Homological mirror symmetry has its origins in theoretical physics but is now of great interest in mathematics due to the deep connections it reveals between different areas of geometry and algebra. This book offers a self-contained and accessible introduction to the subject via the representation theory of algebras and quivers. It is suitable for graduate students and others without a great deal of background in homological algebra and modern geometry. Each part offers a different perspective on homological mirror symmetry. Part I introduces the A-infinity formalism and offers a glimpse of mirror symmetry using representations of quivers. Part II discusses various A- and B-models in mirror symmetry and their connections through toric and tropical geometry. Part III deals with mirror symmetry for Riemann surfaces. The main mathematical ideas are illustrated by means of simple examples coming mainly from the theory of surfaces, helping the reader connect theory with intuition.

The papers collected in this volume are contributions to the 43rd session of the Seminaire de mathematiques superieures (SMS) on "Morse Theoretic Methods in Nonlinear Analysis and Symplectic Topology." This session took place at the Universite de Montreal in July 2004 and was a NATO Advanced Study Institute (ASI). The aim of the ASI was to bring together young researchers from various parts of the world and to present to them some of the most signi cant recent advances in these areas. More than 77 mathematicians from 17 countries followed the 12 series of lectures and participated in the lively exchange of ideas. The lectures covered an ample spectrum of subjects which are re ected in the present volume: Morse theory and related techniques in in nite dim- sional spaces, Floer theory and its recent extensions and generalizations, Morse and Floer theory in relation to string topology, generating functions, structure of the group of Hamiltonian di?eomorphisms and related dynamical problems, applications to robotics and many others. We thank all our main speakers for their stimulating lectures and all p- ticipants for creating a friendly atmosphere during the meeting. We also thank Ms. Diane Belanger, our administrative assistant, for her help with the organi- tion and Mr. Andre Montpetit, our technical editor, for his help in the preparation of the volume."

This work treats an introduction to commutative ring theory and algebraic plane curves, requiring of the student only a basic knowledge of algebra, with all of the algebraic facts collected into several appendices that can be easily referred to, as needed. Kunz's proven conception of teaching topics in commutative algebra together with their applications to algebraic geometry makes this book significantly different from others on plane algebraic curves. The exposition focuses on the purely algebraic aspects of plane curve theory, leaving the topological and analytical viewpoints in the background, with only casual references to these subjects and suggestions for further reading. algebras, their graduated rings and Rees algebras, to deduce basic facts about the intersection theory of plane curves; presents residue theory in the affine plane and its applications to intersection theory; methods of proof for the Riemann-Roch theorem conform to the presentation of curve theory, formulated in the language of filtrations and associated graded rings; and examples, exercises, figures and suggestions for further study round out this fairly self-contained textbook.

The NATO Advanced Study Institute "Axiomatic, enriched and rna tivic homotopy theory" took place at the Isaac Newton Institute of Mathematical Sciences, Cambridge, England during 9-20 September 2002. The Directors were J.P.C.Greenlees and I.Zhukov; the other or ganizers were P.G.Goerss, F.Morel, J.F.Jardine and V.P.Snaith. The title describes the content well, and both the event and the contents of the present volume reflect recent remarkable successes in model categor ies, structured ring spectra and homotopy theory of algebraic geometry. The ASI took the form of a series of 15 minicourses and a few extra lectures, and was designed to provide background, and to bring the par ticipants up to date with developments. The present volume is based on a number of the lectures given during the workshop. The ASI was the opening workshop of the four month programme "New Contexts for Stable Homotopy Theory" which explored several themes in greater depth. I am grateful to the Isaac Newton Institute for providing such an ideal venue, the NATO Science Committee for their funding, and to all the speakers at the conference, whether or not they were able to contribute to the present volume. All contributions were refereed, and I thank the authors and referees for their efforts to fit in with the tight schedule. Finally, I would like to thank my coorganizers and all the staff at the Institute for making the ASI run so smoothly. J.P.C.GREENLEES."

The book provides an introduction to the theory of cluster sets, a branch of topological analysis which has made great strides in recent years. The cluster set of a function at a particular point is the set of limit values of the function at that point which may be either a boundary point or (in the case of a non-analytic function) an interior point of the function's domain. In topological analysis, its main application is to problems arising in the theory of functions of a complex variable, with particular reference to boundary behaviour such as the theory of prime ends under conformal mapping. An important and novel feature of the book is the discussion of more general applications to non-analytic functions, including arbitrary functions. The authors assume a general familiarity with classical function theory but include the more specialised material required for the development of the theory of cluster sets, so making the treatment accessible to graduate students.

Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds, a great geometrical idea due to R. Thom and H. Whitney. These sheaves, generalizing the local systems that are so ubiquitous in mathematics, have powerful applications to the topology of such singular spaces (mainly algebraic and analytic complex varieties).

This introduction to the subject can be regarded as a textbook on modern algebraic topology, treating the cohomology of spaces with sheaf (as opposed to constant)coefficients.

The first 5 chapters introduce derived categories, direct and inverse images of sheaf complexes, Verdier duality, constructible and perverse sheaves, vanishing and characteristic cycles. They also discuss relations to D-modules and intersection cohomology. Later chapters apply this powerful tool to the study of the topology of singularities, polynomial functions and hyperplane arrangements.

Some fundamental results, for which excellent sources exist, are not proved but just stated and illustrated by examples and corollaries. In this way, the reader is guided rather quickly from the basic theory to current research questions, supported in this by examples and exercises.

This book is the result of reworking part of a rather lengthy course of lectures of which we delivered several versions at the Leningrad and Moscow Universities. In these lectures we presented an introduction to the fundamental topics of topology: homology theory, homotopy theory, theory of bundles, and topology of manifolds. The structure of the course was well determined by the guiding term elementary topology, whose main significance resides in the fact that it made us use a rather simple apparatus. tn this book we have retained {hose sections of the course where algebra plays a subordinate role. We plan to publish the more algebraic part of the lectures as a separate book. Reprocessing the lectures to produce the book resulted in the profits and losses inherent in such a situation: the rigour has increased to the detriment of the intuitiveness, the geometric descriptions have been replaced by formulas needing interpretations, etc. Nevertheless, it seems to us tha.t the book retains the main qualities of our lectures: their elementary, systematic, and pedagogical features. The preparation of the reader is assumed to be limi ted to the usual knowledge of set .theory, algebra, and calculus which mathematics students should master after the first year and a half of studies. The exposition is accompanied by examples and exercises. We hope that the book can be used as a topology textbook."

The book gathers the lectures given at the C.I.M.E. summer school "Quantum Cohomology" held in Cetraro (Italy) from June 30th to July 8th, 1997. The lectures and the subsequent updating cover a large spectrum of the subject on the field, from the algebro-geometric point of view, to the symplectic approach, including recent developments of string-branes theories and q-hypergeometric functions.

This book consists essentially of notes which were written for an Advanced Course on Classifying Spaces and Cohomology of Groups. The course took place at the Centre de Recerca Mathematica (CRM) in Bellaterra from May 27 to June 2, 1998 and was part of an emphasis semester on Algebraic Topology. It consisted of two parallel series of 6 lectures of 90 minutes each and was intended as an introduction to new homotopy theoretic methods in group cohomology. The first part of the book is concerned with methods of decomposing the classifying space of a finite group into pieces made of classifying spaces of appropriate subgroups. Such decompositions have been used with great success in the last 10-15 years in the homotopy theory of classifying spaces of compact Lie groups and p-compact groups in the sense of Dwyer and Wilkerson. For simplicity the emphasis here is on finite groups and on homological properties of various decompositions known as centralizer resp. normalizer resp. subgroup decomposition. A unified treatment of the various decompositions is given and the relations between them are explored. This is preceeded by a detailed discussion of basic notions such as classifying spaces, simplicial complexes and homotopy colimits.

This book gives a streamlined introduction to the theory of Seiberg-Witten invariants suitable for second-year graduate students. These invariants can be used to prove that there are many compact topological four-manifolds which have more than one smooth structure, and that others have no smooth structure at all. This topic provides an excellent example of how global analysis techniques, which have been developed to study nonlinear partial differential equations, can be applied to the solution of interesting geometrical problems. In the second edition, some material has been expanded for better comprehension.

From the reviews:"The author has attempted an ambitious and most commendable project. He assumes only a modest knowledge of algebraic topology on the part of the reader to start with, and he leads the reader systematically to the point at which he can begin to tackle problems in the current areas of research centered around generalized homology theories and their applications. ... The author has sought to make his treatment complete and he has succeeded. The book contains much material that has not previously appeared in this format. The writing is clean and clear and the exposition is well motivated. ... This book is, all in all, a very admirable work and a valuable addition to the literature...(S.Y. Husseini in Mathematical Reviews, 1976)

Computational engineering is the treatment of engineering tasks with computers. It is based on computational mathematics, which is presented here in a comprehensive handbook. Engineers and scientists who deal with engineering tasks have to handle large amounts of information, which must be created and structured in a systematic manner. This demands a high level of abstraction and therefore knowledge of the mathematical foundations. From the existing rich repertoire of mathematical theories and methods, the fundamentals of engineering computation are selected and presented in a coherent fashion. They are brought into a suitable order for specific engineering purposes, and their significance for typical applications is shown. The relevant definitions, notations and theories are presented in a durable form which is independent of the fast development of information and communication technology.

This monograph deals with two aspects of the theory of elliptic genus: its topological aspect involving elliptic functions, and its representation theoretic aspect involving vertex operator super-algebras. For the second aspect, elliptic genera are shown to have the structure of modules over certain vertex operator super-algebras. The vertex operators corresponding to parallel tensor fields on closed Riemannian Spin K hler manifolds such as Riemannian tensors and K hler forms are shown to give rise to Virasoro algebras and affine Lie algebras. This monograph is chiefly intended for topologists and it includes accounts on topics outside of topology such as vertex operator algebras.

Using harmonic maps, non-linear PDE and techniques from algebraic geometry this book enables the reader to study the relation between fundamental groups and algebraic geometry invariants of algebraic varieties. The reader should have a basic knowledge of algebraic geometry and non-linear analysis. This book can form the basis for graduate level seminars in the area of topology of algebraic varieties. It also contains present new techniques for researchers working in this area.

This volume contains a re-edition of Max Koecher's famous Minnesota Notes. The main objects are homogeneous, but not necessarily convex, cones. They are described in terms of Jordan algebras. The central point is a correspondence between semisimple real Jordan algebras and so-called omega-domains. This leads to a construction of half-spaces which give an essential part of all bounded symmetric domains. The theory is presented in a concise manner, with only elementary prerequisites. The editors have added notes on each chapter containing an account of the relevant developments of the theory since these notes were first written.

This book describes the representations of Lie superalgebras that are yielded by a graded version of Hudson-Parthasarathy quantum stochastic calculus. Quantum stochastic calculus and grading theory are given concise introductions, extending readership to mathematicians and physicists with a basic knowledge of algebra and infinite-dimensional Hilbert spaces. The develpment of an explicit formula for the chaotic expansion of a polynomial of quantum stochastic integrals is particularly interesting. The book aims to provide a self-contained exposition of what is known about Z_2-graded quantum stochastic calculus and to provide a framework for future research into this new and fertile area.

From the reviews: "... The book under review consists of two monographs on geometric aspects of group theory ... Together, these two articles form a wide-ranging survey of combinatorial group theory, with emphasis very much on the geometric roots of the subject. This will be a useful reference work for the expert, as well as providing an overview of the subject for the outsider or novice. Many different topics are described and explored, with the main results presented but not proved. This allows the interested reader to get the flavour of these topics without becoming bogged down in detail. Both articles give comprehensive bibliographies, so that it is possible to use this book as the starting point for a more detailed study of a particular topic of interest. ..." Bulletin of the London Mathematical Society, 1996

From the reviews: "This volume... consists of two papers. The first, written by V.V. Shokurov, is devoted to the theory of Riemann surfaces and algebraic curves. It is an excellent overview of the theory of relations between Riemann surfaces and their models - complex algebraic curves in complex projective spaces. ... The second paper, written by V.I. Danilov, discusses algebraic varieties and schemes. ... I can recommend the book as a very good introduction to the basic algebraic geometry." "European Mathematical Society" "Newsletter, 1996"
..". To sum up, this book helps to learn algebraic geometry in a short time, its concrete style is enjoyable for students and reveals the beauty of mathematics." Acta Scientiarum Mathematicarum

Any topological theory of knots and links should be based on simple ideas of intersection and linking. In this book, a general theory of link bordism in manifolds and universal constructions of linking numbers in oriented 3-manifolds are developed. In this way, classical concepts of link theory in the 3-spheres are generalized to a certain class of oriented 3-manifolds (submanifolds of rational homology 3-spheres). The techniques needed are described in the book but basic knowledge in topology and algebra is assumed. The book should be of interst to those working in topology, in particular knot theory and low-dimensional topology.

This survey covers groups of homotopy self-equivalence classes of topological spaces, and the homotopy type of spaces of homotopy self-equivalences. For manifolds, the full group of equivalences and the mapping class group are compared, as are the corresponding spaces. Included are methods of calculation, numerous calculations, finite generation results, Whitehead torsion and other areas. Some 330 references are given. The book assumes familiarity with cell complexes, homology and homotopy. Graduate students and established researchers can use it for learning, for reference, and to determine the current state of knowledge.

This is a research monograph covering the majority of known results on the problem of constructing compact symplectic manifolds with no Kaehler structure with an emphasis on the use of rational homotopy theory. In recent years, some new and stimulating conjectures and problems have been formulated due to an influx of homotopical ideas. Examples include the Lupton-Oprea conjecture, the Benson-Gordon conjecture, both of which are in the spirit of some older and still unsolved problems (e.g. Thurston's conjecture and Sullivan's problem). Our explicit aim is to clarify the interrelations between certain aspects of symplectic geometry and homotopy theory in the framework of the problems mentioned above. We expect that the reader is aware of the basics of differential geometry and algebraic topology at graduate level.

From the reviews:
.."., the book must be of great help for a researcher who already has some idea of Lie theory, wants to employ it in his everyday research and/or teaching, and needs a source for customary reference on the subject. From my viewpoint, the volume is perfectly fit to serve as such a source, ... On the whole, it is quite a pleasure, after making yourself comfortable in that favourite office armchair of yours, just to keep the volume gently in your hands and browse it slowly and thoughtfully; and after all, what more on Earth can one expect of any book?"
"The New Zealand Mathematical Society Newsletter"
..". Both parts are very nicely written and can be strongly recommended."
"European Mathematical Society"

The aim of cyclic cohomology theories is the approximation of K-theory by cohomology theories defined by natural chain complexes. The basic example is the approximation of topological K-theory by de Rham cohomology via the classical Chern character. A cyclic cohomology theory for operator algebras is developed in the book, based on Connes' work on noncommutative geometry. Asymptotic cyclic cohomology faithfully reflects the basic properties and features of operator K-theory. It thus becomes a natural target for a Chern character. The central result of the book is a general Grothendieck-Riemann-Roch theorem in noncommutative geometry with values in asymptotic cyclic homology. Besides this, the book contains numerous examples and calculations of asymptotic cyclic cohomology groups.

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 purpose of this book is to present the available (sometimes only partial) solutions to the two fundamental problems: the existence problem and the classification problem for holomorphic structures in a given topological vector bundle over a compact complex surface. Special features of the nonalgebraic surfaces case, like irreducible vector bundles and stability with respect to a Gauduchon metric, are considered. The reader requires a grounding in geometry at graduate student level. The book will be of interest to graduate students and researchers in complex, algebraic and differential geometry.