繁星客栈 - 望月殿 (数学逻辑论坛)

kanex

发表文章数: 860
武功等级: 弹指神通
　　　　　(第六重)
内力值: 343/343

看国外的课程体现的差距

从本科毕业来的学生第一年内就要学这么多东西,真是利害

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

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

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

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

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

PROBABILITY II: Probability and Random Processes
Prerequisite: ANALYSIS AND PROBABILITY I. Can be taken concurrently with ANALYSIS II

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

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

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

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

ALGEBRAIC TOPOLOGY II
Cohomology
Cup products
Pairings homology
Cohomology and homology with coefficients
Universal coefficient theorems
Cech Cohomology
Open coverings and Cech cochains
The coboundary mapping
Cech cohomology
Comparison with singular cohomology
Selected Topics
Group Cohomology
Sheaf Cohomology
de Rham's theorem
Morse functions and Poincaré duality for manifolds
Thom Isomorphism Theorem and cohomology classes Poincaré dual to cycles

COMMUTATIVE ALGEBRA
Basic notions for rings and modules
Rings, ideals, modules
Localization
Primary decomposition
Integrality
Noetherian and Artinian Rings
Noether normalization and Nullstellensatz
Discrete valuation rings, Dedekind domains and curves
Graded Modules and Completions
Dimension theory, Hilbert functions, Regularity
Sheaves and affine schemes

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

ALGEBRAIC NUMBER THEORY
Local fields
Global fields
Valuations
Weak approximation
Chinese Remainder Theorem
Ideal class groups
Minkowski's theorem and Dirichlet's unit theorem
Finiteness of class numbers
Ramification, different and discriminants
Quadratic symbols and quadratic reciprocity law
Zeta functions and L-functions
Chebotarev's density theorem
Preview of class field theory

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发表时间：2005-11-18, 06:21:01