Tag Archives: advance reading

Class presentations

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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.

Suggested reading for lecture 10

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In lecture 10 (tomorrow) I will briefly discuss

  • Diffeomorphism invariance of pseudodifferential operators, and the corollary that the notion of “pseudodifferential operator on a compact manifold” is well defined.
  • L^2 and Sobolev space continuity of pseudodifferential operators.

That should be all the analytical machinery that we need.  I will then go on to explain the outline of the proof of the Lefschetz theorem.  We will fill in the details next week.

Please read Section 2 of Atiyah-Bott paper I, which gives their proof outline, in preparation for the class.

 

Comments for lecture 6

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In this lecture we began the study of the analysis which leads to the Hodge theorem and other key results about elliptic operators.  In subsequent lectures we’ll make use of the work of specialists on pseudodifferential operators (Kohn, Nirenberg, Hormander, Taylor) to prove the estimates we discuss in this lecture.  As Bott said, “If you have an electrical problem, you consult an electrician.  If you have an elliptical problem, you consult an elliptician.”  (Hormander was the go-to elliptician for Atiyah and Bott.)

We began by discussing the Sobolev spaces.   The Sobolev space W^k(M) on a compact manifold \(M\) is the space of functions whose first \(k\) derivatives (considered as distributions) belong to the Hilbert space \(L^2(M)\).  (In order to avoid an explicit discussion of distribution theory we gave in class an alternative, but equivalent, description which is based on Fourier analysis.)  These form a scale of spaces  (that is, \( W^{k+1}\subseteq W^k\) ) with the following key properties.

  • \( W^{k+s}\subseteq C^k \) whenever \(s> {\frac12}\dim M\);
  • A differential operator of order \(s\) extends to a bounded linear map \(W^{k+s}\to W^k\), for every \(k\);
  • The inclusion \(W^{k+\epsilon}\to W^k \) is a compact linear operator for every \(\epsilon>0\) (Rellich-Kondrakov theorem).

As a result of the second bullet point we can define an operator of order \(\le s\) (not necessarily differential!) to be any linear map on smooth functions that extends to a continuous map \(W^{k+s}\to W^k \) for every \(k\).   Every differential operator of order \(\le s\) is an operator of order \(\le s\), but the converse is of course false.  In particular there exist “operators” of negative order.  That makes the following statement sensible.

Fundamental Elliptic Theorem  If \(D\) is elliptic differential of order \(s\), there exists \(Q\) of order \(-s\) such that

\[ I-DQ = R_1,\quad I-QD = R_2 \]

are smoothing operators (i.e. of order \(-\infty\)).

In the next couple of lectures we will see how the pseudodifferential calculus is used to construct the parametrix \(Q\).   You could read about this in the Kohn-Nirenberg papers (see Atiyah-Bott’s bibliography) or in M. Taylor,Pseudodifferential operators, Chapter II (very terse).

 

Comments for lecture 4

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In this session we got around to stating the main theorem of the Atiyah-Bott Lefschetz theorem papers.  We imagine that we have an elliptic complex (as previously defined) and an endomorphism of that complex.  Since an elliptic complex is made up of two (or maybe three) components – manifold, bundles, operators – a geometric endomorphism similarly has several components: a smooth map f\colon M\to M , a bunch of bundle maps from \(f^*E\to E\) for all the bundles \(E\) appearing in the complex, and the whole to give a (co)chain map and therefore an induced map on cohomology.  In these circumstances we can define the Lefschetz number to be the alternating sum of the traces of the induced cohomology maps.  The main theorem expresses this Lefschetz number as a sum over the fixed points of \(f\) (under the assumption that these fixed points are simple).

We reviewed what this theorem tells us in two cases: when we deal with the de Rham complex and an arbitrary smooth map \(M\to M\) (when we recover the classical Lefschetz fixed point theorem), and when we are considering the Dolbeault complex of a complex manifold \(M\), and \(f\) is a holomorphic map \(M\to M\) (this case supplied Atiyah and Bott’s original motivation).  There is a very pretty example on page 460 of the second paper in the series where they consider a self-map of projective space induced by multiplying each homogeneous coordinate by a different scalar factor.  Then the fixed point theorem gives an algebraic identity

\[1 = \sum_{i=0}^n \frac{\gamma_i^n}{\prod_{j\neq i} (\gamma_i-\gamma_j)}\]

for distinct nonzero complex numbers \(\gamma_0,\ldots,\gamma_n \). I wrote a post on my main blog explaining how this identity can be derived from the familiar Lagrange interpolation formula of numerical analysis.

In next class session we will return to the de Rham complex but we will also introduce a Riemannian metric, which makes it possible to get more subtle topological invariants from de Rham theory.  It would be appropriate to read section 6 of paper II in preparation for this session.

Reading for lecture 3

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In lecture 3, in addition to explaining the global interpretation of the symbol (which is where we left things last time), we will also look at the key examples of the de Rham and Dolbeault complexes.

Reading would be section 3 and the beginning of section 4 of the second paper in the Lefschetz theorem series.  These of course do more than describe these examples, they also apply the main theorem to the examples; but it is only the first of these, the description of the examples, that will interest us in this lecture.