Правительство Российской ФедерацииФедеральное государственное автономное образовательное учреждение высшего профессионального образования "Национальный исследовательский университет "Высшая школа экономики"Факультет МатематикиПрограмма дисциплины Foundations of homotopical algebra.для направления 010100.68 «Математика» подготовки магистрамагистерская программа «Математика»Автор программы: Brav, Chris, доцент, chris.i.brav@gmail.com FILLIN "MERGEFORMAT" Рекомендована секцией УМС по математике «___»____________ 2014 г.Председатель С.М. ХорошкинУтверждена УС факультета FILLIN "MERGEFORMAT" математики «___»_____________2014 г.Ученый секретарь Ю.М. Бурман_____________________Москва, 2014 Настоящая программа не может быть использована другими подразделениями университета и другими вузами без разрешения кафедры-разработчика программы.Explanatory remarksPrerequisites: A minimal acquaintance with basic ideas in algebraic topology, homological algebra, and category theory, such as chain complexes, homotopy, categories, functors, and natural transformations. These topics will be reviewed as necessary.Annotation.The course 'Foundations of homotopical algebra' is a part of the MS program 010100.68 Mathematics.Homotopical algebra provides a general foundation for treating mathematical objects up to 'weak equivalence' (for example nice topological spaces up to homotopy or chain complexes up to homology isomorphism) and for showing that different mathematical objects are nothing but different models of the same underlying notion (for example, so-called simplicial sets provide a purely combinatorial model for homotopy types of nice topological spaces). While homotopical algebra has traditionally been the domain of topologists, it is becoming increasingly important in algebraic geometry.In the first module of the course we review the singular chain complex of a topological space and note how it is a kind of linearisation of a more basis object, namely the singular set of a topological space. We then discuss the topological realisation of a simplicial set, which produces a topological space from a simplicial set, and begin to explore basic examples and constructions in simplicial homotopy theory (Kan complexes and fibrations, the classifying space of a group, simplicial homotopy groups). Finally, we explain the precise sense in which topological spaces and simplicial sets are different models of the same basic notion, namely homotopy type. We show in particular that topologically defined homotopy groups and simplicially defined homotopy groups are isomorphic.In the second module of the course we generalise the relation between topological spaces and simplicial sets and see it as a 'Quillen equivalence of model categories'. We shall illustrate the basics of the theory of model categories (weak-equivalences, fibrations, cofibrations, the homotopy category of a model category, Quillen adjunctions and equivalences, homotopy limits and colimits) with various algebraic examples (chain complexes, differential graded algebras, differential graded Lie algebras). If time permits, we shall formulate the basic statements of rational homotopy theory, which provides an efficient means of computing certain homotopy invariants of a topological spaces in terms of differential graded Lie algebras.Goals of the course and its position within the MS program as a wholeTargets of the course:Introducing students to simplicial sets as models for homotopy theory of topological spaces.Introducing students to the generalised notion of homotopy theory, which studies mathematical objects up to 'weak equivalence'.Goals achieved by the course:Study of basic simplicial constructions such as the singular set of a topological space, the geometric realisation of a simplicial set, simplicial homotopy groups, and classifying spaces. Proving that the simplicial set of a topological space is a faithful combinatorial model of its homotopy theory.Introduction of the general theory of model categories (homotopy categories, Quillen adjunctions and equivalences, homotopy limits and colimits) and basic examples (topological spaces, simplicial sets, chain complexes, differential graded Lie algebras).Study of the rational homotopy theory of topological spaces in terms of differential graded Lie algebras.Thematic plan of the courseNTopicTotal hoursClass hoursHomework1Review of the singular chain complex and introduction of simplicial sets and basic examples9452Review of category theory, functorial relation between topological spaces and simplicial sets9453Simplicial homotopy, Kan complexes, simplicial homotopy groups9454Comparison with topological homotopy groups, first examples of model structures9455General definition of model category and example of chain complexes9456Homotopy categories, Quillen adjunctions and equivalences, homotopy limits and colimits2110117The model category of differential graded Lie algebras14688Introduction to rational homotopy theory1046 Total:904050Testing students’ knowledge of the courseWritten homework every other week.A written exam at the end of each module, consisting of a selection of the homework problems.Scoring64 percent for the homework36 percent for the exams (18 percent each). SyllabusThe singular set of a topological space and its geometric realisationReview of the singular chain complex of a topological space. Definition of the singular set of a topological space and the general definition of a simplicial set and its geometric realisation. The singular set/geometric realisation adjunction. Review of adjoint functors, as necessary.Kan complexes, simplicial groups, examplesKan complexes, simplicial homotopy of maps between simplicial sets, the classifying space of agroup, simplicial homotopy groupsComparison between homotopy theory of simplicial sets and of topological spacesProof of the isomorphism between homotopy groups of a topological space and of its singular set. Definition of the model structures on topological spaces and on simplicial sets and stating of the equivalence between them. Proof perhaps post-poned.General theory of model categoriesDefinition of a model category (weak equivalences, fibrations, cofibrations, and relations between them) and of its homotopy category. Derived functors. Example of chain complexes in full detail. Quillen adjunctions and equivalences. Example of groupoids and 1-truncated simplicial sets. Homotopy limits and colimits.Rational homotopy theory.Model category of differential graded Lie algebras. Rational homotopy type of a topological space in terms of differential graded Lie algebras.ReferencesP. Goerss and R. Jardine, Simplicial homotopy theory, BirkhauserA. Joyal and M. Tierney, Notes on simplicial homotopy theory, http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern47.pdfAQAAAGwAAAAAAAAAAAAAABoDAABjBAAAAAAAAAAAAACiUQAAnnMAACBFTUYAAAEAQEsAALQBAAAH
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Normal0610536008Microsoft Office Word05014falseНазвание1Министерство экономического развития и торговли НИУ ВШЭfalse7047falsefalse14.0000Министерство экономического развития и торговли tzelentcovauser22009-11-23T10:58:00Z2014-12-23T08:17:00Z2014-12-23T08:17:00Z