The articles in this volume are an outgrowth of an International Confer- ence in Intersection Theory that took place in Bologna, Italy (December 1997). In a somewhat unorthodox format aimed at both the mathematical community as well as summer school students, talks were research-oriented as well as partly expository. There were four series of expository talks by the following people: M. Brion, University of Grenoble, on Equivariant Chow groups and applications; H. Flenner, University of Bochum, on Joins and intersections; E. M. Friedlander, Northwestern University, on Intersection products for spaces of algebraic cycles; R. Laterveer, University of Strasbourg, on Bigraded Chow (co)homology. Four introductory papers cover the following topics and bring the reader to the forefront of research: 1) the excess intersection algorithm of Stuckrad and Vogel, combined with the deformation to the normal cone, together with many of its geo- metric applications; 2) new and very important homotopy theory techniques that are now used in intersection theory; 3) the Bloch-Beilinson filtration and the theory of motives; 4) algebraic stacks, the modern language of moduli theory. Other research articles concern such active fields as stable maps and Gromov-Witten invariants, deformation theory of complex varieties, and others. Organizers of the conference were Rudiger Achilles, Mirella Manaresi, and Angelo Vistoli, all from the University of Bologna; the scientific com- mittee consisted of Geir Ellingsrud, University of Oslo, William Fulton, University of Michigan at Ann Arbor, and Angelo Vistoli. The conference was financed by the European Union (contract no.

In various contexts of topology, algebraic geometry, and algebra (e.g. group representations), one meets the following situation. One has two contravariant functors K and A from a certain category to the category of rings, and a natural transformation p: K--+A of contravariant functors. The Chern character being the central exam ple, we call the homomorphisms Px: K(X)--+ A(X) characters. Given f: X--+ Y, we denote the pull-back homomorphisms by and fA: A(Y)--+ A(X). As functors to abelian groups, K and A may also be covariant, with push-forward homomorphisms and fA: A( X)--+ A(Y). Usually these maps do not commute with the character, but there is an element r f E A(X) such that the following diagram is commutative: K(X) A(X) fK j J A K( Y) ------p;-+ A( Y) The map in the top line is p x multiplied by r f. When such commutativity holds, we say that Riemann-Roch holds for f. This type of formulation was first given by Grothendieck, extending the work of Hirzebruch to such a relative, functorial setting. Since then viii INTRODUCTION several other theorems of this Riemann-Roch type have appeared. Un derlying most of these there is a basic structure having to do only with elementary algebra, independent of the geometry. One purpose of this monograph is to describe this algebra independently of any context, so that it can serve axiomatically as the need arises."

The articles in this volume are an outgrowth of an International Confer- ence in Intersection Theory that took place in Bologna, Italy (December 1997). In a somewhat unorthodox format aimed at both the mathematical community as well as summer school students, talks were research-oriented as well as partly expository. There were four series of expository talks by the following people: M. Brion, University of Grenoble, on Equivariant Chow groups and applications; H. Flenner, University of Bochum, on Joins and intersections; E. M. Friedlander, Northwestern University, on Intersection products for spaces of algebraic cycles; R. Laterveer, University of Strasbourg, on Bigraded Chow (co)homology. Four introductory papers cover the following topics and bring the reader to the forefront of research: 1) the excess intersection algorithm of Stuckrad and Vogel, combined with the deformation to the normal cone, together with many of its geo- metric applications; 2) new and very important homotopy theory techniques that are now used in intersection theory; 3) the Bloch-Beilinson filtration and the theory of motives; 4) algebraic stacks, the modern language of moduli theory. Other research articles concern such active fields as stable maps and Gromov-Witten invariants, deformation theory of complex varieties, and others. Organizers of the conference were Rudiger Achilles, Mirella Manaresi, and Angelo Vistoli, all from the University of Bologna; the scientific com- mittee consisted of Geir Ellingsrud, University of Oslo, William Fulton, University of Michigan at Ann Arbor, and Angelo Vistoli. The conference was financed by the European Union (contract no.

Schubert varieties and degeneracy loci have a long history in mathematics, starting from questions about loci of matrices with given ranks. These notes, from a summer school in Thurnau, aim to give an introduction to these topics, and to describe recent progress on these problems. There are interesting interactions with the algebra of symmetric functions and combinatorics, as well as the geometry of flag manifolds and intersection theory and algebraic geometry.

From the ancient origins of algebraic geometry in the solutions of polynomial equations, through the triumphs of algebraic geometry during the last two centuries, intersection theory has played a central role. The aim of this book is to develop the foundations of this theory, and to indicate the range of classical and modern applications. Although a comprehensive history of this vast subject is not attempted, the author points out some of the striking early appearances of the ideas of intersection theory. A suggested prerequisite for the reading of this book is a first course in algebraic geometry. Fulton's introduction to intersection theory has been well used for more than 10 years. It is still the only existing complete modern treatise of the subject and received the Steele Prize for best exposition in August 1996.

Rather than choosing one point of view of modern topology, the author concentrates on concrete problems in spaces with a few dimensions, introducing only as much algebraic machinery as necessary. This makes it possible to see a wider variety of important features in the subject than is common in introductory texts; it is also in line with the historical development of the subject. Aimed at students not necessarily intending to specialise in algebraic topology, the first part of the book emphasises relations with calculus and uses these ideas to prove the Jordan curve theorem, before going on to study fundamental groups and covering spaces so as to emphasise group actions. A final section gives a taste of the generalisation to higher dimensions.

The primary goal of these lectures is to introduce a beginner to the finite-dimensional representations of Lie groups and Lie algebras. Intended to serve non-specialists, the concentration of the text is on examples. The general theory is developed sparingly, and then mainly as useful and unifying language to describe phenomena already encountered in concrete cases. The book begins with a brief tour through representation theory of finite groups, with emphasis determined by what is useful for Lie groups. The focus then turns to Lie groups and Lie algebras and finally to the heart of the course: working out the finite dimensional representations of the classical groups. The goal of the last portion of the book is to make a bridge between the example-oriented approach of the earlier parts and the general theory.

There are few more powerful questions than, “Where are you from” or “Where do you live?” People feel intensely connected to cities as places and to other people who feel that same connection. In order to understand place – and understand human settlements generally – it is important to understand that places are not created by accident. They are created in order to further a political or economic agenda. Better cities emerge when the people who shape them think more broadly and consciously about the places they are creating. In Place and Prosperity: How Cities Help Us to Connect and Innovate, urban planning expert William Fulton takes an engaging look at the process by which these decisions about places are made, how cities are engines of prosperity, and how place and prosperity are deeply intertwined. Fulton has been writing about cities over his forty-year career that includes working as a journalist, professor, mayor, planning director, and the director of an urban think tank in one of America’s great cities. Place and Prosperity is a curated collection of his writings with new and updated selections and framing material. Though the essays in Place and Prosperity are in some ways personal, drawing on Fulton’s experience in learning and writing about cities, their primary purpose is to show how these two ideas – place and prosperity – lie at the heart of what a city is and, by extension, what our society is all about. Fulton shows how, over time, a successful place creates enduring economic assets that don’t go away and lay the groundwork for prosperity in the future. But for urbanism to succeed, all of us have to participate in making cities great places for everybody. Because cities, imposing though they may be as physical environments, don’t work without us. Cities are resilient. They’ve been buffeted over the decades by White flight, decay, urban renewal, unequal investment, increasingly extreme weather events, and now the worst pandemic in a century, and they’re still going strong. Fulton shows that at their best, cities not only inspire and uplift us, but they make our daily life more convenient, more fulfilling – and more prosperous.

The primary goal of these lectures is to introduce a beginner to the finite dimensional representations of Lie groups and Lie algebras. Since this goal is shared by quite a few other books, we should explain in this Preface how our approach differs, although the potential reader can probably see this better by a quick browse through the book. Representation theory is simple to define: it is the study of the ways in which a given group may act on vector spaces. It is almost certainly unique, however, among such clearly delineated subjects, in the breadth of its interest to mathematicians. This is not surprising: group actions are ubiquitous in 20th century mathematics, and where the object on which a group acts is not a vector space, we have learned to replace it by one that is {e. g. , a cohomology group, tangent space, etc. }. As a consequence, many mathematicians other than specialists in the field {or even those who think they might want to be} come in contact with the subject in various ways. It is for such people that this text is designed. To put it another way, we intend this as a book for beginners to learn from and not as a reference. This idea essentially determines the choice of material covered here. As simple as is the definition of representation theory given above, it fragments considerably when we try to get more specific.

Toric varieties are algebraic varieties arising from elementary geometric and combinatorial objects such as convex polytopes in Euclidean space with vertices on lattice points. Since many algebraic geometry notions such as singularities, birational maps, cycles, homology, intersection theory, and Riemann-Roch translate into simple facts about polytopes, toric varieties provide a marvelous source of examples in algebraic geometry. In the other direction, general facts from algebraic geometry have implications for such polytopes, such as to the problem of the number of lattice points they contain. In spite of the fact that toric varieties are very special in the spectrum of all algebraic varieties, they provide a remarkably useful testing ground for general theories.

The aim of this mini-course is to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications. The text concludes with Stanley's theorem characterizing the numbers of simplicies in each dimension in a convex simplicial polytope. Although some general theorems are quoted without proof, the concrete interpretations via simplicial geometry should make the text accessible to beginners in algebraic geometry.

Geechi Suede and Sonny Cheeba are Camp Lo. These two emcees from the Bronx, NY entered the American hip hop scene with an insider slang that bewildered listeners as they radiated the look of a bygone era of black culture. In 1996, they collaborated with producer Ski and a host of other contributors to create Uptown Saturday Night, featuring the seminal single "Luchini (a.k.a. This is It)." While other 1990s rappers referred to 1970s Blaxploitation culture, Camp Lo were self-described "time travelers" who weaved the slang and style of a soulful past into state-of-the-art lyrical flows. Uptown Saturday Night is a tapestry of 1970s black popular culture and 1990s New York City hip hop. This volume will detail how the album's fantastic world of "Coolie High" reflected classic films like Cooley High and the Sidney Poitier film from which the album's title is derived, and promoted vintage slang and fashion. The book features new interviews with Camp Lo, producer Ski, Trugoy the Dove from De La Soul, Ish from Digable Planets, and others, and offers musical and cultural analyses that detail the development of the album and its essential contributions to a post-soul aesthetic.

Talk City is a collection of the remarkable blogs the distinguished urban planner Bill Fulton wrote while serving as a member of the City Council in the California beach town of Ventura. The blog started out as a way to explain what had happened at the weekly council meetings. Before long, however, it turned into an evocative, real-time chronicle of what it was like to serve as an underpaid, overstressed, part-time local elected official during hard times. If you like local government and politics, you'll love how Talk City reveals the stresses and strains of serving as an elected official in a typical American city.

This is a reproduction of a book published before 1923. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ++++ The below data was compiled from various identification fields in the bibliographic record of this title. This data is provided as an additional tool in helping to ensure edition identification: ++++ Lockhart's Book Of Instructions For Locomotive Firemen William Fulton Lockhart Author, 1908 Transportation; Railroads; General; Collection locomotives; Railroads; Trainmen; Transportation / Railroads / General; Transportation / Railroads / History

How did Federal Express decide to locate at the Memphis Airport? Why is China also losing manufacturing jobs? Do artists really help turn around a struggling neighborhood? What should you do with a declining auto mall - save it or let it die and start over again? What's better - subsidizing an business or subsidizing the infrastructure such a business requires? These are the kinds of questions that cities and states deal with all the time in their economic development. Bill Fulton's new book, ROMANCING THE SMOKESTACK: HOW CITIES AND STATES PURSUE PROSPERITY, is a collection of economic development columns from GOVERNING magazine that covers deals with these questions - and reveals the good, the bad, and the ugly about how economic development is practiced in the United States. Bill Fulton is a veteran author (GUIDE TO CALIFORNIA PLANNING, THE RELUCTANT METROPOLIS), urban planning and economic development consultant (with the firm Design, Community & Environment), and currently also mayor of Ventura, California, one of the most innovative communities in America. This book discusses economic development efforts that are sometimes shrewd and sometimes stupid - but shows that cities and states are tireless in their efforts to find the next economic engine. You can read an excerpt from the introduction here: https: //www.createspace.com/Preview/1073034

Equivariant cohomology has become an indispensable tool in algebraic geometry and in related areas including representation theory, combinatorial and enumerative geometry, and algebraic combinatorics. This text introduces the main ideas of the subject for first- or second-year graduate students in mathematics, as well as researchers working in algebraic geometry or combinatorics. The first six chapters cover the basics: definitions via finite-dimensional approximation spaces, computations in projective space, and the localization theorem. The rest of the text focuses on examples – toric varieties, Grassmannians, and homogeneous spaces – along with applications to Schubert calculus and degeneracy loci. Prerequisites are kept to a minimum, so that one-semester graduate-level courses in algebraic geometry and topology should be sufficient preparation. Featuring numerous exercises, examples, and material that has not previously appeared in textbook form, this book will be a must-have reference and resource for both students and researchers for years to come.

This book develops the combinatorics of Young tableaux and shows them in action in the algebra of symmetric functions, representations of the symmetric and general linear groups, and the geometry of flag varieties. The first part of the book is a self-contained presentation of the basic combinatorics of Young tableaux, including the remarkable constructions of "bumping" and "sliding", and several interesting correspondences. In Part II the author uses these results to study representations with geometry on Grassmannians and flag manifolds, including their Schubert subvarieties, and the related Schubert polynomials. Much of this material has never before appeared in book form. There are numerous exercises throughout, with hints and answers provided. Researchers in representation theory and algebraic geometry as well as in combinatorics will find this book interesting and useful, while students will find the intuitive presentation easy to follow.

This book introduces some of the main ideas of modern intersection theory, traces their origins in classical geometry and sketches a few typical applications. It requires little technical background: much of the material is accessible to graduate students in mathematics. A broad survey, the book touches on many topics, most importantly introducing a powerful new approach developed by the author and R. MacPherson. It was written from the expository lectures delivered at the NSF-supported CBMS conference at George Mason University, held June 27-July 1, 1983.The author describes the construction and computation of intersection products by means of the geometry of normal cones. In the case of properly intersecting varieties, this yields Samuel's intersection multiplicity; at the other extreme it gives the self-intersection formula in terms of a Chern class of the normal bundle; in general it produces the excess intersection formula of the author and R. MacPherson. Among the applications presented are formulas for degeneracy loci, residual intersections, and multiple point loci; dynamic interpretations of intersection products; Schubert calculus and solutions to enumerative geometry problems; and Riemann-Roch theorems.