This week’s schedule

Both classes this week are student presentations.   My understanding is that Qijun will present on Tuesday and Angel on Thursday (but if you guys want to switch around, that would be fine with me).

Qijun is going to present on “Analytic K-homology”.  Here the idea (as usual, due to Atiyah) is that by abstracting the functional-analytic properties of pseudodifferential operators, one can develop a model for a homology theory that is dual to K-theory.

Angel is going to present on “Elliptic operators on homogeneous spaces”, a famous paper by Bott.  What does the index theorem say in the “most symmetrical case”, where the underlying manifold is a homogeneous space of a compact Lie group?

 

Comments on Lecture 22 (today)

So today we discussed the multiplicative axiom, the last (and most important) of the three “B” axioms that characterize the analytic index.   This axiom says (in its simplest version) that if \(M=X\times Y\) is a product of manifolds and \(a\in K(TX)\), \(b\in K(TY)\), then

\[ \newcommand{\Index}{{\rm Index}} \Index(ab) = \Index(a)\Index(b). \]

In class we discussed the case where the symbols \(a,b\) are represented by differential operators and, following the original paper, we argued using the “double complex” definition of the product on K-theory.  (Jinpeng’s notes, which I’ll post in a moment, give an account of what we said in class.)  Here I’d like to explain how the formalism of \(\newcommand{\Z}{{\mathbb Z}} \Z_2\)-graded operators and vector spaces allows one to make the algebra more concise and conceptual.   (This then becomes part of the formalism that is taken over for Kasparov’s KK-theory.) Continue reading

Lecture 19

Here are the notes for lecture 19

Lecture 19

Tomorrow we have the first student presentation, by Damien.  This is about the index of families and its use in proving Bott periodicity.  The key paper can be found online at this link. The reference is

Atiyah, M.F. 1968. “Bott Periodicity and the Index of Elliptic Operators.” Oxford Quarterly Journal of Mathematics 19: 113–40.

 

 

Class presentations

Hi all.  Here is a PRELIMINARY schedule of student class presentations and so on, for the rest of the semester.  I will try to update as we go along.

November 4th – Damien – Families index and Bott periodicity

November 11th and 13th (I will be away) – David and Jinpeng – Characteristic classes and genera

December 2 – Alok (Vector fields on manifolds) AND Angel (Elliptic operators on homogeneous spaces)

December 9 – Qijun – Analytic K-homology

 

References

 

Atiyah, M. F., and I. M. Singer. 1971. “The Index of Elliptic Operators: IV.” The Annals of Mathematics, Second Series, 93 (1): 119–38. (section 1)

Atiyah, M.F. 1968. “Bott Periodicity and the Index of Elliptic Operators.” Oxford Quarterly Journal of Mathematics 19: 113–40.

Atiyah, M. F., and I. M. Singer. 1968. “The Index of Elliptic Operators: III.” The Annals of Mathematics, Second Series, 87 (3): 546–604. (Sections 1 and 2)

Milnor, J.W., and J.D. Stasheff. 1974. Characteristic Classes. Vol. 76. Annals of Mathematics Studies. Princeton University Press, Princeton, N.J.

Atiyah, M.F. 1970. “Vector Fields on Manifolds.” Arbeitsgemeinschaft Für Forschung Des Landes Nordrhein Westfalen 200: 7–24.

Bott, R. 1965. “The Index Theorem for Homogeneous Differential Operators.” In Differential and Combinatorial Topology: A Symposium in Honor of Marston Morse, edited by S.S. Cairns, 167–86. Princeton University Press, Princeton, N.J.

Atiyah, M.F. 1969. “Global Theory of Elliptic Operators.” In Proceedings of the International Symposium on Functional Analysis, Tokyo, 21–30. University of Tokyo Press.

Brown, L. G., R. G. Douglas, and P. A. Fillmore. 1977. “Extensions of C*-Algebras and K-Homology.” Annals of Mathematics 105 (2): 265–324. doi:10.2307/1970999.

Kasparov, G.G. 1975. “Topological Invariants of Elliptic Operators I: K-Homology.” Mathematics of the USSR — Izvestija 9: 751–92.