Symplectic and quantum geometry

Wednesdays, 12:40-2, 736 Evans, CCN 54560

Thanks to all speakers. Please contact me at your convenience with any questions about your talk,
or help with additional references if needed.

Tentative schedule of lectures


Topic
Speaker
References
9/3
Introduction and outline of the seminar
Teleman

9/10
Symplectic manifolds, geometric quantization
Catherine
5, 12b, 21
9/17
Quantum harmonic oscillator. The metaplectic representation
Robert?
1 §14-16, 3
9/24
Hamiltonian actions. Moment maps. Quantization commutes with reduction Morgan
4, 6, 8
10/1
Duistermaat-Heckman formulas and equivariant cohomology
Ryan
1, 7, 15 (beginning)
10/8
The index formula for the Dirac operator (after Witten)
QC
14, 15 (part)
10/15
Holomorphic quantization of the moduli of flat connections
Daniel
8, 12 a,b
10/22
Real polarizations and integrable systems Michael
5, 10
10/29
Deformation quantization Alex
11
11/5
Toeplitz quantization and the semiclassical limit Ammar
9
11/12
Quantization of the space of flat connections
Teleman
11/19
Loop groups and group-valued moment maps
Tim Weelinck

11/24 M Tian's almost isometry theorem Kieran?Teleman?
16, 17, 18
12/3 Gromov-Tischler embedding theorem Benjamin
20
12/10 Donaldson divisor theorem Jeff
18?, 19


References

(Math Reviews, Berkeley Library and sometimes source links)

  1. Guillemin and Sternberg, Symplectic Techniques in Physics, CUP 1984
  2. Kostant, On the definition of quantization, Géométrie symplectique et physique mathématique (Colloq. Internat. CNRS, No. 237, Aix-en-Provence, 1974)
  3. Folland, Harmonic analysis in Phase Space, Annals of Math. Studies 122, Princeton
  4. Guillemin and Sternberg, A normal form for the moment map. In: Differential geometric methods in mathematical physics (Jerusalem, 1982), 161–175, Math. Phys. Stud., 6
  5. Guillemin and Sternberg, The Gelʹfand-Cetlin system and quantization of the complex flag manifolds. J. Funct. Anal. 52 (1983)
  6. Guillemin and Sternberg, Geometric quantization and multiplicities of group representations. Invent. Math. 67 (1982)
  7. Atiyah and Bott, The moment map and equivariant cohomology, Topology 23 (1984)
  8. Atiyah and Bott, The Yang-Mills equation on Riemann surfaces Philos. Trans. Royal Society A 308 (1983)
  9. M. Schlichenmaier, Berezin-Toeplitz quantization, arxiv.org/abs/math/0009219
  10. Jeffrey and Weitsman: Half density quantization of the moduli space of flat connections and Witten's semiclassical manifold invariants. Topology 32 (1993)
  11. Fedosov, A simple geometrical construction of deformation quantization. J. Differential Geom. 40 (1994)
  12. Hitchin, a. Flat connections and geometric quantization. Comm. Math. Phys. 131 (1990);
    b. Geometric quantization of spaces of connections. In: Geometry of low-dimensional manifolds, 2 (Durham, 1989), LMS Lecture Note Ser., 151
  13. M. Gross, Mirror Symmetry and the Strominger-Yau-Zaslow conjecture, arXiv:1212.4220
  14. Atiay, Circular symmetry and stationary-phase approximation, Asterisque 1985
  15. J.M.Bismut, Duistermaat-Heckman formulas and index theory, in: Geometric aspects of analysis and mechanics, Birkhaeuser 2011
  16. Tian, On a set of polarized Kaehler metrics on algebraic manifolds, Journal Diff. Geom. 32 (1990)
  17. Zelditch, Szego kernel and a theorem of Tian, IMRN 6 (1998)
  18. Shiffmann and Zelditch, Asymptotics of almost holomorphic sections of ample line bundes on symplectic manifolds
  19. Auroux, A remark about Donaldson's construction of symplectic submanifolds
  20. M. de Araujo, Symplectic embeddings, MA thesis, Lisbon
  21. Hurt, Geometric quantization in action, Mathematics and its applications 8, Reidel 1982