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This self-contained treatment of Morse Theory focuses on applications and is intended for a graduate course on differential or algebraic topology. The book is divided into three conceptually distinct parts. The first part contains the foundations of Morse theory (over the reals). The second part consists of applications of Morse theory over the reals, while the last part describes the basics and some applications of complex Morse theory, a.k.a. Picard-Lefschetz theory. This is the first textbook to include topics such as Morse-Smale flows, min-max theory, moment maps and equivariant cohomology, and complex Morse theory. The exposition is enhanced with examples, problems, and illustrations, and will be of interest to graduate students as well as researchers. The reader is expected to have some familiarity with cohomology theory and with the differential and integral calculus on smooth manifolds. Liviu Nicolaescu is Associate Professor of Mathematics at University of Notre Dame.

Differential geometry. Global analysis --- Differential topology --- differentiaal geometrie --- topologie

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Differential geometry. Global analysis --- Differential topology --- differentiaal geometrie --- topologie

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The recent revolution in differential topology related to the discovery of non-standard ("exotic") smoothness structures on topologically trivial manifolds such as R4 suggests many exciting opportunities for applications of potentially deep importance for the spacetime models of theoretical physics, especially general relativity. This rich panoply of new differentiable structures lies in the previously unexplored region between topology and geometry. Just as physical geometry was thought to be trivial before Einstein, physicists have continued to work under the tacit - but now shown to be inco

Differential topology. --- Space and time --- Mathematical physics. --- Mathematical models.

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The São Carlos Workshop on Real and Complex Singularities is the longest running workshop in singularities. It is held every two years and is a key international event for people working in the field. This volume contains papers presented at the eighth workshop, held at the IML, Marseille, July 19-23, 2004. The workshop offers the opportunity to establish the state of the art and to present new trends, new ideas and new results in all of the branches of singularities. This is reflected by the contributions in this book. The main topics discussed are equisingularity of sets and mappings, geometry of singular complex analytic sets, singularities of mappings, characteristic classes, classification of singularities, interaction of singularity theory with some of the new ideas in algebraic geometry imported from theoretical physics, and applications of singularity theory to geometry of surfaces in low dimensional euclidean spaces, to differential equations and to bifurcation theory.

Differential geometry. Global analysis --- Differential topology --- Geometry --- Ergodic theory. Information theory --- differentiaal geometrie --- geometrie --- topologie --- informatietheorie

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This collection of surveys is an out?ow from the 2006 conference Carthapos06 in Tunis (Tunisia). Apart from regular conference talks, ?ve survey talks formed the core of a workshop in Positivity, supported by the National Science Foun- tion.Theconferenceorganizers(KarimBoulabiar,GerardBuskes,andAbdelmajid Triki) decided to expand on the idea of core surveys and the nine surveys in this book are the harvest from that idea. Positivity derives from an order relation. Order relations are the mathem- ical tool for comparison. It is no surprise that seen in such very general light, the historyof Positivityis ancient.Archimedes,certainly,hadthe veryessenceofpo- tivityinmindwhenhediscoveredthelawofthelever.Hismethodofexhaustionto calculate areas uses a principle that nowadays carries his name, the Archimedean property. The surveys in this book are slanted into the direction that Archimedes took.Functionalanalysisisheavilyrepresented.Butthereismore.Latticeordered groupsappearinthearticlebyMartinezinthemodernjacketofframes.Henriksen and Banerjee write their survey on rings of continuous functions. Blecher and de Pagterineachoftheirpaperssurveypartsofnon-commutativefunctionalanalysis. Positiveoperatorsare the main topic in the papers by Curbera and Ricker, Schep, and Wickstead. And positive bilinear maps are the protagonists in the survey by Bu, Buskes, and Kusraev. The conference organizers (and editors of this volume) write about f-algebras. Carthapos06 was more than just a conference and workshop in Africa. It brought together researchers in Positivity from many directions of Positivity and form many corners of the world. This book can be seen as a culmination of their paths meeting in Tunisia, Africa.

Economics --- Ordered algebraic structures --- Differential topology --- Operator theory --- Functional analysis --- Mathematical analysis --- algebra --- analyse (wiskunde) --- functies (wiskunde) --- economie --- topologie