The Index of Elliptic Operators

Graduate Seminar on Global Analysis (S4B3)

Prof. Dr. Matthias Lesch, Dr. Boris Vertman

The Atiyah-Singer Index Theorem is one of the great achievements in Global Analysis. It has led to striking applications in geometry and topology, revealing deep connections between the two fields. In this seminar we will formulate and study the various forms of the Atiyah-Singer Index Theorem

  • The Index Theorem for elliptic pseudodifferential operators on closed manifolds
  • The equivariant Index Theorem
  • The Index Theorem for families of operators
  • The Index Theorem for operators equivariant with respect to the action of a Clifford algebra

We will follow the Chapter 3 of the book by Lawson and Michelson [LM], after recalling some basic prerequisites. Participants are also encouraged to study the original papers by Atiyah and Singer [AS1]-[AS5] which are a landmark of mathematical writing.

Prerequisites

Basic knowledge of Global Analysis and Elliptic Operators.

Literatur

  • [A] M. F. Atiyah "K-theory", Advanced Book Classics (2nd ed.), Addison-Wesley (1989)
  • [AS1] M. F. Atiyah, I. M. Singer "The Index of Elliptic Operators I", Ann. Math, Vol. 87, No. 3 (1968a)
  • [AS2] M. F. Atiyah, I. M. Singer "The Index of Elliptic Operators II", Ann. Math, Vol. 87 (2), No. 3 (1968)
  • [AS3] M. F. Atiyah, I. M. Singer "The Index of Elliptic Operators III", Ann. Math, Vol. 87 (2), No. 3 (1968)
  • [AS4] M. F. Atiyah, I. M. Singer "The Index of Elliptic Operators IV", Ann. Math, Vol. 93 (2), No. 1 (1971)
  • [AS5] M. F. Atiyah, I. M. Singer "The Index of Elliptic Operators V", Ann. Math, Vol. 93 (2), No. 1 (1971)
  • [BB] B. Booss, D. D. Bleeker "Topology and Analysis", New York (1985)
  • [H] D. Husemoller "Fibre bundles", McGraw-Hill (1966)
  • [KN] S. Kobayashi, K. Nomizu "Foundations of differential geometry", Vol I, Interscience Publ. (1963)
  • [LM] B. Lawson, L. Michelson "Spin geometry", Princeton University Press (1989)

Time and Venue

Tuesdays, 8:15, Mathematikzentrum, Room 0.008

Breakdown of Talks (with Handout)

  • Principal Bundles and the Spin Connection
    Definition of principal G-bundles and G-connections, the spin connection.
    Literature: Chapter 2, Section 1 and 4 of Lawson-Michelson [LM], [KN], [H].
    assigned to Jose Miguel Zapata Rolon

  • K-Theory
    Definition and properties, vector bundles, Atiyah-Bott periodicity theorem.
    Literature: Chapter 3, Section 9 of Lawson-Michelson [LM], Atiyah [A], [BB].
    assigned to Guglielmo Albanese

  • Elliptic Pseudodifferential operators and topological invariance of the index
    Chapter 3, Sections 2-5 and 7 in Lawson-Michelson [LM]
    assigned to Benedikt Sauer

  • Index of a family of elliptic operators and the equivariant G-index
    Chapter 3, Sections 8 and 9 in Lawson-Michelson [LM]
    assigned to Markus Land

  • The Clifford-index
    Chapter 3, Section 10 in Lawson-Michelson [LM]
    assigned to Markus Land

  • Multiplicative Sequences, the Chern Character,
    Thom isomorphisms and the Chern Character defect (2 talks)
    Chapter 3, Sections 11 and 12 in Lawson-Michelson [LM]
    assigned to Mark Pedron

  • The Atiyah-Singer Index Theorem (2 talks)
    Chapter 3, Section 13 in Lawson-Michelson [LM]
    assigned to Niklas Kulke

  • Fixed Point formulas for elliptic operators
    Chapter 3, Section 14 in Lawson-Michelson [LM]
    assigned to Christian Stümer

  • The index theorem for families and the Clifford-Index Theorem
    Chapter 3, Sections 15-16 in Lawson-Michelson [LM]
    assigned to Tristan Buckmaster



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