Second course
Algebraic topology (Fuchs-Fomenko). Cohomology (simplicial, singular, de Rham), their equivalence, Poincaré duality, homotopy groups. Dimension. Bundles (in the sense of Serre), spectral sequences (Mishchenko, "Vector bundles ..."). Calculation of the cohomology of classical Lie groups and projective space.
Vector bundles, connection, Gauss-Bonnet formula, Euler, Chern, Pontryagin, Stiefel-Whitney classes. Chern's multiplicativity. Classifying spaces ("Characteristic Classes", Milnor and Stashef).
Differential geometry. Levi-Civita connection, curvature, algebraic and differential Bianchi identity. Killing Fields. Gaussian curvature of a two-dimensional Riemannian manifold. Cellular decomposition of loop space in terms of geodesics. Morse theory on the space of loops (based on the book "Morse Theory" by Milnor and "Einstein Manifolds" by Arthur Besse). Principal bundles and connections in them.
Commutative algebra (Atiyah-Macdonald). Noetherian rings, Krull dimension, Nakayama lemma, adic completion, integral closedness, discrete valuation rings. Plane modules, local plane criterion.
Beginnings of algebraic geometry. (the first chapter of Hartshorne is either Shafarevich or green Mumford). An affine variety, a projective variety, a projective morphism, the image of a projective variety is projective (via resultants). Bunches. Zariski topology. Algebraic variety as a ringed space. Hilbert's Zeros Theorem. The spectrum of the ring.
Beginnings of homological algebra. Ext, Tor groups for modules over a ring, resolvents, projective and injective modules (Atiyah-Macdonald). Construction of injective modules. Grothendieck Duality (from Springer Lecture Notes in Math, Grothendieck Duality, nos. 21 and 40).
Number theory; local and global fields, discriminant, norm, ideal class group (blue book by Cassels and Frohlich).
Reductive groups, root systems, semisimple group representations, weights, Killing form.
