1. Manifold Theory
   1. Point Set Topology
   2. Homeomorphism and Homotopy
   3. Compact, Connected, Hausdorff Spaces
   4. Topological, Differentiable, Complex Manifolds
   5. Physical Examples from Classical Mechanics 
2. Differential Forms, Tensors, and Curvature
   1. Differential Forms
   2. Parallel Transport and Affine Connections
   3. Riemann, Ricci, Scalar Curvature Tensors
   4. Riemannian and Pseudo-Riemannian Metrics
   5. Integration on curved manifolds
   6. Physical Examples: Electromagnetic Theory in Curved Spacetime, General Relativity 
3. Homology and Cohomology
   1. Chains, Cycles, Boundary Operator
   2. Physical Examples: Hamiltonian Mechanics, Dirac Monopoles, Electromagnetic Duality in Arbitrary Dimensions

      
       

    

   MODERN GEOMETRY II: Global Structures

    

4. Lie Groups and Lie Algebras
   1. Review of the Classical Lie Groups and their Algebras
   2. Differential Geometric Aspects of Group Manifolds
   3. Basic Representation Theory
   4. Physical Examples: Lie Groups in Particle Theory; The Structure of the Standard Model; The Eight-fold Way; Particle Mass Relations and Sum Rules
5. Fiber Bundles
   1. The Classical Groups
   2. Vector Bundles
   3. Principal Bundles
   4. Aspects of Bundle Classification, Characteristic Classes
   5. Physical Examples: Dirac and `t Hooft-Polyakov Magnetic Monopoles, Instantons
   6. Aspects of Anomalies and aspects of Index Theory
6. Homotopy Theory and Defects in Quantum Field Theory
   1. Fundamental Group, Higher Homotopy Groups
   2. Further Application to Bundle classification
   3. Physical Examples: Monopoles, Strings, Domain Walls, Textures
   4. Quantum Mechanics on Topologically Nontrivial Spacetimes
7. Geometry and String Theory
   1. General Relativity in Arbitrary Dimensions
   2. Mathematical Aspects of Kaluza-Klein Theory
   3. Kähler Manifolds
   4. Harmonic Analysis
   5. Calabi-Yau Manifolds and Low Energy String Theory
   6. Mirror Symmetry

1. Holomorphic Functions
   1. Holomorphic functions, Cauchy-Riemann equations
   2. Conformal mappings
   3. Cauchy integral formula, residues
2. Analytic Continuation
   1. Gamma and zeta functions
   2. Hypergeometric functions and monodromy
   3. Braid group representations
   4. Correlation functions in conformal field theory
3. Riemann Surfaces
   1. The Riemann surface y²=x(x-1)(x-l)
   2. Holomorphic and meromorphic differentials
   3. Homology, fundamental group, surface classification
   4. Weierstrass elliptic functions
   5. Theta functions
   6. The moduli space of tori
   7. Introduction to Riemann surfaces of arbitrary genera
   8. Fields of meromorphic functions, field extensions, Galois theory
4. Theta Functions and Modular Forms
   1. Modular transformations and modular forms
   2. Eisenstein series, Dedekind eta-function, Kronecker limit formula
   3. Hecke operators
   4. Poisson summation, theta-functions of lattices
   5. Exact formulas for heat kernels
5. Selected Topics, chosen from
   1. Integrable models, spectral curves, and solitons
   2. Modular forms and infinite-dimensional algebras
   3. Geometry of the moduli space of Riemann surfaces
   4. Solvable models in statistical mechanics or conformal field theory
   5. Introduction to L-functions

1. Measure Theory
   1. Construction of the integral, limits and integration
   2. L^p spaces of functions
   3. Construction of measures, Lebesgue-Stieltjes product measures
   4. Examples: ergodicity, Liouville measure, Hausdorff measure
2. Elements of Probability
   1. The coin-tossing or random walk model
   2. Independent events and independent random variables
   3. The Khintchin weak law and the Kolmogorov strong law of large numbers
   4. Notions of convergence of random variables
   5. The Central Limit Theorem
3. Elements of Fourier Analysis
   1. Fourier transforms of measures, Fourier-Lévy Inversion Formula
   2. Convergence of distributions and characteristic functions
   3. Proof of the Central Limit and Lindeberg Theorems
   4. Fourier transforms on Euclidean spaces
   5. Fourier series, the Poisson summation formula
   6. Spectral decompositions of the Laplacian
   7. The heat equation and heat kernel
4. Brownian Motion
   1. Brownian motion as a Gaussian process
   2. Brownian motion as scaling limit of random walks
   3. Brownian motion as random Fourier series
   4. Brownian motion and the heat equation
   5. Elementary properties of Brownian paths

1. First Order Partial Differential Equations
   1. Cauchy's Theorem for first order real partial differential equations
   2. Completely integrable first order equations
2. Implicit Function Theorems
   1. Basic examples of linear and non-linear partial differential equations
   2. The functional analytic framework, Banach and Hilbert spaces
   3. Bounded linear operators, spectrum, invertibility
   4. Implicit function theorems in Banach spaces
   5. Sketch of subsequent applications to the basic examples
3. Second Order Partial Differential Equations
   1. Qualitative description: elliptic, parabolic, hyperbolic equations
   2. The Cauchy problem
   3. Maximum principles
   4. Sobolev and Schauder spaces
   5. A priori estimates and Green's functions
   6. Riesz-Schauder theory of compact operators
   7. Detailed treatment of basic examples
   8. The Laplace and heat equations on compact manifolds
   9. Applications to de Rham and Hodge theory
4. Selected Topics, chosen from
   1. Riemann-Roch and index theorems
   2. Determinants of Laplacians, modular forms
   3. Integral representations, Hilbert transforms, singular integral operators
   4. Subelliptic equations
   5. Nash-Moser implicit function theorems
   6. Non-linear equations from geometry or physics

Prerequisite: ANALYSIS AND PROBABILITY I. Can be taken concurrently with ANALYSIS II 

1. Rare Events
   1. Cramér's Theorem
   2. Introduction to the Theory of Large Deviations
   3. The Shannon-Breiman-McMillan Theorem
2. Conditional Distributions and Expectations
   1. Absolute continuity and singularity of measures
   2. Radon-Nikodým theorem. Conditional distributions
   3. Conditional expectations as least-square projections
   4. Notion of conditional independence
   5. Introduction to Markov Chains. Harmonic functions
3. Martingales
   1. Definitions, basic properties, examples, transforms
   2. Optional sampling and upcrossings theorems, convergence
   3. Burkholder-Gundy and Azuma inequalities
   4. Doob decomposition, square-integrable martingales
   5. Strong laws of large numbers and central limit theorems
4. Applications
   1. Optimal stopping
   2. Branching processes and their limiting behavior. Urn schemes
   3. Stochastic approximation. Probabilistic analysis of algorithms
5. Stochastic Integrals and Stochastic Differential Equations
   1. Detailed study of Brownian motion
   2. Martingales in continuous time
   3. Doob-Meyer decomposition, stopping times
   4. Integration with respect to continuous martingales, Itô's rule
   5. Girsanov's theorem and its applications
   6. Introduction to stochastic differential equations. Diffusion processes
6. Elements of Potential Theory
   1. The Dirichlet problem. Poisson integral formula
   2. Solution in terms of Brownian motion
   3. Detailed study of the heat equation; Cauchy and boundary-value problems
   4. Feynman-Kac theorems, applications

1. Basic Notions
   1. Abstract groups, algebraic groups over a field, topological groups, Lie groups
   2. Subgroups, normal subgroups, quotient groups
   3. Homomorphisms of groups - image, kernel, exact sequences
   4. Cyclic groups, abelian groups, nilpotent groups
   5. Conjugacy classes, left and right cosets of a subgroup
2. Algebraic Examples
   1. Units of a ring, k* for k a field, roots of unity in a commutative ring, R*, S¹ in C*
   2. GL(n, R) as the group of units of n x n-matrices over a commutative ring R
   3. The determinant and SL(n, R), O(n, R), Sympl(2n, R) when there is (-1) in R
   4. Algebraic groups of the above types over a field, definition of linear algebraic groups
   5. Group structure on an elliptic curve
   6. Group of p-adic integers, and its multiplicative group of units
3. Geometric Examples and Symmetry
   1. Permutation groups
   2. Symmetries of regular plane figures and of Platonic solids
   3. The Lie groups SL(n, R), SO(n, R), SO(p, q), Sympl(2n, R)
   4. Isometries of the line, the plane, and higher dimensional Euclidean spaces
   5. Isometries of spheres and of Minkowski space. The Poincaré group
   6. Isometries of the hyperbolic plane, conformal isomorphisms of S², relation with SL(2, R) and SL(2, C)
   7. Clifford algebras and the spin groups
   8. The Heisenberg group
4. Lie Algebras
   1. Definition, examples of the Lie algebra of an associative algebra
   2. The Lie algebra of a Lie group. The universal enveloping algebra and the Poincaré-Birkhoff-Witt theorem
5. Representations
   1. Definition in the various categories of groups, representations of a Lie algebra
   2. Infinitesimal generators for the action of a Lie group
   3. The infinitesimal representation associated to a linear representation of a Lie group
   4. Turning actions into linear representations on the functions
   5. Classification of the (finite dimensional) representations of sl(2, C), SU(2), and SO(3)
   6. Representations of the Heisenberg algebra
6. Representations of Finite and Compact Lie Groups
   1. Complete reducibility, Schur's lemma, characters, orthogonality relations for characters of a finite group
   2. Dimension of the space of characters of a finite group
   3. The decomposition of the regular representation of a finite group
   4. Characters of a compact group - complete reducibility, Schur's lemma, orthogonality of characters
   5. Peter-Weyl theorem (except the proof of the decomposibility of a Hilbert space representation into finite dimensional sub representations)
   6. Example of L²(S¹) and Fourier analysis
   7. Example of L²(S²) as a module over SO(3) and spherical harmonics
7. Finite Groups and Counting Principles
   1. Orders of elements and subgroups
   2. Groups of order pⁿ are nilpotent
   3. Subgroups of index 2 are normal
   4. The Sylow theorems
   5. Classification of groups of order pq for p, q distinct primes. Groups of order 12

     

    

   GROUPS AND REPRESENTATIONS II

    

8. Lie Groups and Lie Algebras: the Exponential Mapping
   1. Baker-Campbell-Hausdorff formula
   2. A Lie group is determined by its Lie algebra up to covering
   3. Action of a Lie group is determined by its infinitesimal action
9. Maximal Tori of a Compact Lie Group
   1. Existence and uniqueness up to conjugation
   2. Every element is contained in a maximal torus
   3. Regular elements
   4. The Weyl group
   5. Weyl group action on the maximal torus and on corresponding abelian Lie algebra
   6. Decomposition of the adjoint representation root spaces. Weyl chambers
   7. Groups generated by reflection
   8. Positive roots, dominant root and alcove
   9. Dynkin diagrams
   10. The classical examples SU(n), SO(n), Sympl(2n)
10. Complex Semi-Simple Lie Groups and Lie Algebras
11. Irreducible Representations of Compact Groups
    1. Weight spaces, dominant weights
    2. Examples for SU(n), Sympl(2n) and SO(n)
12. Selected Topics, chosen from
    1. Borel-Weil-Bott theory
    2. Infinite-dimensional representations of SL(2, R)
    3. Kac-Moody algebras
    4. The Virasoro algebra
    5. Supersymmetry

1. Homology Theory
   1. Singular homology -- definition, simple computations
   2. Cellular homology -- definition
   3. Eilenberg-Steenrod Axioms for homology
   4. Computations: Sⁿ, RPⁿ, CPⁿ, Tⁿ, S²^S³, Grassmannians, X*Y
   5. Alexander duality -- Jordan curve theorem and higher dimensional analogues
   6. Applications: Winding number, degree of maps, Brouwer fixed point theorem
   7. Lefschetz fixed point theorem
2. Homotopy Theory
   1. Homotopy of maps, of pointed maps
   2. The homotopy category and homotopy functors --examples
   3. p₁(X, x₀)
   4. Van Kampen's theorem
   5. Higher homotopy groups and the Hurewicz theorem
   6. p₃(S²)
   7. Higher homotopy groups of the sphere
3. Covering Spaces
   1. Definition of a covering projection
   2. Examples -- Coverings of S¹, Sⁿ covering RPⁿ, Spin(n) covering SO(n)
   3. Homotopy path lifting
   4. Classification of coverings of a reasonable space
4. Homology with Local Coefficients
   1. Local coefficient systems
   2. Relation with covering spaces
   3. Obstruction theory
   4. The Alexander polynomial of a knot

     

    

   ALGEBRAIC TOPOLOGY II

    

5. Cohomology
   1. Cup products
   2. Pairings homology
   3. Cohomology and homology with coefficients
   4. Universal coefficient theorems
6. Cech Cohomology
   1. Open coverings and Cech cochains
   2. The coboundary mapping
   3. Cech cohomology
   4. Comparison with singular cohomology
7. Selected Topics
   1. Group Cohomology
   2. Sheaf Cohomology
   3. de Rham's theorem
   4. Morse functions and Poincaré duality for manifolds
   5. Thom Isomorphism Theorem and cohomology classes Poincaré dual to cycles

1. Basic notions for rings and modules
   1. Rings, ideals, modules
   2. Localization
   3. Primary decomposition
   4. Integrality
   5. Noetherian and Artinian Rings
   6. Noether normalization and Nullstellensatz
   7. Discrete valuation rings, Dedekind domains and curves
   8. Graded Modules and Completions
   9. Dimension theory, Hilbert functions, Regularity
   10. Sheaves and affine schemes

     

    

   ALGEBRAIC GEOMETRY

    

2. Varieties
   1. Projective Varieties
   2. Morphisms and Rational Maps
   3. Nonsingular Varieties
   4. Intersections of Varieties
3. Schemes
   1. Basic properties of schemes
   2. Separated and proper morphisms
   3. Quasi-coherent sheaves
   4. Weil and Cartier divisors, line bundles and ampleness
   5. Differentials
   6. Sheaf cohomology
4. Curves
   1. Residues and duality
   2. Riemann-Roch
   3. Branched coverings
   4. Projective embeddings
   5. Canonical curves and Clifford's Theorem

     

    

   ALGEBRAIC NUMBER THEORY

    

   1. Local fields
   2. Global fields
   3. Valuations
   4. Weak approximation
   5. Chinese Remainder Theorem
   6. Ideal class groups
   7. Minkowski's theorem and Dirichlet's unit theorem
   8. Finiteness of class numbers
   9. Ramification, different and discriminants
   10. Quadratic symbols and quadratic reciprocity law
   11. Zeta functions and L-functions
   12. Chebotarev's density theorem
   13. Preview of class field theory