Symplectic geometry

The purpose of the lectures was to discuss various properties of the Hilbert schemes of points on surfaces. Although it was not noticed until recently, the Hilbert schemes have relationship with many other branch of mathematics, such as topology, hyper-Kahler geometry, symplectic geometry, singularities, and representation theory. This is reflected to this note each chapter, which roughly corresponds to one lecture, discusses different topics. 

Feng, K. On difference schemes and symplectic geometry. In Proceedings of the 1984 Beijing Symposium on Differential Geometry and Differential Equations, pp. 42-58. Science Press, Beijing (1985)... 

Feng, K. Difference schemes for Hamiltonian formalism and symplectic geometry. J. Comput. Math. 4, 279-289 (1986)... 

Feng, K. Symplectic geometry and numerical methods in fluid dynamics. In Zhuang, F., Zhu, Y. (eds.) Tenth International Conference on Numerical Methods in Fluid Dynamics. Lecture Notes in Physics, vol. 264, pp. 1-7. Springer, Berlin/Heidelberg (1986). doi 10.1007/ BFb0041762. ISBN 978-3-540-17172-0... 

In line with this theory, a symplectic geometry in a linear space TxM s constructed whose properties are determined by setting a skew-symmetric ( ) nondegenerate scalar product (, ). This cannot be done in all cases but only when the dimension of the manifold M is even. Then an even-dimensional space TxM transforms into a symplectic space In what follows, we shall assume for convenience that a symplectic space is modelled on a Euclidean space (on which, therefore, two forms are simultaneously given, namely, symmetric and skew-symmetric). 

Weinstein, A. "Symplectic Geometry. Bull Amer. Math. Soc. 5 (1981), 1-13. Guillemin, V., and Sternberg, S. "Convexity properties of the moment mapping. Invent. Math. 67 (1982), 491-513. 

Cohomology of Quotients in Symplectic and Algebraic Geometry, Princeton University Press, Princeton, New Jersey 1984. 

F. Kirwan, Cohomology of quotients in symplectic and algebraic geometry . Mathematical Notes, Princeton Univ. Press, 1985. 

In previous papers (Froeschle et al. 2000, Guzzo et al. 2002, Lega et al. 2002) we used the FLI to describe the geometry of the resonances, integrating orbits of the Hamiltonian system of equation 6 and of the following 4-dimensional symplectic map ... 

Q j simultaneously at all the points of a certain (maybe small) open neighbourhood of any point on a symplectic manifold. This fact constitutes the content of the known Darhoux theorem which has no analogue (in the indicated sense) in Riemannian geometry. 

Tatarinov, Ya. V. "Geometric formalism of classical dynamics canonical originals. Vestnik Mosk. Gos. Univers., ser. mat. mekh.. No. 4 (1983), 85-95. Tischler, D. "Closed 2-forms and an embedding theorem for symplectic manifolds. J. Diff. Geometry 12 (1977), 229-235. 

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