Math curriculum

>School program
Euclidean geometry, complex numbers, scalar multiplication, the Cauchy-Schwarz inequality. Principles of Quantum Mechanics (Kostrikin-Manin). Group of transformations of the plane and space. Conclusion of trigonometric identities. The geometry of the upper half plane (hyperbolic). Properties of inversion. Effect of linear fractional transformations.
Rings and fields. Linear algebra, finite groups, Galois theory. Proof of Abel. Basis, rank, determinant, classical Lie group. Dedekind. Determination of the real numbers. The definition of the tensor product of vector spaces.
Set theory. Zorn's lemma. A well-ordered set. Cauchy-Hamel basis. Cantor-Bernstein theorem. Uncountable set of real numbers.
Metric spaces. The set-theoretic topology (defined continuous maps, compact, custom maps). Countable base. The definition of compactness in terms of convergent sequences for spaces with a countable base. Homotopy, the fundamental group, homotopy equivalence.
p-adic numbers, Ostrowski theorem, multiplication, and division of the p-adic numbers in a column
Differentiation, integration, the Newton-Leibniz. Delta epsilon formalism lemma policeman.

>First year
Analysis on $ R ^ n $. Differential display. lemma contracting maps. Implicit Function Theorem. Riemann and Lebesgue. ("Analysis" of Laurent Schwartz, "Analysis" Zorich, "Problems and theorems of functions. Analysis" Kirillov-Gvishiani)
Hilbert spaces, Banach spaces (definition). The existence of a basis in the Hilbert space. Continuous and discontinuous linear operators. Criteria for continuity. Examples of compact operators. ("Analysis" of Laurent Schwartz, "Analysis" Zorich, "Problems and theorems of functions. Analysis" Kirillov-Gvishiani)
Smooth manifolds, submersion immersion Sard's theorem. Partition of unity. Differential Topology (Milnor-Wallace). Transversality. Degree of the map as a topological invariant.
Differential forms, de Rham operator, Stokes theorem, Maxwell's equations of the electromagnetic field. Gauss's theorem as a particular example.
Comprehensive analysis of a single variable (based on the book by Henri Cartan, or the first volume of Shabbat). Contour integrals, Cauchy, Riemann mapping theorem of any simply connected subset of $ C $ in a circle theorem on the extension of borders, Picard's theorem on the achievement of an entire function of all values ​​except three. Multivalent functions (for example, the logarithm).
Category theory, the definition of the functors, equivalence, adjoint functors (McLane, Categories for working mathematician, Gelfand-Manin, the first chapter.)
Lie groups and algebras. Lie groups. Lie algebra as linearization. Universal enveloping algebra, the Poincare-Birkhoff-Witt. Free Lie algebra. Campbell-Hausdorff series and the construction of a Lie group by its algebra (yellow Serre, first half).

>Second year
Algebraic Topology (Fuchs-Fomenko). Cohomology (simplicial, singular, de Rham), their equivalence, Poincaré duality, the homotopy groups. Dimension. Bundle (Serre), the spectral sequence (Mishchenko, "Vector bundles ..."). Computation of the cohomology of classical Lie groups and projective space.
Vector bundles, connection, the Gauss-Bonnet, the Euler class, Black, Pontryagin, Stiefel-Whitney test. Multiplicativity of the Chern character. Classifying spaces ("Characteristic Classes", Milnor and Stashef).
Differential Geometry. Levi-Civita connection, the curvature of the algebraic and differential Bianchi identity. Killing Fields. Gaussian curvature of two-dimensional Riemannian manifold. Cell decomposition of the loop space in terms of geodesics. Morse theory on the loop space (based on the book of Milnor, "Morse theory" and Arthur Besse "Einstein manifolds"). Principal bundles and connections in them.
Commutative algebra (Atiyah-Macdonald). Noetherian, Krull dimension, Lemma Nakayama adic completion, integrally closed, discrete valuation ring. Flat modules, local criterion plane.
Beginning of algebraic geometry. (Chapter Hartshorne either green or Shafarevich Mumford). Affine variety, projective variety, projective morphism, the image of a projective variety is projective (via resultants). Beams. Zariski topology. Algebraic variety as a ringed space. Hilbert's theorem on zeros. The spectrum of the ring.
Beginning of homological algebra. Group Ext, Tor for modules over a ring, resolutions, projective and injective modules (Atiyah-Macdonald). Construction of injective modules. Grothendieck duality (in the book Springer Lecture Notes in Math, Grothendieck Duality, number about 21 and 40).
Number theory, local and global field discriminant rule the ideal class group (blue book Cassels and Frohlich).
Reductive groups, root systems, representations of semisimple groups, the weight, the Killing form. Groups generated by reflections, their classification. Lie algebra cohomology. Computation of the cohomology in terms of invariant forms. Singular cohomology of a compact Lie group and its cohomology algebra. Invariants of classical Lie groups. (Yellow Serre, the second half, Hermann Weyl, "Invariants of classical groups"). Design of special Lie groups. Hopf algebra. Quantum groups (definition).

>Third year
K-theory as a cohomological functor, Bott periodicity, Clifford algebra. Spinors (Atiyah's book "K-theory" or Mishchenko "the vector bundles and their application"). Spectra. Eilenberg-Mac Lane. Infinite loop space (the book Switzer or yellow book or Adams Adams "Lectures on generalized cohmology", 1972).
Differential operators, pseudodifferential operators, symbol, elliptic operators. Properties of the Laplace operator. Self-adjoint operators with discrete spectrum. Green operator and applications to Hodge theory on Riemannian manifolds. Quantum mechanics. (Book R.Uellsa analysis or Mishchenko "vectors of the bundle and its application").
The index formula (Atiyah-Bott-Patodi, Mishchenko), the Riemann-Roch. Zeta function of the operator with discrete spectrum and its asymptotic behavior.
Homological algebra (Gelfand-Manin, all in the last intermediate). Cohomology of Sheaves, derived category, triangulated category, derived functor, the spectral sequence of a double. The composition of triangulated functors and the corresponding spectral sequence. Verdier duality. The formalism of the six functors and perverse sheaves.
Scheme algebraic geometry, the scheme over the ring, the projective spectra, derivatives, Serre duality, coherent beams, replacement base. Own and separated scheme, valyuativny property and separability criterion (Hartshorne). Functors representability, the moduli space. Direct and inverse images of sheaves, higher direct images. With proper mapping higher direct images are coherent.
Cohomological methods in algebraic geometry, semicontinuity cohomology theorem of Zariski connectedness theorem Stein decomposition.
Kahler manifolds, Lefschetz, Hodge theory, the ratio of Kodaira, properties of the Laplace operator (zero head head Griffiths-Harris, clearly stated in the book André Weil "Kahler manifolds"). Hermitian bundle. Line bundles and their curvature. Line bundles with positive curvature. Kodaira-Nakano Vanishing on (Griffiths-Harris).
Holonomy, the Ambrose-Singer theorem, special holonomy holonomy classification, Calabi-Yau, hyper, theorem of Calabi-Yau manifolds.
Spinors on the manifold, the Dirac operator, the Ricci curvature, the formula Weitzenbock-Lichnerowicz, Bochner's theorem. Bogomolov's theorem on the decomposition of manifolds with zero canonical class (Arthur Besse, "Einstein manifolds").
Tate cohomology class field theory (Cassels-Frohlich, blue book). The calculation of the factor Galois number field commutator. Brauer and its applications.
Ergodic theory. Ergodicity of billiards.
Complex curves, pseudo-conformal mappings, Teichmüller space, the theory of Ahlfors-Bers (the book Ahlfors thin).

>Fourth year.
Rational homotopy type and profinite étale cover nerve cell complex homotopy equivalent to its profinite type. Topological definition of étale cohomology. Galois action on profinite homotopy type (Sullivan, "Geometric Topology").
Étale cohomology in algebraic geometry, the comparison functor, Henselian, geometric point. Base change. Any smooth variety over a field locally in the étale topology is isomorphic to $ A ^ n $. Étale fundamental group (Milne, review of Daniel VINITI and SGA 4 1/2, the first article of Deligne).
Elliptic curves, j-invariant, automorphic forms, Taniyama-Weil conjecture and its applications to number theory (Fermat's theorem).
Rational homotopy (in the last chapter of the book or the Gelfand-Manin article Griffiths-Morgan-Long-Sullivan). Massey operations and rational homotopy type. Vanishing Massey operations on a Kahler manifold.
Chevalley group of generators and relations (the book Steinberg). K_2 computation of the field (Milnor, Algebraic K-theory).
Algebraic K-theory of Quillen, $ BGL ^ + $ and $ Q $-structure (review Suslin in the 25th volume of VINITI, lectures Quillen - Lecture Notes in Math. 341).
Complex analytic manifold, coherent beams, the coherence theorem of Oka, Hilbert's Nullstellensatz for ideals in the sheaf of holomorphic functions. Noetherian ring of germs of functions, Weierstrass theorem on division, Weierstrass preparation theorem. Theorem on a branched covering. Grauert-Remmert (image of a compact analytic space under a holomorphic analytic morphism). Hartogs' theorem on the continuation of an analytic function. Multidimensional Cauchy formula and its applications (uniform limit of holomorphic functions is holomorphic).

>Fifth Year
Kodaira-Spencer theory. Deformation of a manifold and solutions Maurer-Cartan equations. Solvability of the Maurer-Cartan and Massey operations on the DG-Lie algebra cohomology of vector fields. Moduli space and finite (see lecture Kontsevich, or the collected works of Kodaira). Theorem Bogomolov-Tian-Todorov on deformations of Calabi-Yau manifolds.
Symplectic reduction. Moment map. Kempf-Ness theorem.
Deformations of coherent bundles and bundles in algebraic geometry. Geometric invariant theory. Moduli space of bundles on a curve. Stability. Uhlenbeck compactification, Gieseker and Maruyama. Geometric invariant theory is the symplectic reduction (third edition of Mumford's geometric invariant theory, application, Frances Kirwan).
Instantons in the four-dimensional geometry. Donaldson theory. Donaldson invariants. Instantons on Kahler surfaces.
The geometry of complex surfaces. Kodaira classification, Kahler and nekelerovy surface of the Hilbert scheme of points on the surface. Castelnuovo-Enriques criterion, the Riemann-Roch inequality Bogomolov-Miyaoka-Yau. The ratio of invariants of the surface. Elliptic surfaces, the surface Kummer K3 surfaces and Enriques.
Elements of the program Mori: Kawamata-Viehweg zero freedom theorem of base points, the theorem of Mori cone (Clemens-Kollar-Mori, "Multidimensional complex geometry", plus not translated Kollar-Mori and Kawamata-Matsuki-Massoud) .
Stable bundles as instantons. Yang-Mills equation on a Kahler manifold. Theorem Donaldson-Uhlenbeck-Yau metrics on the Yang-Mills theory on a stable bundle. Its interpretation in terms of the symplectic reduction. Stable bundles and instantons on hyperkähler manifolds, an explicit solution of the Maurer-Cartan equation in terms of the Green's operator.
Pseudoholomorphic curves in symplectic manifold. Gromov-Witten. Quantum cohomology. Mirror conjecture and interpretation. The structure of the group of symplectomorphisms (under the Kontsevich-Manin book Polterovich, "Symplectic geometry" green book of pseudoholomorphic curves and lecture notes McDuff and Salamon).
Complex spinors, Seiberg-Witten, Seiberg-Witten. Why Seiberg-Witten invariants are Gromov-Witten.
Hyperkähler reduction. Flat bundles, and the equation of the Yang-Mills theory. Hyperkähler structure on the moduli space of flat bundles (Hitchin-Simpson).
Mixed Hodge structure. Mixed Hodge structure on the cohomology of algebraic varieties. Mixed Hodge structures on Malcev completion of the fundamental group. Variations of mixed Hodge structures. Theorem on a nilpotent orbit. Theorem on $ SL (2)-orbit. Close and vanishing cycles. The exact sequence of the Clemens-Schmid (the red book Griffiths "Transcendental methods in algebraic geometry").
Nonabelian Hodge theory. Variations of Hodge structures as fixed points of $ C ^ * $-action on the moduli space of Higgs bundles (dissertation Simpson).
Weil conjecture and proof. L-adic sheaves, perverse sheaves, the Frobenius automorphism, its weight, purity theorem (Beilinson, Bernstein, Deligne, plus Deligne, Weil Conjectures II).
Quantitative algebraic topology Gromov (the book Gromov "Metric structures for Riemannian and non-Riemannian spaces"). Gromov-Hausdorff metric, prekompaktnost set of metric spaces, hyperbolic manifolds and hyperbolic groups, harmonic maps into hyperbolic space, the proof of Mostow rigidity (two compact Kahler manifolds, covered by one and the same symmetric space X of negative curvature, isometric if their fundamental groups isomorphic, and dim X> 1).
Variety of general type, Kobayashi and Bergman metrics, analytical stiffness (Sioux).