Algebraisk topologi

III. Latin American School of Mathematics

These proceedings contain the contributions of some of the participants in the "e;intensive research period"e; held at the De Giorgi Research Center in Pisa, during the period May-June 2010. The central theme of this research period was the study of configuration spaces from various points of view. This topic originated from the intersection of several classical theories: Braid groups and related topics, configurations of vectors (of great importance in Lie theory and representation theory), arrangements of hyperplanes and of subspaces, combinatorics, singularity theory. Recently, however, configuration spaces have acquired independent interest and indeed the contributions in this volume go far beyond the above subjects, making it attractive to a large audience of mathematicians.

This volume of research papers is an outgrowth of the Manin Seminar at Moscow University, devoted to K-theory, homological algebra and algebraic geometry. The main topics discussed include additive K-theory, cyclic cohomology, mixed Hodge structures, theory of Virasoro and Neveu-Schwarz algebras.

During the Winter and spring of 1985 a Workshop in Algebraic Topology was held at the University of Washington. The course notes by Emmanuel Dror Farjoun and by Frederick R. Cohen contained in this volume are carefully written graduate level expositions of certain aspects of equivariant homotopy theory and classical homotopy theory, respectively. M.E. Mahowald has included some of the material from his further papers, represent a wide range of contemporary homotopy theory: the Kervaire invariant, stable splitting theorems, computer calculation of unstable homotopy groups, and studies of L(n), Im J, and the symmetric groups.

The aim of this international conference the third of its type was to survey recent developments in Geometric Topology and Shape Theory with an emphasis on their interaction. The volume contains original research papers and carefully selected survey of currently active areas. The main topics and themes represented by the papers of this volume include decomposition theory, cell-like mappings and CE-equivalent compacta, covering dimension versus cohomological dimension, ANR's and LCn-compacta, homology manifolds, embeddings of continua into manifolds, complement theorems in shape theory, approximate fibrations and shape fibrations, fibered shape, exact homologies and strong shape theory.

These proceedings reflect the main activities of the Paris Seminaire d'Algebre 1989-1990, with a series of papers in Invariant Theory, Representation Theory and Combinatorics. It contains original works from J. Dixmier, F. Dumas, D. Krob, P. Pragacz and B.J. Schmid, as well as a new presentation of Derived Categories by J.E. Bjork and as introduction to the deformation theory of Lie equations by J.F. Pommaret. J. Dixmier: Sur les invariants du groupe symetrique dans certaines representations II.- B.J. Schmid: Finite groups and invariant theory.- J.E. Bjork: Derived categories.- P. Pragacz: Algebro-Geometric applications of Schur S- and Q-polynomials.- F. Dumas: Sous-corps de fractions rationnelles des corps gauches de series de Laurent.- D. Krob: Expressions rationnelles sur un anneau.- J.F. Pommaret: Deformation theory of algebraic and Geometric structures.- M. van den Bergh: Differential operators on semi-invariants for tori and weighted projective spaces.

The Symposium on the Current State and Prospects ofMathematics was held in Barcelona from June 13 to June 18,1991. Seven invited Fields medalists gavetalks on thedevelopment of their respective research fields. Thecontents of all lectures were collected in the volume,together witha transcription of a round table discussionheld during the Symposium. All papers are expository. Someparts include precise technical statements of recentresults, but the greater part consists of narrative textaddressed to a very broad mathematical public. CONTENTS: R. Thom: Leaving Mathematics for Philosophy.- S. Novikov: Role of Integrable Models in the Development ofMathematics.- S.-T. Yau: The Current State and Prospects ofGeometry and Nonlinear Differential Equations.- A. Connes:Noncommutative Geometry.- S. Smale: Theory of Computation.-V. Jones: Knots in Mathematics and Physics.- G. Faltings:Recent Progress in Diophantine Geometry.

This volume (a sequel to LNM 1108, 1214, 1334 and 1453)continues the presentation to English speaking readers ofthe Voronezh University press series on Global Analysis andIts Applications. The papers are selected fromtwo Russianissues entitled "e;Algebraic questions of Analysis andTopology"e; and "e;Nonlinear Operators in Global Analysis"e;. CONTENTS: YuE. Gliklikh: Stochastic analysis, groups ofdiffeomorphisms and Lagrangian description of viscousincompressible fluid.- A.Ya. Helemskii: From topologicalhomology: algebras with different properties of homologicaltriviality.- V.V. Lychagin, L.V. Zil'bergleit: Duality instable Spencer cohomologies.- O.R. Musin: On some problemsof computational geometry and topology.- V.E. Nazaikinskii,B.Yu. Sternin, V.E.Shatalov: Introduction to Maslov'soperational method (non-commutative analysis anddifferential equations).- Yu.B. Rudyak: The problem ofrealization of homology classes from Poincare up to thepresent.- V.G. Zvyagin, N.M. Ratiner: Oriented degree ofFredholm maps of non-negativeindex and its applications toglobal bifurcation of solutions.- A.A. Bolibruch: Fuchsiansystems with reducible monodromy and the Riemann-Hilbertproblem.- I.V. Bronstein, A.Ya. Kopanskii: Finitely smoothnormal forms of vector fields in the vicinity of a restpoint.- B.D. Gel'man: Generalized degree of multi-valuedmappings.- G.N. Khimshiashvili: On Fredholmian aspects oflinear transmission problems.- A.S. Mishchenko: Stationarysolutions of nonlinear stochastic equations.- B.Yu. Sternin,V.E. Shatalov: Continuation of solutions to ellipticequations and localisation of singularities.- V.G. Zvyagin,V.T. Dmitrienko: Properness of nonlinear ellipticdifferential operators in H|lder spaces.

Since the subject of Groups of Self-Equivalences was first discussed in 1958 in a paper of Barcuss and Barratt, a good deal of progress has been achieved. This is reviewed in this volume, first by a long survey article and a presentation of 17 open problems together with a bibliography of the subject, and by a further 14 original research articles.

Proceedings 1972

The papers in this collection, all fully refereed, originalpapers, reflect many aspects of recent significant advancesin homotopy theory and group cohomology. From the Contents: A. Adem: On the geometry and cohomologyof finite simple groups.- D.J. Benson: Resolutions andPoincar duality for finite groups.- C. Broto and S. Zarati:On sub-A*-algebras of H*V.- M.J. Hopkins, N.J. Kuhn, D.C. Ravenel: Morava K-theories of classifying spaces andgeneralized characters for finite groups.- K. Ishiguro:Classifying spaces of compact simple lie groups and p-tori.-A.T. Lundell: Concise tables of James numbers and somehomotopyof classical Lie groups and associated homogeneousspaces.- J.R. Martino: Anexample of a stable splitting: theclassifying space of the 4-dim unipotent group.- J.E. McClure, L. Smith: On the homotopy uniqueness of BU(2) atthe prime 2.- G. Mislin: Cohomologically central elementsand fusion in groups.

With one exception, these papers are original and fullyrefereed research articles on various applications ofCategory Theory to Algebraic Topology, Logic and ComputerScience. The exception is an outstanding and lengthy surveypaper by Joyal/Street (80 pp) on a growing subject: it givesan account of classical Tannaka duality in such a way as tobe accessible to the general mathematical reader, and toprovide a key for entry to more recent developments andquantum groups. No expertise in either representation theoryor category theory is assumed. Topics such as the Fouriercotransform, Tannaka duality for homogeneous spaces, braidedtensor categories, Yang-Baxter operators, Knot invariantsand quantum groups are introduced and studies. From the Contents: P.J. Freyd: Algebraically completecategories.- J.M.E. Hyland: First steps in synthetic domaintheory.- G. Janelidze, W. Tholen: How algebraic is thechange-of-base functor?.- A. Joyal, R. Street: Anintroduction to Tannaka duality and quantum groups.- A. Joyal, M. Tierney: Strong stacks andclassifying spaces.- A. Kock: Algebras for the partial map classifier monad.- F.W. Lawvere: Intrinsic co-Heyting boundaries and the Leibnizrule in certain toposes.- S.H. Schanuel: Negative sets haveEuler characteristic and dimension.-

This book demonstrates the lively interaction between algebraic topology, very low dimensional topology and combinatorial group theory. Many of the ideas presented are still in their infancy, and it is hoped that the work here will spur others to new and exciting developments. Among the many techniques disussed are the use of obstruction groups to distinguish certain exact sequences and several graph theoretic techniques with applications to the theory of groups.

These are proceedings of an International Conference on Algebraic Topology, held 28 July through 1 August, 1986, at Arcata, California. The conference served in part to mark the 25th anniversary of the journal Topology and 60th birthday of Edgar H. Brown. It preceded ICM 86 in Berkeley, and was conceived as a successor to the Aarhus conferences of 1978 and 1982. Some thirty papers are included in this volume, mostly at a research level. Subjects include cyclic homology, H-spaces, transformation groups, real and rational homotopy theory, acyclic manifolds, the homotopy theory of classifying spaces, instantons and loop spaces, and complex bordism.

Categorical algebra and its applications contain several fundamental papers on general category theory, by the top specialists in the field, and many interesting papers on the applications of category theory in functional analysis, algebraic topology, algebraic geometry, general topology, ring theory, cohomology, differential geometry, group theory, mathematical logic and computer sciences. The volume contains 28 carefully selected and refereed papers, out of 96 talks delivered, and illustrates the usefulness of category theory today as a powerful tool of investigation in many other areas.

* Contains research and survey articles by well known and respected mathematicians on recent developments and research trends in differential geometry and topology* Dedicated in honor of Lieven Vanhecke, as a tribute to his many fruitful and inspiring contributions to these fields* Papers include all necessary introductory and contextual material to appeal to non-specialists, as well as researchers and differential geometers

Algebraic K-theory encodes important invariants for several mathematical disciplines, spanning from geometric topology and functional analysis to number theory and algebraic geometry. As is commonly encountered, this powerful mathematical object is very hard to calculate. Apart from Quillen's calculations of finite fields and Suslin's calculation of algebraically closed fields, few complete calculations were available before the discovery of homological invariants offered by motivic cohomology and topological cyclic homology. This book covers the connection between algebraic K-theory and Bökstedt, Hsiang and Madsen's topological cyclic homology and proves that the difference between the theories are ¿locally constant¿. The usefulness of this theorem stems from being more accessible for calculations than K-theory, and hence a single calculation of K-theory can be used with homological calculations to obtain a host of ¿nearby¿ calculations in K-theory. For instance, Quillen's calculation of the K-theory of finite fields gives rise to Hesselholt and Madsen's calculations for local fields, and Voevodsky's calculations for the integers give insight into the diffeomorphisms of manifolds. In addition to the proof of the full integral version of the local correspondence between K-theory and topological cyclic homology, the book provides an introduction to the necessary background in algebraic K-theory and highly structured homotopy theory; collecting all necessary tools into one common framework. It relies on simplicial techniques, and contains an appendix summarizing the methods widely used in the field. The book is intended for graduate students and scientists interested in algebraic K-theory, and presupposes a basic knowledge of algebraic topology.

At the heart of the topology of global optimization lies Morse Theory: The study of the behaviour of lower level sets of functions as the level varies. Roughly speaking, the topology of lower level sets only may change when passing a level which corresponds to a stationary point (or Karush-Kuhn- Tucker point). We study elements of Morse Theory, both in the unconstrained and constrained case. Special attention is paid to the degree of differentiabil- ity of the functions under consideration. The reader will become motivated to discuss the possible shapes and forms of functions that may possibly arise within a given problem framework. In a separate chapter we show how certain ideas may be carried over to nonsmooth items, such as problems of Chebyshev approximation type. We made this choice in order to show that a good under- standing of regular smooth problems may lead to a straightforward treatment of "e;just"e; continuous problems by means of suitable perturbation techniques, taking a priori nonsmoothness into account. Moreover, we make a focal point analysis in order to emphasize the difference between inner product norms and, for example, the maximum norm. Then, specific tools from algebraic topol- ogy, in particular homology theory, are treated in some detail. However, this development is carried out only as far as it is needed to understand the relation between critical points of a function on a manifold with structured boundary. Then, we pay attention to three important subjects in nonlinear optimization.