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Обычная версия сайта
2019/2020

Научно-исследовательский семинар "Введение в эллиптические операторы"

Лучший по критерию «Полезность курса для Вашей будущей карьеры»
Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»
Лучший по критерию «Новизна полученных знаний»
Статус: Дисциплина общефакультетского пула
Когда читается: 1, 2 модуль
Язык: английский
Кредиты: 6

Course Syllabus

Abstract

In this seminar we will cover the basics of the theory of elliptic operators, examples of which include the de Rham and Dolbeault operators, as well as their generalisations such as the Dirac operator. We will explain how several seemingly unrelated results in geometry and topology (e.g. the Hirzebruch and Rokhlin signature theorems and the Riemann-Roch theorem) all follow from the general index formula by M. Atiyah and I. Singer, and sketch a proof of the latter.
Learning Objectives

Learning Objectives

  • Learn the basic theory and classical examples of elliptic differential operators, as well as related techniques and ideas
Expected Learning Outcomes

Expected Learning Outcomes

  • To be able to calculate the index of standard elliptic operators using Chern classes and to apply index theorems and related results in their own research
Course Contents

Course Contents

  • Vector bundles and characteristic classes: a summary of results
  • Differential operators: the definition and first examples
  • Elliptic operators: the definition and basic properties
  • Riemannian metrics on manifolds and the de Rham operator
  • The signature operator
  • Complex manifolds and the Dolbeault operator
  • Clifford algebras and their representations
  • Reduction of the structure group of a vector bundle. Spin structures on vector bundles
  • Dirac operators. Constructing Dirac operators using Spin structures
  • Elliptic regularity and related results about elliptic operators on compact manifolds
  • First applications: the de Rham and Hodge decomposition theorems; the Serre duality
  • The Atiyah-Singer index formula
  • Applications of the index formula: the Riemann–Roch theorem, the Hirzebruch signature theorem, V. Rokhlin’s signature theorem
Assessment Elements

Assessment Elements

  • blocking home exam
  • non-blocking Работа на семинаре
Interim Assessment

Interim Assessment

  • Interim assessment (2 module)
    100% home examination
Bibliography

Bibliography

Recommended Core Bibliography

  • Liviu I. Nicolaescu, Instructor Liviu, & I. Nicolaescu. (2005). Notes on the Atiyah-Singer Index Theorem. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.CACC3B54
  • Liviu I. Nicolaescu. (2013). Notes on the Atiyah-Singer Index Theorem. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.C8AE2AFD

Recommended Additional Bibliography

  • Allen Hatcher. (2001). Algebraic topology. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.5FA4491E
  • Lawson, H. B., & Michelsohn, M.-L. (1989). Spin Geometry (PMS-38), Volume 38. Princeton, N.J.: Princeton University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1232568
  • Milnor, J. W., & Stasheff, J. D. (1974). Characteristic Classes. (AM-76), Volume 76. Princeton, N.J.: Princeton University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1432981
  • Palais, R. S., Atiyah, M. F., & Institute for Advanced Study, P. N. J. . (1965). Seminar on Atiyah-Singer Index Theorem. (AM-57), Volume 57. Princeton, N.J.: Princeton University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1432909
  • Wells, R. O., & American Mathematical Society. (1986). Several Complex Variables. Providence, R.I.: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=772701