Topology Books

Book SynopsisThis detailed yet accessible text introduces the advanced mathematical methods at the core of theoretical physics. Based on a course for senior undergraduate students of physics, it is written in a clear, pedagogical style and would also be valuable to students in other areas of science and engineering.Trade Review'The recent explosive development of topological quantum matter requires a deep systematic understanding of modern mathematics. Quantum many-body entanglement in topological quantum matter is a new phenomenon that requires new mathematical language to describe. This is a rare book that provides systematic and in-depth coverage of some of the most important mathematical concepts, such as groups, geometry, topology and algebra, among others. Many abstract mathematical notions are explained in an easy, explicit fashion. This is an in-depth, friendly book on modern mathematics. Very timely and highly recommended.' Xiao-Gang Wen, Massachusetts Institute of TechnologyTable of Contents1. Introduction; 2. Group theory; 3. Representation theory of groups; 4. Differentiable manifolds; 5. Riemannian geometry; 6. Semisimple Lie algebras and their unitary representations; Appendix A; References; Index.

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Book SynopsisThis text is an introduction to some of the mathematical wonders of Maxwell''s equations. These equations led to the prediction of radio waves, the realization that light is a type of electromagnetic wave, and the discovery of the special theory of relativity. In fact, almost all current descriptions of the fundamental laws of the universe can be viewed as deep generalizations of Maxwell''s equations. Even more surprising is that these equations and their generalizations have led to some of the most important mathematical discoveries of the past thirty years. It seems that the mathematics behind Maxwell''s equations is endless. The goal of this book is to explain to mathematicians the underlying physics behind electricity and magnetism and to show their connections to mathematics. Starting with Maxwell''s equations, the reader is led to such topics as the special theory of relativity, differential forms, quantum mechanics, manifolds, tangent bundles, connections, and curvature.Table of Contents1. A brief history; 2. Maxwell's equations; 3. Electromagnetic waves; 4. Special relativity; 5. Mechanics and Maxwell's equations; 6. Mechanics, Lagrangians, and the calculus of variations; 7. Potentials; 8. Lagrangians and electromagnetic forces; 9. Differential forms; 10. The Hodge * operator; 11. The electromagnetic two-form; 12. Some mathematics needed for quantum mechanics; 13. Some quantum mechanical thinking; 14. Quantum mechanics of harmonic oscillators; 15. Quantizing Maxwell's equations; 16. Manifolds; 17. Vector bundles; 18. Connections; 19. Curvature; 20. Maxwell via connections and curvature; 21. The Lagrangian machine, Yang–Mills, and other forces.

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Book SynopsisDifferential geometry is the study of curved spaces using the techniques of calculus. It is a mainstay of undergraduate mathematics education and a cornerstone of modern geometry. It is also the language used by Einstein to express general relativity, and so is an essential tool for astronomers and theoretical physicists. This introductory textbook originates from a popular course given to third year students at Durham University for over twenty years, first by the late L. M. Woodward and later by John Bolton (and others). It provides a thorough introduction by focusing on the beginnings of the subject as studied by Gauss: curves and surfaces in Euclidean space. While the main topics are the classics of differential geometry - the definition and geometric meaning of Gaussian curvature, the Theorema Egregium, geodesics, and the GaussBonnet Theorem - the treatment is modern and student-friendly, taking direct routes to explain, prove and apply the main results. It includes many exercisesTrade Review'An excellent introduction to the subject, suitable for learners and lecturers alike. The authors strike a perfect balance between clear prose and clean mathematical style and provide plenty of examples, exercises and intuitive diagrams. The choice of material stands out as well: covering the essentials and including interesting further topics without cluttering. This wonderful book again reminded me of the beauty of this topic!' Karsten Fritzsch, Gottfried Wilhelm Leibniz Universität Hannover, Germany'How to present a coherent and stimulating introduction to a mathematical subject without getting carried away into bloating it by our love for the subject? This book not only expresses the authors' enthusiasm for differential geometry but also condenses decades of teaching experience: it focuses on few milestones, covering the required theory in an efficient and stimulating way. It will be a pleasure to teach/learn alongside this text.' Udo Hertrich-Jeromin, Technische Universität Wien, Austria'This is an attractive candidate as a text for an undergraduate course in classical differential geometry and should certainly be given serious consideration by any instructor teaching such a course.' Mark Hunacek, Department of Mathematics, Iowa State UniversityTable of ContentsPreface; 1. Curves in Rn; 2. Surfaces in Rn; 3. Smooth maps; 4. Measuring how surfaces curve; 5. The Theorema Egregium; 6. Geodesic curvature and geodesics; 7. The Gauss–Bonnet theorem; 8. Minimal and CMC surfaces; 9. Hints or answers to some exercises; Index.

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Book SynopsisAfter the development of manifolds and algebraic varieties in the previous century, mathematicians and physicists have continued to advance concepts of space. This book and its companion explore various new notions of space, including both formal and conceptual points of view, as presented by leading experts at the New Spaces in Mathematics and Physics workshop held at the Institut Henri Poincaré in 2015. This volume covers a broad range of topics in mathematical physics, including noncommutative geometry, supergeometry, derived symplectic geometry, higher geometric quantization, intuitionistic quantum logic, problems with the continuum description of spacetime, twistor theory, loop quantum gravity, and geometry in string theory. It is addressed primarily to mathematical physicists and mathematicians, but also to historians and philosophers of these disciplines.Trade Review'The collection would be of interest to any physicist, mathematician, historian, or philosopher seeking a survey of the approaches to dealing with the modern concept of space in physics … Recommended.' E. Kincanon, Choice MagazineTable of ContentsIntroduction Mathieu Anel and Gabriel Catren; Part I. Noncommutative and supercommutative geometries: 1. Noncommutative geometry, the spectral standpoint Alain Connes; 2. The logic of quantum mechanics (revisited) Klaas Landsman; 3. Supergeometry in mathematics and physics Mikhail Kapranov; Part II. Symplectic geometry: 4. Derived stacks in symplectic geometry Damien Calaque; 5. Higher prequantum geometry Urs Schreiber; Part III. Spacetime: 6. Struggles with the continuum John C. Baez; 7. Twistor theory: a geometric perspective for describing the physical world Roger Penrose; 8. Quantum geometry of space Muxin Han; 9. Stringy geometry and emergent space Marcos Mariño.

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Book SynopsisAfter the development of manifolds and algebraic varieties in the previous century, mathematicians and physicists have continued to advance concepts of space. This book and its companion explore various new notions of space, including both formal and conceptual points of view, as presented by leading experts at the New Spaces in Mathematics and Physics workshop held at the Institut Henri Poincaré in 2015. The chapters in this volume cover a broad range of topics in mathematics, including diffeologies, synthetic differential geometry, microlocal analysis, topos theory, infinity-groupoids, homotopy type theory, category-theoretic methods in geometry, stacks, derived geometry, and noncommutative geometry. It is addressed primarily to mathematicians and mathematical physicists, but also to historians and philosophers of these disciplines.Trade Review'The essays are self-contained, providing motivation to read selectively. Examples in each chapter illustrate the usefulness of these new notions of space … Recommended.' M. Clay, Choice MagazineTable of ContentsIntroduction Mathieu Anel and Gabriel Catren; Part I. Differential geometry: 1. An Introduction to diffeology Patrick Iglesias-Zemmour; 2. New methods for old spaces: synthetic differential geometry Anders Kock; 3. Microlocal analysis and beyond Pierre Schapira; Part II. Topology and algebraic topology: 4. Topo-logie Mathieu Anel and André Joyal; 5. Spaces as infinity-groupoids Timothy Porter; 6. Homotopy type theory: the logic of space Michael Shulman; Part III. Algebraic geometry: 7. Sheaves and functors of points Michel Vaquié; 8. Stacks Nicole Mestrano and Carlos Simpson; 9. The geometry of ambiguity: an introduction to the ideas of derived geometry Mathieu Anel; 10. Geometry in dg categories Maxim Kontsevich.

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Book SynopsisThe continued and dramatic rise in the size of data sets has meant that new methods are required to model and analyze them. This timely account introduces topological data analysis (TDA), a method for modeling data by geometric objects, namely graphs and their higher-dimensional versions: simplicial complexes. The authors outline the necessary background material on topology and data philosophy for newcomers, while more complex concepts are highlighted for advanced learners. The book covers all the main TDA techniques, including persistent homology, cohomology, and Mapper. The final section focuses on the diverse applications of TDA, examining a number of case studies drawn from monitoring the progression of infectious diseases to the study of motion capture data. Mathematicians moving into data science, as well as data scientists or computer scientists seeking to understand this new area, will appreciate this self-contained resource which explains the underlying technology and how it can be used.Table of ContentsPart I. Background: 1. Introduction; 2. Data; Part II. Theory: 3. Topology; 4. Shape of data; 5. Structures on spaces of barcodes; Part III. Practice: 6. Case studies; References; Index.

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Book SynopsisThis book is about the interplay between algebraic topology and the theory of infinite discrete groups. It is a hugely important contribution to the field of topological and geometric group theory, and is bound to become a standard reference in the field.Trade ReviewFrom the reviews:"The author of this book has done a great service to the geometric group theory community by writing a very useful and well-written book on many topics in geometric group theory that every neophyte and researcher in the field should know. … This book is suitable as a textbook for a graduate course, with many good examples and exercises. The reviewer highly recommends this book as a basic reference book for topological methods in group theory." (John G. Ratcliffe, Mathematical Reviews, Issue 2008 j)"This is an interesting book on the interplay between algebraic topology and the theory of infinite discrete groups written for graduate students and group theorists who need to learn more in geometric and homological group theory. … It is a beautiful text in algebraic topology, with modern topics and which points the reader towards new research directions." (Corina Mohorianu, Zentralblatt MATH, Vol. 1142, 2008)“This book is an invaluable resource for anyone wanting a deep understanding of topics related to the ends of groups. … there is a good deal of material in this book that does not appear anywhere else in the literature. … Geoghegan’s book provides a well-presented, concrete development of geometric group theory focused on a topological approach.” (John Meier, Bulletin of the American Mathematical Society, July, 2012)Table of ContentsAlgebraic Topology for Group Theory.- CW Complexes and Homotopy.- Cellular Homology.- Fundamental Group and Tietze Transformation.- Some Techniques in Homotopy Theory.- Elementary Geometric Topology.- Finiteness Properties of Groups.- The Borel Construction and Bass-Serre Theory.- Topological Finiteness Properties and Dimension of Groups.- Homological Finiteness Properties of Groups.- Finiteness Properties of Some Important Groups.- Locally Finite Algebraic Topology for Group Theory.- Locally Finite CW Complexes and Proper Homotopy.- Locally Finite Homology.- Cohomology of CW Complexes.- Topics in the Cohomology of Infinite Groups.- Cohomology of Groups and Ends of Covering Spaces.- Filtered Ends of Pairs of Groups.- Poincaré Duality in Manifolds and Groups.- Homotopical Group Theory.- The Fundamental Group At Infinity.- Higher homotopy theory of groups.- Three Essays.- Three Essays.

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Book SynopsisOffers a collection of articles written in memory of Boris Dubrovin (1950-2019). The authors express their admiration for his remarkable personality and for the contributions he made to mathematical physics. For many of the authors, Dubrovin was a friend, colleague, inspiring mentor, and teacher.Table of Contents K. Aleshkin and K. Saito, Primitive forms without higher residue structure and integrable hierarchies (I) M. Alkadhem, G. Antoniou, and M, Feigin, Solutions of $BC_n$ type of WDVV equations P. P. Boalch, Topology of the Stokes phenomenon G. Cotti and A. Varchenko, Equivariant quantum differential equation and qKZ equations for a projective space: Stokes bases as exceptional collections, Stokes matrices as Gram matrices, and $\mathcyr{B}$-theorem L. David and C. Hertling, Meromorphic connections over F-manifolds G. Jorjadze and S. Theisen, Canonical maps and integrability in $T\bar{T}$ deformed 2d CFTs I. Krichever and A. Varchenko, Incarnations of XXX $\widehat{\mathfrak{sl}_N}$ Bethe ansatz equations and integrable hierarchies F. Magri, The Kowalewski separability conditions O. Mokhov and N. A. Strizhova, On the Liouville integrable reduction of the associativity equations in the case of three primary fields S. M. Natanzon and A. Yu. Orlov, Hurwitz numbers from matrix integrals over Guassian measure N. Reshetikhin, Spin Calogero-Moser models on symmetric spaces M. Semenov-Tian-Shansky, Quantum toda lattice: A challenge for representation theory A. O. Smirnov, M. V. Pavlov, V. B. Matveev, and V. S. Gerdjikov, Finite-gap solutions of the Mikhalev equation I. A. B. Strachan, Flat coordinates on orbit spaces: From Novikov algebras to cyclic quotient singularities K. Takasaki, Cubic Hodge integrals and integrable hierarchies of Volterra type G. Tian and G. Xu, Gauged Witten equation and adiabatic limit.

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Book SynopsisOffers a collection of articles written in memory of Boris Dubrovin (1950-2019). The authors express their admiration for his remarkable personality and for the contributions he made to mathematical physics. For many of the authors, Dubrovin was a friend, colleague, inspiring mentor, and teacher.Table of Contents T. Bridgeland, Geometry from Donaldson-Thomas invariants V. M. Buchstaber and A. P. Veselov, Fricke identities, Frobenius $K$-characters and Markov equation I. Cherednik, On Harish-Chandra theory of global nonsymmetric functions N. C. Combe and Y. I. Manin, Symmetries of genus zero modular operad R. Coquereaux and J.-B. Zuber, On Schur problem and Kostka numbers P. Etingof, E. Frenkel, and D. Kazhdan, An analytic version of the Langlands correspondence for complex curves D. Gaiotto, T. Johnson-Freyd, and E. Witten, A note on some minimally supersymmetric models in two dimensions O. Garcia-Prada and D. Salamon, A moment map interpretation of the Ricci form, Kahler-Einstein structures, and Teichmuller spaces E. Getzler and S. W. Pohorence, Global gauge conditions in the Batalin-Vilkovisky formalism V. Golyshev and D. Zagier, Interpolated Apery numbers, quasiperiods of modular forms, and motivic gamma functions Y.-P. Lee, H.-W. Lin, and C.-L. Wang, Quantum flips I: Local model M. Marcolli, Aspects of $p$-adic geometry related to entanglement entropy S. Merkulov, Grothendieck-Teichmuller group, operads and graph complexes: A survey O. Ogievetsky and S. Shlosman, Platonic compounds of cylinders.

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Book SynopsisFocuses on group-theoretic aspects of topological dynamics such as studying groups using their actions on topological spaces, using group theory to study symbolic dynamics, and other connections between group theory and dynamical systems.Table of Contents Dynamical systems Group actions Groupoids Iterated monodromy groups Groups from groupoids Growth and amenability Bibliography Index

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Table of Contents1. Basic definitions.- 2. The invariant bilinear form and the generalized Casimir operator.- 3. Integrable representations and the Weyl group of a Kac-Moody algebra.- 4. Some properties of generalized Cartan matrices.- 5. Real and imaginary roots.- 6. Affine Lie algebras: the normalized invariant bilinear form, the root system and the Weyl group.- 7. Affine Lie algebras: the realization (case k = 1).- 8. Affine Lie algebras: the realization (case k = 2 or 3). Application to the classification of finite order automorphisms.- 9. Highest weight modules over the Lie algebra g(A).- 10. Integrable highest weight modules: the character formula.- 11. Integrable highest weight modules: the weight system, the contravariant Hermitian form and the restriction problem.- 12. Integrable highest weight modules over affine Lie algebras. Application to ?-function identities.- 13. Affine Lie algebras, theta functions and modular forms.- 14. The principal realization of the basic representation. Application to the KdV-type hierarchies of non-linear partial differential equations.- Index of notations and definitions.- References.

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Book SynopsisUntil the mid-twentieth century, topological studies were focused on the theory of suitable structures on sets of points. The concept of open set exploited since the twenties offered an expression of the geometric intuition of a "realistic" place (spot, grain) of non-trivial extent.Imitating the behaviour of open sets and their relations led to a new approach to topology flourishing since the end of the fifties.It has proved to be beneficial in many respects. Neglecting points, only little information was lost, while deeper insights have been gained; moreover, many results previously dependent on choice principles became constructive. The result is often a smoother, rather than a more entangled, theory.No monograph of this nature has appeared since Johnstone's celebrated Stone Spaces in 1983. The present book is intended as a bridge from that time to the present. Most of the material appears here in book form for the first time or is presented from new points of view. Two appendices provide an introduction to some requisite concepts from order and category theories.Trade ReviewFrom the reviews:“The book covers, in a comprehensive and self-contained way, the development of point-free topology in the last 30 years … . The book ends with two short appendices containing those facts about partially ordered sets and categories needed in the book. In conclusion, this is a very good book; it is nicely written and is highly recommended for anyone wishing to gain an overview of point-free topology.” (Javier Gutiérrez García, Mathematical Reviews, Issue 2012 j)“The book starts with a recount of sobriety and the TD-axiom, which is then quickly followed by the definition of the category of locales, various facets of spectra, the usual adjunctions and several criteria for spatiality. … this book is an erudite account of the current status of pointfree topology, written beautifully in the authors’ inimitable style. It is both easily accessible to a complete beginner, and an excellent source of reference for the mature pointfree practitioner.” (Themba Dube, Zentralblatt MATH, Vol. 1231, 2012)“In a total of fifteen chapters the reader is taken from spaces and lattices of open sets, through a thorough discussion of the notions of frames and locales … . book ends (modulo its appendices) with a discussion of localic groups … together with frames, that’s what this entire book is ultimately all about. … is quite-well written: it’s a clear and thorough treatment of what is really rather an important subject, which is to say, a greatly underappreciated one.” (Michael Berg, The Mathematical Association of America, April, 2012)Table of ContentsPreface.- Introduction.- I. Spaces and lattices of open sets.- II. Frames and locales. Spectra.- III. Sublocales.- IV. Structure of localic morphisms. The categories Loc and Frm.- V. Separation axioms.- VI. More on sublocales.-VII. Compactness and local compactness.- VIII. (Symmetric) uniformity and nearness.- IX. Paracompactness.- X. More about completion.- XI. Metric frames.- XII. Entourages, non-symmetric uniformity.- XIII. Connectedness.- XIV. The frame of reals and real functions.- XV. Localic groups.- Appendix I: Posets.- Appendix II: Categories.- Bibliography.- Index of Notation.- Index.

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Book SynopsisThis book presents the relationship between ultrafilters and topologies on groups. It shows how ultrafilters are used in constructing topologies on groups with extremal properties and how topologies on groups serve in deriving algebraic results about ultrafilters. The contents of the book fall naturally into three parts. The first, comprising Chapters 1 through 5, introduces to topological groups and ultrafilters insofar as the semigroup operation on ultrafilters is not required. Constructions of some important topological groups are given. In particular, that of an extremally disconnected topological group based on a Ramsey ultrafilter. Also one shows that every infinite group admits a nondiscrete zero-dimensional topology in which all translations and the inversion are continuous. In the second part, Chapters 6 through 9, the Stone-Cêch compactification βG of a discrete group G is studied. For this, a special technique based on the concepts of a local left group and a local homomorphism is developed. One proves that if G is a countable torsion free group, then βG contains no nontrivial finite groups. Also the ideal structure of βG is investigated. In particular, one shows that for every infinite Abelian group G, βG contains 22 G minimal right ideals. In the third part, using the semigroup βG, almost maximal topological and left topological groups are constructed and their ultrafilter semigroups are examined. Projectives in the category of finite semigroups are characterized. Also one shows that every infinite Abelian group with finitely many elements of order 2 is absolutely ω-resolvable, and consequently, can be partitioned into ω subsets such that every coset modulo infinite subgroup meets each subset of the partition. The book concludes with a list of open problems in the field. Some familiarity with set theory, algebra and topology is presupposed. But in general, the book is almost self-contained. It is aimed at graduate students and researchers working in topological algebra and adjacent areas.

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Book SynopsisThis monograph gives a systematic presentation of classical and recent results obtained in the last couple of years. It comprehensively describes the methods concerning the topological structure of fixed point sets and solution sets for differential equations and inclusions. Many of the basic techniques and results recently developed about this theory are presented, as well as the literature that is disseminated and scattered in several papers of pioneering researchers who developed the functional analytic framework of this field over the past few decades. Several examples of applications relating to initial and boundary value problems are discussed in detail. The book is intended to advanced graduate researchers and instructors active in research areas with interests in topological properties of fixed point mappings and applications; it also aims to provide students with the necessary understanding of the subject with no deep background material needed. This monograph fills the vacuum in the literature regarding the topological structure of fixed point sets and its applications.Table of Contents1 TOPOLOGICAL STRUCTURE OF FIXED POINT SETS 11 1.1 Case of single-valued mappings . . . . . . . . . . . . . . . . . . . . . . 11 1.1.1 Fundamental ¯xed point theorems . . . . . . . . . . . . . . . . . 11 1.1.2 Approximation theorems . . . . . . . . . . . . . . . . . . . . . . 14 1.1.3 Browder{Gupta Theorems . . . . . . . . . . . . . . . . . . . . . 16 1.1.4 Acyclicity of the solution sets of operator equation . . . . . . . 21 1.1.5 Solution sets for nonexpansive maps . . . . . . . . . . . . . . . . 24 1.2 Case of multi-valued mappings . . . . . . . . . . . . . . . . . . . . . . . 25 1.2.1 Fixed point theorems . . . . . . . . . . . . . . . . . . . . . . . . 25 1.2.2 Multivalued contractions . . . . . . . . . . . . . . . . . . . . . . 27 1.2.3 Fixed point sets of multi-valued contractions . . . . . . . . . . . 29 1.2.4 Fixed point sets of multivalued condensing maps . . . . . . . . . 32 1.2.5 Approximation of multi-valued maps . . . . . . . . . . . . . . . 37 1.3 Admissible maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 1.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 1.3.2 Fixed point theorems for admissible multivalued maps . . . . . 48 1.3.3 Browder{Gupta type results for admissible mappings . . . . . . 54 1.4 Topological structure of ¯xed point sets of inverse limit maps . . . . . . 58 1.4.1 De¯nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 1.4.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 1.4.3 Multi-maps of inverse systems . . . . . . . . . . . . . . . . . . . 60 1.5 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 1.5.1 Semi-compactness in L1 . . . . . . . . . . . . . . . . . . . . . . 63 1.5.2 Decomposability in L1(T;E) . . . . . . . . . . . . . . . . . . . . 64 1.5.3 Michael family of subsets . . . . . . . . . . . . . . . . . . . . . . 66 2 EXISTENCE THEORY FOR DIFFERENTIAL EQUATIONS AND INCLUSIONS 71 2.1 Case of di®erential equations . . . . . . . . . . . . . . . . . . . . . . . . 71 2.1.1 Existence and uniqueness results . . . . . . . . . . . . . . . . . 71 2.1.2 Picard-LindelÄof Theorem . . . . . . . . . . . . . . . . . . . . . . 72 2.1.3 Peano and Carath¶eodory theorems . . . . . . . . . . . . . . . . 77 2.1.4 Global existence theorems . . . . . . . . . . . . . . . . . . . . . 79 2.1.5 Existence results on non-compact intervals . . . . . . . . . . . . 82 2.1.6 A boundary value problem on the half-line . . . . . . . . . . . . 89 2.2 Case of di®erential inclusions . . . . . . . . . . . . . . . . . . . . . . . 94 2.2.1 Initial value problem . . . . . . . . . . . . . . . . . . . . . . . . 94 2.2.2 A boundary value problem . . . . . . . . . . . . . . . . . . . . . 99 3 SOLUTIONS SETS FOR DIFFERENTIAL EQUATIONS AND IN- CLUSIONS 105 3.1 Solutions sets for di®erential equations . . . . . . . . . . . . . . . . . . 105 3.1.1 Problems on bounded intervals . . . . . . . . . . . . . . . . . . 105 3.1.2 Problems on unbounded intervals . . . . . . . . . . . . . . . . . 107 3.1.3 Kneser-Hukuhara Theorem . . . . . . . . . . . . . . . . . . . . . 109 3.2 Aronszajn-type results for di®erential inclusions . . . . . . . . . . . . . 111 3.3 Application to neutral di®erential inclusions . . . . . . . . . . . . . . . 118 3.3.1 The convex case . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.3.2 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.3.3 Solutions sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 3.4 Application to second order di®erential inclusions . . . . . . . . . . . . 136 3.4.1 The convex case . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 3.4.2 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . . . 141 3.4.3 Solution sets to second-order di®erential equations . . . . . . . . 144 3.4.4 Solution sets to second-order di®erential inclusions . . . . . . . 146 3.5 Application to a nonlocal problem . . . . . . . . . . . . . . . . . . . . . 150 3.5.1 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . 150 3.5.2 Solutions set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 3.6 Application to a nonlocal viability problem . . . . . . . . . . . . . . . . 152 3.6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 3.6.2 Viable solutions on proximate retracts . . . . . . . . . . . . . . 154 3.7 Application to hyperbolic di®erential inclusions . . . . . . . . . . . . . 158 3.7.1 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . 158 3.7.2 Solution sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 3.8 Application to abstract Volterra operators . . . . . . . . . . . . . . . . 166 4 IMPULSIVE DIFFERENTIAL INCLUSIONS: EXISTENCE AND SOLUTION SETS 169 4.1 Impulsive di®erential inclusions . . . . . . . . . . . . . . . . . . . . . . 169 4.1.1 C0¡Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 4.1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 4.1.3 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . 174 4.1.4 Structure of solution sets . . . . . . . . . . . . . . . . . . . . . . 190 4.2 A periodic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 4.2.1 Existence results: 1 2 ½(T(b)) . . . . . . . . . . . . . . . . . . . 203 4.2.2 The convex case: direct approach . . . . . . . . . . . . . . . . . 204 4.2.3 The convex case: MNC approach . . . . . . . . . . . . . . . . . 211 4.2.4 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . . . 216 4.2.5 The parameter-dependant case . . . . . . . . . . . . . . . . . . 219 4.2.6 Filippov's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 224 4.2.7 Existence of solutions: 1 62 ½(T(b)) . . . . . . . . . . . . . . . . 232 4.3 Impulsive Functional Di®erential Inclusions . . . . . . . . . . . . . . . . 238 4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 4.3.2 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . 239 4.3.3 Structure of the solution set . . . . . . . . . . . . . . . . . . . . 247 4.4 Impulsive di®erential inclusions on the half-line . . . . . . . . . . . . . 251 4.4.1 Existence results and compactness of solution sets . . . . . . . . 252 4.4.2 Topological structure via the projective limit . . . . . . . . . . . 266 4.4.3 Using solution sets to prove existence results . . . . . . . . . . . 282 I SUPPLEMENTS 287 5 PRELIMINARY NOTIONS OF TOPOLOGY 289 5.1 Extension and embedding properties . . . . . . . . . . . . . . . . . . . 289 5.2 Homotopical properties of spaces . . . . . . . . . . . . . . . . . . . . . 296 5.3 ·Cech homology (cohomology) functor . . . . . . . . . . . . . . . . . . . 303 5.4 Maps of spaces of ¯nite type . . . . . . . . . . . . . . . . . . . . . . . . 304 5.5 ·Cech homology functor with compact carriers . . . . . . . . . . . . . . 311 5.6 Acyclic sets and Vietoris maps . . . . . . . . . . . . . . . . . . . . . . . 313 5.7 Homology of open subsets of Euclidean spaces . . . . . . . . . . . . . . 317 5.8 Lefschetz number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 5.9 Coincidence problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 6 BACKGROUND IN MULTI-VALUED ANALYSIS 335 6.1 Continuity of multivalued mappings . . . . . . . . . . . . . . . . . . . . 337 6.1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 6.1.2 Upper semi-continuity . . . . . . . . . . . . . . . . . . . . . . . 339 6.1.3 Lower semi-continuity . . . . . . . . . . . . . . . . . . . . . . . 344 6.1.4 Hausdor® continuity . . . . . . . . . . . . . . . . . . . . . . . . 347 6.2 Selection theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 6.2.1 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . 349 6.2.2 Michael's selection theorem . . . . . . . . . . . . . . . . . . . . 350 6.2.3 ¾¡selectionable mappings . . . . . . . . . . . . . . . . . . . . . 353 6.2.4 The Kuratowski-Ryll-Nardzewski selection theorem . . . . . . . 356 6.2.5 Hausdor®-measurable multivalued maps . . . . . . . . . . . . . 371 6.2.6 The Scorza-Dragoni property . . . . . . . . . . . . . . . . . . . 373 6.2.7 The Bressan-Colombo-Fryszkowski selection theorem . . . . . . 379 6.3 The Bochner integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 6.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 6.3.2 Nemytski·i operators . . . . . . . . . . . . . . . . . . . . . . . . 383 6.3.3 Integration of multivalued maps . . . . . . . . . . . . . . . . . . 386 6.4 Compactness in C([a; b];E) and PC([a; b];E) . . . . . . . . . . . . . . . 388 6.5 Further auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . 391

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Book SynopsisPeriodic differential equations appear in many contexts such as in the theory of nonlinear oscillators, in celestial mechanics, or in population dynamics with seasonal effects. The most traditional approach to study these equations is based on the introduction of small parameters, but the search of nonlocal results leads to the application of several topological tools. Examples are fixed point theorems, degree theory, or bifurcation theory. These well-known methods are valid for equations of arbitrary dimension and they are mainly employed to prove the existence of periodic solutions. Following the approach initiated by Massera, this book presents some more delicate techniques whose validity is restricted to two dimensions. These typically produce additional dynamical information such as the instability of periodic solutions, the convergence of all solutions to periodic solutions, or connections between the number of harmonic and subharmonic solutions. The qualitative study of periodic planar equations leads naturally to a class of discrete dynamical systems generated by homeomorphisms or embeddings of the plane. To study these maps, Brouwer introduced the notion of a translation arc, somehow mimicking the notion of an orbit in continuous dynamical systems. The study of the properties of these translation arcs is full of intuition and often leads to "non-rigorous proofs". In the book, complete proofs following ideas developed by Brown are presented and the final conclusion is the Arc Translation Lemma, a counterpart of the Poincaré–Bendixson theorem for discrete dynamical systems. Applications to differential equations and discussions on the topology of the plane are the two themes that alternate throughout the five chapters of the book.

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Book SynopsisIntroduction to Dynamical Systems and Geometric Mechanics provides a comprehensive tour of two fields that are intimately entwined: dynamical systems is the study of the behavior of physical systems that may be described by a set of nonlinear first-order ordinary differential equations in Euclidean space, whereas geometric mechanics explore similar systems that instead evolve on differentiable manifolds. The first part discusses the linearization and stability of trajectories and fixed points, invariant manifold theory, periodic orbits, Poincaré maps, Floquet theory, the Poincaré-Bendixson theorem, bifurcations, and chaos. The second part of the book begins with a self-contained chapter on differential geometry that introduces notions of manifolds, mappings, vector fields, the Jacobi-Lie bracket, and differential forms.

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Table of ContentsStatement of the theorem.- Applications of the index theorem.- Outline of the proof.- The atiyah-singer fixed point theorem.- Applications of the fixed point theorem.

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Book SynopsisThis book offers a comprehensive introduction to the general theory of C*-algebras and von Neumann algebras. Beginning with the basics, the theory is developed through such topics as tensor products, nuclearity and exactness, crossed products, K-theory, and quasidiagonality. The presentation carefully and precisely explains the main features of each part of the theory of operator algebras; most important arguments are at least outlined and many are presented in full detail.Trade ReviewFrom the reviews: "The volume under review is supposed to cover basics on operator algebras … . Blackadar’s book is very well written and pleasant to read. It is especially suited to readers who already know the basics of operator algebras but who need a reference for some result or who wish to have a unified approach to topics treated by them." (Paul Jolissaint, Mathematical Reviews, Issue 2006 k) "This volume is an important and useful contribution to the literature on C*-algebras and von Neumann algebras. … The book is extremely well written. It can be recommended as a reference to graduate students working in operator algebra theory and to other mathematicians who want to bring themselves up-to-date on the subject." (V. M. Manuilov, Zentralblatt MATH, Vol. 1092 (18), 2006)Table of ContentsOperators on Hilbert Space.- C*-Algebras.- Von Neumann Algebras.- Further Structure.- K-Theory and Finiteness.

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Book SynopsisLes Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Ce livre est le cinquième du traité ; il est consacré aux bases de l’analyse fonctionnelle. Il contient en particulier le théorème de Hahn-Banach et le théorème de Banach-Steinhaus. Il comprend les chapitres: -1. Espaces vectoriels topologiques sur un corps value; -2. Ensembles convexes et espaces localement convexes; -3. Espaces d’applications linéaires continues; -4. La dualité dans les espaces vectoriels topologiques; -5. Espaces hilbertiens (théorie élémentaire). Il contient également des notes historiques. Ce volume a été publié en 1981.Table of ContentsEspaces vectoriels topologiques sur un corps valué.- Ensembles convexes et espaces localement convexes.- Espaces d'applications linéaires continues.- La dualité dans les espaces vectoriels topologiques.- Espaces hilbertiens (théorie élémentaire).

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Book SynopsisSince its inception by von Neumann 70 years ago, the theory of operator algebras has become a rapidly developing area of importance for the understanding of many areas of mathematics and theoretical physics. Accessible to the non-specialist, this first part of a three-volume treatise provides a clear, carefully written survey that emphasizes the theory's analytical and topological aspects.Trade Review"... These three bulky volumes [EMS 124, 125, 127], written by one of the most prominent researchers of the area, provide an introduction to this repidly developing theory. ... These books can be warmly recommended to every graduate student who wants to become acquainted with this exciting branch of matematics. Furthermore, they should be on the bookshelf of every researcher of the area." (László Kérchy, Acta Scientiarum Mathematicarum, Vol. 69, 2003)Table of ContentsFundaments of Banach Algebras and C*-Algebras.- Topologies and Density Theorems in Operator Algebras.- Conjugate Spaces.- Tensor Products of Operator Algebras and Direct Integrals.- Types of von Neumann Algebras and Traces.- Appendix: Polish Spaces and Standard Borel Spaces.

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Book SynopsisThis is an introduction to a very active field of research, on the boundary between mathematics and physics. It is aimed at graduate students and researchers in geometry and string theory. Proofs or sketches are given for many important results. From the reviews: "An excellent introduction to current research in the geometry of Calabi-Yau manifolds, hyper-Kähler manifolds, exceptional holonomy and mirror symmetry....This is an excellent and useful book." --MATHEMATICAL REVIEWS Trade ReviewFrom the reviews: "Summer schools in Nordfjordeid, Norway, have been organized regularly since 1996. Topics vary each year but there are always three series of lectures by invited experts with evening exercises. […] The themes of all three contributions are interrelated and together they give a nice introduction into a very interesting field of research on the border between mathematics and physics. l would like to strongly recommend the book to anybody interested in the topic." (jbu) European Mathematical Society Newsletter, Sept. 2004, p. 44 "[...] This book is an excellent introduction to current research in the geometry of Calabi-Yau manifolds, hyper-Kähler manifolds, exceptional holonomy and mirror symmetry. [...] This is an excellent and useful book. The different chapters mostly fit together reasonably well [...]. It will be popular with anyone trying to learn about Kähler manifolds with c1=0 and mirror symmetry." Richard P. Thomas, Mathematical Reviews, Clippings from Issue 2004c "Summer schools in Nordfjordeid, Norway, have been organized regularly since 1996. … The school held in June 2001 was devoted to recent interaction between differential and algebraic geometry. The book consists of notes written by lecturers of the corresponding three series of lectures. … they give a nice introduction into a very interesting field of research on the border between mathematics and physics. I would like to strongly recommend the book to anybody interested in the topic." (EMS, September, 2004) "This book is an excellent introduction to current research in the geometry of Calabi-Yau manifolds, hyper-Kähler manifolds, exceptional holonomy and mirror symmetry. … the choice of topics is a sensible one. … This is an excellent and useful book. … It will be popular with anyone trying to learn about Kähler manifolds with c1 = 0 and mirror symmetry." (Richard P. Thomas, Mathematical Reviews, 2004 c)Table of ContentsI. Riemannian Holonomy Groups and Calibrated Geometry.- 1 Introduction.- 2 Introduction to Holonomy Groups.- 3 Berger’s Classification of Holonomy Groups.- 4 Kahler Geometry and Holonomy.- 5 The Calabi Conjecture.- 6 The Exceptional Holonomy Groups.- 7 Introduction to Calibrated Geometry.- 8 Calibrated Submanifolds in ?n.- 9 Constructions of SL m-folds in ?m.- 10 Compact Calibrated Submanifolds.- 11 Singularities of Special Lagrangian m-folds.- 12 The SYZ Conjecture, and SL Fibrations.- II. Calabi-Yau Manifolds and Mirror Symmetry.- 13 Introduction.- 14 The Classical Geometry of Calabi-Yau Manifolds.- 15 Kahler Moduli and Gromov-Witten Invariants.- 16 Variation and Degeneration of Hodge Structures.- 17 A Mirror Conjecture.- 18 Mirror Symmetry in Practice.- 19 The Strominger-Yau-Zaslow Approach to Mirror Symmetry.- III. Compact Hyperkähler Manifolds.- 20 Introduction.- 21 Holomorphic Symplectic Manifolds.- 22 Deformations of Complex Structures.- 23 The Beauville-Bogomolov Form.- 24 Cohomology of Compact Hyperkähler Manifolds.- 25 Twistor Space and Moduli Space.- 26 Projectivity of Hyperkähler Manifolds.- 27 Birational Hyperkähler Manifolds.- 18 The (Birational) Kahler Cone.- References.

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Book SynopsisEinstein's equations stem from General Relativity. In the context of Riemannian manifolds, an independent mathematical theory has developed around them. This is the first book which presents an overview of several striking results ensuing from the examination of Einstein’s equations in the context of Riemannian manifolds. Parts of the text can be used as an introduction to modern Riemannian geometry through topics like homogeneous spaces, submersions, or Riemannian functionals.Trade ReviewFrom the reviews: "[...] an efficient reference book for many fundamental techniques of Riemannian geometry. [...] despite its length, the reader will have no difficulty in getting the feel of its contents and discovering excellent examples of all interaction of geometry with partial differential equeations, topology, and Lie groups. Above all, the book provides a clear insight into the scope and diversity of problems posed by its title."S.M. Salamon in MathSciNet 1988 "It seemed likely to anyone who read the previous book by the same author, namely Manifolds all of whose geodesic are closed, that the present book would be one of the most important ever published on Riemannian geometry. This prophecy is indeed fulfilled."T.J. Wilmore in Bulletin of the London Mathematical Society 1987 "Einstein Manifolds is accordingly described as Besse’s second book … . there is no doubt that Einstein Manifolds is a magnificient work of mathematical scholarship. … It is truly a seminal work on an incomparably fascinating and important subject." (Michael Berg, MathDL, March, 2008) "The present book is intended to be a complete reference book. … The book under review serves several purposes. It is an efficient reference for many fundamental techniques of Riemannian geometry as well as excellent examples of the interaction of geometry with partial differential equations, topology and Lie groups. Certainly the monograph provides a clear insight into the scope and diversity of problems posed by its title." (Adela-Gabriela Mihai, Zentralblatt MATH, Vol. 1147, 2008)Table of ContentsBasic Material.- Basic Material (Continued): Kähler Manifolds.- Relativity.- Riemannian Functionals.- Ricci Curvature as a Partial Differential Equation.- Einstein Manifolds and Topology.- Homogeneous Riemannian Manifolds.- Compact Homogeneous Kähler Manifolds.- Riemannian Submersions.- Holonomy Groups.- Kähler-Einstein Metrics and the Calabi Conjecture.- The Moduli Space of Einstein Structures.- Self-Duality.- Quaternion-Kähler Manifolds.- A Report on the Non-Compact Case.- Generalizations of the Einstein Condition.

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Book SynopsisOver the years, this book has become a standard reference and guide in the set theory community. It provides a comprehensive account of the theory of large cardinals from its beginnings and some of the direct outgrowths leading to the frontiers of contemporary research, with open questions and speculations throughout.Table of ContentsPreliminaries.- Beginnings.- Partition Properties.- Forcing and Sets of Reals.- Aspects of Measurability.- Strong Hypotheses.- Determinacy.

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Book SynopsisIn many areas of mathematics some “higher operations” are arising. These havebecome so important that several research projects refer to such expressions. Higher operationsform new types of algebras. The key to understanding and comparing them, to creating invariants of their action is operad theory. This is a point of view that is 40 years old in algebraic topology, but the new trend is its appearance in several other areas, such as algebraic geometry, mathematical physics, differential geometry, and combinatorics. The present volume is the first comprehensive and systematic approach to algebraic operads. An operad is an algebraic device that serves to study all kinds of algebras (associative, commutative, Lie, Poisson, A-infinity, etc.) from a conceptual point of view. The book presents this topic with an emphasis on Koszul duality theory. After a modern treatment of Koszul duality for associative algebras, the theory is extended to operads. Applications to homotopy algebra are given, for instance the Homotopy Transfer Theorem. Although the necessary notions of algebra are recalled, readers are expected to be familiar with elementary homological algebra. Each chapter ends with a helpful summary and exercises. A full chapter is devoted to examples, and numerous figures are included. After a low-level chapter on Algebra, accessible to (advanced) undergraduate students, the level increases gradually through the book. However, the authors have done their best to make it suitable for graduate students: three appendices review the basic results needed in order to understand the various chapters. Since higher algebra is becoming essential in several research areas like deformation theory, algebraic geometry, representation theory, differential geometry, algebraic combinatorics, and mathematical physics, the book can also be used as a reference work by researchers.Trade ReviewFrom the reviews:“It is a welcome addition to the existing literature and will, no doubt, become a standard reference for many authors working in this quickly developing field. … it is an impressive piece of work, which gives a comprehensive account of the foundations of the theory of algebraic operads, starting from the most basic notions, such as associative algebras and modules. It will be of interest to a broad swath of mathematicians: from undergraduate students to experts in the field.” (Andrey Yu. Lazarev, Mathematical Reviews, March, 2013)Table of ContentsPreface.- 1.Algebras, coalgebras, homology.- 2.Twisting morphisms.- 3.Koszul duality for associative algebras.- 4.Methods to prove Koszulity of an algebra.- 5.Algebraic operad.- 6 Operadic homological algebra.- 7.Koszul duality of operads.- 8.Methods to prove Koszulity of an operad.- 9.The operads As and A\infty.- 10.Homotopy operadic algebras.- 11.Bar and cobar construction of an algebra over an operad.- 12.(Co)homology of algebras over an operad.- 13.Examples of algebraic operads.- Apendices: A.The symmetric group.- B.Categories.- C.Trees.- References.- Index.- List of Notation.

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Book SynopsisSignal processing is the discipline of extracting information from collections of measurements. To be effective, the measurements must be organized and then filtered, detected, or transformed to expose the desired information. Distortions caused by uncertainty, noise, and clutter degrade the performance of practical signal processing systems.In aggressively uncertain situations, the full truth about an underlying signal cannot be known. This book develops the theory and practice of signal processing systems for these situations that extract useful, qualitative information using the mathematics of topology -- the study of spaces under continuous transformations. Since the collection of continuous transformations is large and varied, tools which are topologically-motivated are automatically insensitive to substantial distortion. The target audience comprises practitioners as well as researchers, but the book may also be beneficial for graduate students.Trade ReviewFrom the book reviews:“This text provides a nice exposition of the topological ideas used to extract information from signals and the practical details of signal processing. … Robinson’s intended audience is first year graduate students in both engineering and mathematics, and advanced undergraduates. … Throughout the text there are numerous examples and diagrams. Each chapter also ends with some open questions. These features make the book quite readable.” (Michele Intermont, MAA Reviews, February, 2015)“Three major goals for this book: firstly to show that topological invariants provide qualitative information about signals that is both relevant and practical, second to show that the signal processing concepts of filtering, detection, and noise correspond respectively to the concepts of sheaves, functoriality and sequences, and third to advocate for the use of sheaf theory in signal processing. … The target audience is practitioners so that the theoretical notions are covered with the practitioner in mind with motivations emphasized.” (Jonathan Hodgson, zbMATH, Vol. 1294, 2014)Table of ContentsIntroduction and informal discussion.- Parametrization.- Signals.- Detection.- Transforms.- Noise.

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Book SynopsisDieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.Table of ContentsErster Teil. Grundbegriffe der mengentheoretischen Topologie.- Erstes Kapitel: Topologische und metrische Räume.- Zweites Kapitel: Kompakte Räume.- Zweiter Teil. Topologie der Komplexe.- Drittes Kapitel: Polyeder und ihre Zellenzerlegungen.- Viertes Kapitel: Eckpunkt- und Koeffizientenbereiche.- Fünftes Kapitel: Bettische Gruppen.- Sechstes Kapitel: Zerspaltungen und Unterteilungen von Komplexen.- Siebentes Kapitel: Spezielle Fragen aus der Theorie der Komplexe.- Dritter Teil. Topologische Invarianzsätze und anschließende Begriffsbildungen.- Achtes Kapitel: Simpliziale Approximationen stetiger Abbildungen. Stetige Zyklen.- Neuntes Kapitel: Kanonische Verschiebungen. Nochmals Invarianz der Dimensionszahl und der Bettischen Gruppen. Allgemeiner Dimensionsbegriff.- Zehntes Kapitel: Der Zerlegungssatz für den Euklidischen Raum. Weitere Invarianzsätze.- Vierter Teil. Verschlingungen im Euklidischen Raum. Stetige Abbildungen von Polyedern.- Elftes Kapitel: Verschlingungstheorie. Der Alexandersche Dualitätssatz.- Zwölftes Kapitel: Der Brouwersche Abbildungsgrad. Die Kroneckersche Charakteristik.- Dreizehntes Kapitel: Homotopie- und Erweiterungssätze für Abbildungen.- Vierzehntes Kapitel: Fixpunkte.- Anhang I. Abelsche Gruppen.- § 1 Allgemeine Begriffe und Sätze.- § 2. Moduln (Freie Gruppen).- § 4. Gruppen mit endlich-vielen Erzeugenden.- § 5. Charaktere.- § 2. Konvexe Mengen.- § 3. Konvexe und baryzentrische Hüllen. Simplexe.- § 4. Konvexe Raumstücke. Konvexe Zellen.- 1. Nachtrag: Zentralprojektion.- 2. Nachtrag: Der Schwerpunkt.- Verzeichnis der topologischen Bücher.

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Book SynopsisSome years ago a conference on l-adic cohomology in Oberwolfach was held with the aim of reaching an understanding of Deligne's proof of the Weil conjec­ tures. For the convenience of the speakers the present authors - who were also the organisers of that meeting - prepared short notes containing the central definitions and ideas of the proofs. The unexpected interest for these notes and the various suggestions to publish them encouraged us to work somewhat more on them and fill out the gaps. Our aim was to develop the theory in as self­ contained and as short a manner as possible. We intended especially to provide a complete introduction to etale and l-adic cohomology theory including the monodromy theory of Lefschetz pencils. Of course, all the central ideas are due to the people who created the theory, especially Grothendieck and Deligne. The main references are the SGA-notes [64-69]. With the kind permission of Professor J. A. Dieudonne we have included in the book that finally resulted his excellent notes on the history of the Weil conjectures, as a second introduction. Our original notes were written in German. However, we finally followed the recommendation made variously to publish the book in English. We had the good fortune that Professor W. Waterhouse and his wife Betty agreed to translate our manuscript. We want to thank them very warmly for their willing involvement in such a tedious task. We are very grateful to the staff of Springer-Verlag for their careful work.Table of ContentsI. The Essentials of Etale Cohomology Theory.- II. Rationality of Weil ?-Functions.- III. The Monodromy Theory of Lefschetz Pencils.- IV. Deligne’s Proof of the Weil Conjecture.- Appendices.- A I. The Fundamental Group.- A II. Derived Categories.- A III. Descent.

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Book SynopsisCe livre des Éléments de mathématique est consacré à la Topologie algébrique. Les quatre premiers chapitres présentent la théorie des revêtements d'un espace topologique et du groupe de Poincaré. On construit le revêtement universel d'un espace connexe pointé délaçable et on établit l'équivalence de catégories entre revêtements de cet espace et actions du groupe de Poincaré. On démontre une version générale du théorème de van Kampen exprimant le groupoïde de Poincaré d'un espace topologique comme un coégalisateur de diagrammes de groupoïdes. Dans de nombreuses situations géométriques, on en déduit une présentation explicite du groupe de Poincaré.Table of ContentsMode d'Emploi.- Introduction.- Chapitre I. Revêtements.- 1. Produits fibrés et carrés cartésiens.- 2. Applications étales.- 3. Faisceaux.- 4. Revêtements.- 5. Revêtements principaux.- 6. Espaces simplement connexes.- Exercices.- Chapitre II. Groupoïdes.- 1. Carquois.- 2. Graphes.- 3. Groupoïdes.- 4. Homotopies.- 5. Coégalisateur.- Exercices.- Chapitre III. Homotopie et Groupoïdes de Poincaré.- 1. Homotopies, homéotopies.- 2. Homotopie et chemins.- 3. Groupoïde de Poincaré.- 4. Homotopie et revêtements.- 5. Homotopie et revêtements (cas des espaces localement connexes par arcs).- Exercices.- Chapitre IV. Espaces Delaçables.- 1. Espaces délaçables.- 2. Groupes de Poincaré des espaces délaçables.- 3. Groupes de Poincaré des groupes topologiques.- 4. Théorie de la descente.- 5. Théorème de van Kampen.- 6. Espaces classifiants.- Exercices.- Index des notations.- Index terminologique.

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