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Mathematical Methods for Modern Physics
Lorenzo Cornalba - Università di Milano Bicocca
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| The course will focus on the geometry of spaces (topological spaces, differentiable and complex manifolds) and its relation to differential calculus. We shall first focus on the purely topological aspects, introducing the notions of simplicial and singular homology and cohomology, and discussing Poincaré duality. We shall then move to the study of smooth manifolds, starting with de Rham cohomology of differential forms. After introducing Cech cohomology, we shall focus on complex manifolds and Dolbeault cohomology. If time permits, we shall conclude with an introduction to fiber bundles and characteristic classes, which will be used in the description of the Atiyah-Singer theorem. |
| Lecture notes |
Recommended bibliography includes:
A. Hatcher, "Algebraic Topology" (Cambridge University Press)
R. Bott and L.W. Tu, "Differential Forms in Algebraic Topology" (GTM Springer-Verlag)
P. Griffiths and J. Harris, "Principles of Algebraic Geometry" (J. Wiley)
T. Eguchi , P.Gilkey and A. Hanson, "Gravitation, Gauge Theories and Differential Geometry", Phys. Rept. 66 (1980) 213
M. Atiyah, R. Bott and V.K. Patodi, "On the Heat Equation and the Index Theorem", Inventiones Math. 19 (1973) 279
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