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2017/2018

Научно-исследовательский семинар "Основные спектральные последовательности 2"

Статус: Дисциплина общефакультетского пула
Когда читается: 1, 2 модуль
Язык: английский

Программа дисциплины

Аннотация

A spectral sequence is a tool of homological algebra that has many applications in algebra, algebraic geometry, and algebraic topology. Roughly speaking, a spectral sequence is a system for keeping track of collections of exact sequences that have maps between them. There are many definitions of spectral sequences and many slight variations that are useful for certain purposes. The most common type is a "first quadrant cohomological spectral sequence". Our goal is to cover in more detail the Adams and Adams-Novikov spectral sequences, as well as some prerequisites for constructing these sequences, and some applications. This course is elective. Pre-requisites: a working knowledge of basic algebraic topology plus exact couples and the "usual" Leray-Serre spectral sequence.
Цель освоения дисциплины

Цель освоения дисциплины

  • To introduce Main Spectral Sequences
  • To offer an opportunity to solve advanced problems in this area
Результаты освоения дисциплины

Результаты освоения дисциплины

  • Know fundamental facts, constructions and concepts about spectral sequences
  • Have an experience of problem solving
Содержание учебной дисциплины

Содержание учебной дисциплины

  • Stable Adams spectral sequence: overview and a sketch of the construction
  • The Steenrod algebra. Cohomology of Eilenberg-MacLane spaces with field coefficients
  • Spectra and their basic properties
  • A digression: duality for spectra
  • The Adams resolution and the Adams spectral sequence modulo a prime for spectra
  • A digression: basics of cobordism theory. The Pontrjagin-Thom construction
  • An application of the stable Adams spectral sequence: calculating cobordism rings
  • Сonstructions of the unstable Adams spectral sequence
  • The bar construction and the Adams-Novikov spectral sequence
Элементы контроля

Элементы контроля

  • неблокирующий Created with Sketch. Homeworks
  • неблокирующий Created with Sketch. Final Exam
Промежуточная аттестация

Промежуточная аттестация

  • Промежуточная аттестация (2 модуль)
    0.5 * Final Exam + 0.5 * Homeworks
Список литературы

Список литературы

Рекомендуемая основная литература

  • McLean, M. (2010). A spectral sequence for symplectic homology. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.1011.2478

Рекомендуемая дополнительная литература

  • James, I. M. Handbook of Algebraic Topology: North Holland: p.1324 , 1995. - ISBN 978-0-444-81779-2