibit invariance under translations or motions. Stochastic and Integral Geometry provides the mathematically oriented reader with a rigorous and detailed introduction to the basic stationary models used in stochastic geometry random sets, point processes, random mosaics and to the integral geometry that is needed for their investigation. The interplay between both disciplines is demonstrated by various fundamental results. A chapter on selected problems about geometric probabilities and an outlook to non-stationary models are included, and much additional information is given in the section notes."

The problem of compactifying the moduli spaceA of principally polarized g abelian varieties has a long and rich history. The majority of recent work has focusedonthe toroidal compacti?cations constructed over C by Mumford and his coworkers, and over Z by Chai and Faltings. The main drawback of these compacti?cations is that they are not canonical and do not represent any r- sonable moduli problem on the category of schemes. The starting point for this work is the realization of Alexeev and Nakamura that there is a canonical compacti?cation of the moduli space of principally polarized abelian varieties. Indeed Alexeev describes a moduli problem representable by a proper al- braic stack over Z which containsA as a dense open subset of one of its g irreducible components. In this text we explain how, using logarithmic structures in the sense of Fontaine, Illusie, and Kato, one can de?ne a moduli problem "carving out" the main component in Alexeev's space. We also explain how to generalize the theory to higher degree polarizations and discuss various applications to moduli spaces for abelian varieties with level structure.

Field Arithmetic explores Diophantine fields through their absolute Galois groups. This largely self-contained treatment starts with techniques from algebraic geometry, number theory, and profinite groups. Graduate students can effectively learn generalizations of finite field ideas. We use Haar measure on the absolute Galois group to replace counting arguments. New Chebotarev density variants interpret diophantine properties. Here we have the only complete treatment of Galois stratifications, used by Denef and Loeser, et al, to study Chow motives of Diophantine statements.

Progress from the first edition starts by characterizing the finite-field like P(seudo)A(lgebraically)C(losed) fields. We once believed PAC fields were rare. Now we know they include valuable Galois extensions of the rationals that present its absolute Galois group through known groups. PAC fields have projective absolute Galois group. Those that are Hilbertian are characterized by this group being pro-free. These last decade results are tools for studying fields by their relation to those with projective absolute group. There are still mysterious problems to guide a new generation: Is the solvable closure of the rationals PAC; and do projective Hilbertian fields have pro-free absolute Galois group (includes Shafarevich's conjecture)?

The third edition improves the second edition in two ways: First it removes many typos and mathematical inaccuracies that occur in the second edition (in particular in the references). Secondly, the third edition reports on five open problems (out of thirtyfour open problems of the second edition) that have been partially or fully solved since that edition appeared in 2005.

"This book revives and vastly expands the classical theory of resultants and discriminants. Most of the main new results of the book have been published earlier in more than a dozen joint papers of the authors. The book nicely complements these original papers with many examples illustrating both old and new results of the theory." Mathematical Reviews

This second edition is a corrected and extended version of the first. It is a textbook for students, as well as a reference book for the working mathematician, on cohomological topics in number theory. In all it is a virtually complete treatment of a vast array of central topics in algebraic number theory. New material is introduced here on duality theorems for unramified and tamely ramified extensions as well as a careful analysis of 2-extensions of real number fields.

Aimed primarily at graduate students and beginning researchers, this book provides an introduction to algebraic geometry that is particularly suitable for those with no previous contact with the subject; it assumes only the standard background of undergraduate algebra. The book starts with easily-formulated problems with non-trivial solutions and uses these problems to introduce the fundamental tools of modern algebraic geometry: dimension; singularities; sheaves; varieties; and cohomology. A range of exercises is provided for each topic discussed, and a selection of problems and exam papers are collected in an appendix to provide material for further study.

Schubert varieties provide an inductive tool for studying flag varieties. This book is mainly a detailed account of a particularly interesting instance of their occurrence: namely, in relation to classical invariant theory. More precisely, it is about the connection between the first and second fundamental theorems of classical invariant theory on the one hand and standard monomial theory for Schubert varieties in certain special flag varieties on the other.