Sub-Riemannian Spectral Geometry

Vorlesung: Mo., Mi. 10-11.30 Uhr, (12.04 - 23.07)

Übungen: Mi. 14-15.30 Uhr

Lectures will be on zoom. Please contact us if interested.

Lecture Notes

Lecture outlines -

Part I
1. Motivations for sR geometry: Problems in mechanics (parking cars, rolling balls, falling cats), Geometric problems (dual isoperimetric problem), Hypoelliptic PDE's
2. Bracket generating distributions I: Simple examples (contact, even-contact, Engel and Martinet planes), normal forms.
3. Bracket generating distributions II: Chow-Rashevski theorem (comparison with Frobenius theorem)
4. Bracket generating distributions III: Canonical flag, Growth and weight vectors, Non-holonomic order, Privileged coordinates & their existence
5. Bracket generating distributions IV: Privileged coordinate dilation, sR tangent space, nilpotent Lie algebra & nilpotentization
6. sR metrics and distance, distance estimates, Ball-box theorem.
7. Hausdorff measure and dimension. Popp's measure. Mitchell's measure theorem.
8. Gromov-Hausdorff convergence. Mitchell's tangent cone theorem.
9. Bismut's chart theorem and existence of sR geodesics.
10. Normal geodesics and their minimality, Hamilton-Jacobi theory
11. Montgomery's abnormal: proof of minimality.
12. Montgomery's abnormal: failure of Hamilton equations, C1 rigidity.
13. Endpoint mapping and its differential, singular curves, abnormal geodesics are singular.
14. Hsu's theorem: microlocal characterization of singular curves.
15. Principal G-bundles: sR structures, normal geodesics for bi-invariant metrics.
16. Principal G-bundles: normal geodesics and Wong's equations.

Part II
17. sR Laplacian: definition and basic properties.
18. Pseudodifferential calculus I: asymptotic summation, residual terms, reduction.
19. Pseudodifferential calculus II: adjoint, composition, Sobolev boundedness.
20. Pseudodifferential calculus III: ellipticity, parametrix and elliptic estimate.
21. Kohn-Radkevic subelliptic estimate, Hormander's theorem.
22. Partial homomorphisms, Rothschild-Stein lifting theorem.
23. Lifting operators, reduction of RS subelliptic estimate to nilpotent group.
24. Filtered calculus I: composition, adjoint, Sobolev boundedness.
25. Filtered calculus II: principal symbol, ellipticity, elliptic & subelliptic estimates.
26. Filtered calculus III: Glowacki and Rockland theorems.
27. sR heat kernel expansion, Metevier's Weyl law
28. Bochner Laplacian, smallest eigenvalue.

References:
1. R. Montgomery, A tour of subriemannian geometries, their geodesics and appli- cations, vol. 91 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2002.
2. A. Bellaïche and J.-J. Risler, eds., Sub-Riemannian geometry, vol. 144 of Progress in Mathematics, Birkhäuser Verlag, Basel, 1996.
3. R. W. Goodman, Nilpotent Lie groups: structure and applications to analysis, Lecture Notes in Mathematics, Vol. 562, Springer-Verlag, Berlin-New York, 1976.
4. L. P. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math., 137 (1976), pp. 247-320

Contacts:
Lecturer - Nikhil Savale (nsavale@math.uni-koeln.de)
TA - Paul Creutz (pcreutz@math.uni-koeln.de)