|
|
We meet on Fridays at 4:00 in RLM 10.176.
- January 28. Pavel Safronov. Brief overview of various approaches to integrable systems.
- February 11. Thomas Mainiero. Lax representation for KdV and KP hierarchies, tau-function.
- February 25. Thomas Mainiero. Hirota equation, soliton solutions and vertex operators.
- March 4. Aswin Balasubramanian. Grassmannians and Hirota equations I.
- March 11. Aswin Balasubramanian. Grassmannians and Hirota equations II.
- March 25. Pavel Safronov. Spectral data attached to a differential operator.
- April 1. Hendrik Orem. Geometric inverse scattering theory I.
- April 8. Andrew Kontaxis. Isospectral deformations and the KP system I.
- April 15. Andrew Kontaxis. Isospectral deformations and the KP system II.
Literature
Classical inverse scattering method:
- C. Gardner, J. Greene, M. Kruskal, R. Miura, Method for solving the Korteweg-deVries equation, Phys. Rev. Lett. 19 (1967), 1095-1097.
- L.D. Faddeev, L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer, 2007. Google Books.
Quantum inverse scattering method:
- E.K. Sklyanin, L.A. Takhtajan, L.D. Faddeev, Quantum Inverse Problem Method I, Theor. Math. Phys. 40 (1979), 688-706.
- N.M. Bogolyubov, A.G. Izergin, V.E. Korepin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press, 1993. Google Books.
- M.A. Semenov-Tian-Shansky, Quantum and Classical Integrable Systems, arXiv:q-alg/9703023.
Connections to algebraic geometry:
- I.M. Krichever, Methods of algebraic geometry in the theory of non-linear equations, Russ. Math. Surv. 32 (1977), 185-213.
- T. Miwa, M. Jimbo, E. Date, Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras, Cambridge University Press, 2000. Google Books.
- G. Segal, G. Wilson, Loop groups and equations of KdV type, Publications mathématiques de l'I.H.É.S. 61 (1985), 5-65. Numdam.
- N. Kawamoto, Yu. Namikawa, A. Tsuchiya, Ya. Yamada, Geometric realization of conformal field theory on Riemann surfaces, Comm. Math. Phys. 116 (1988), 247-308. ProjectEuclid.
- M. Mulase, Algebraic theory of the KP equations, 1994.
- N. M. J. Woodhouse, The symplectic and twistor geometry of the general isomonodromic deformation problem, J. Geom. Phys. 39 (2001), 97-128. arXiv:nlin/0007024v1.
Hitchin system:
- N. Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987) 91-114.
- N. Hitchin, Flat connections and geometric quantization, Comm. Math. Phys. 131 (1990) 347-380. ProjectEuclid.
- R. Donagi, E. Markman, Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles, arXiv:alg-geom/9507017.
- A. Beilinson, V. Drinfeld, Quantization of Hitchin's integrable system and Hecke eigensheaves.
Overviews of various integrable systems:
- M. Audin, Lectures on integrable systems and gauge theory, 1995.
- O. Babelon, D. Bernard, M. Talon, Introduction to Classical Integrable Systems, Cambridge monographs on mathematical physics, 2003.
- P. Etingof, Lectures on Calogero-Moser systems, arXiv:math/0606233.
KdV and quantum gravity:
- E. Witten, Two dimensional gravity and intersection theory on moduli space, Surveys in Diff. Geom. 1 (1991), 243-310.
- M. Kontsevich, Intersection Theory on the Moduli Space of Curves and the Matrix Airy Function, Comm. Math. Phys. 147 (1992), 1-23. ProjectEuclid.
|