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2017/2018

Научно-исследовательский семинар "Основные спектральные последовательности 2"

Статус: Дисциплина общефакультетского пула
Когда читается: 1, 2 модуль
Язык: английский
Кредиты: 3

Course Syllabus

Abstract

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.
Learning Objectives

Learning Objectives

  • To introduce Main Spectral Sequences
  • To offer an opportunity to solve advanced problems in this area
Expected Learning Outcomes

Expected Learning Outcomes

  • Know fundamental facts, constructions and concepts about spectral sequences
  • Have an experience of problem solving
Course Contents

Course Contents

  • 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
Assessment Elements

Assessment Elements

  • non-blocking Homeworks
  • non-blocking Final Exam
Interim Assessment

Interim Assessment

  • Interim assessment (2 module)
    0.5 * Final Exam + 0.5 * Homeworks
Bibliography

Bibliography

Recommended Core Bibliography

  • 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

Recommended Additional Bibliography

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