丘成桐大学生数学竞赛数学专业大纲.doc

S．T．YAU College Student Mathematics Contests Algebra, Number Theory and Combinatorics (second draft)Linear Algebra Abstract vector spaces; subspaces; dimension; matrices and linear transformations; matrix algebras and groups; determinants and traces; eigenvectors and eigenvalues, characteristic and minimal polynomials; diagonalization and triangularization of operators; invariant subspaces and canonical forms; inner products and orthogonal bases; reduction of quadratic forms; hermitian and unitary operators, bilinear forms; dual spaces; adjoints. tensor products and tensor algebras; Integers and polynomialsIntegers, Euclidean algorithm, unique decomposition; congruence and the Chinese Remainder theorem; Quadratic reciprocity ; Indeterminate Equations. Polynomials, Euclidean algorithm, uniqueness decomposition, zeros; The fundamental theorem of algebra; Polynomials of integer coefficients, the Gauss lemma and the Eisenstein criterion; Polynomials of several variables, homogenous and symmetric polynomials, the fundamental theorem of symmetric polynomials.Group Groups and homomorphisms, Sylow theorem, finitely generated abelian groups. Examples: permutation groups, cyclic groups, dihedral groups, matrix groups, simple groups, Jordan-Holder theorem, linear groups (GL(n, F) and its subgroups), p-groups, solvable and nilpotent groups, group extensions, semi-direct products, free groups, amalgamated products and group presentations.RingBasic properties of rings, units, ideals, homomorphisms, quotient rings, prime and maximal ideals, fields of fractions, Euclidean domains, principal ideal domains and unique factorization domains, polynomial and power series rings, Chinese Remainder Theorem, local rings and localization, Nakayama's lemma, chain conditions and Noetherian rings, Hilbert basis theorem, Artin rings, integral ring extensions, Nullstellensatz, Dedekind domains,algebraic sets, Spec(A).Module Modules and algebra Free and projective; tensor products; irreducible modules and Schur’s lemma; semisimple, simple and primitive rings; density and Wederburn theorems; the structure of finitely generated modules over principal ideal domains, with application to abelian groups and canonical forms; categories and functors; complexes, injective modues, cohomology; Tor and Ext. Field Field extensions, algebraic extensions, transcendence bases; cyclic and cyclotomic extensions; solvability of polynomial equations; finite fields; separable and inseparable extensions; Galois theory, norms and traces, cyclic extensions, Galois theory of number fields, transcendence degree, function fields.Group representationIrreducible representations, Schur's lemma, characters, Schur orthogonality, character tables, semisimple group rings, induced representations, Frobenius reciprocity, tensor products, symmetric and exterior powers, complex, real, and rational representations.Lie AlgebraBasic concepts, semisimple Lie algebras, root systems, isomorphism and conjugacy theorems, representation theory. Combinatorics (TBA)References:Strang, Linear algebra, Academic Press.I.M. Gelfand, Linear Algebra《整数与多项式》冯克勤 余红兵著 高等教育出版社Jacobson, Nathan Basic algebra. I. Second edition. W. H. Freeman and Company, New York, 1985. xviii+499 pp.Jacobson, Nathan Basic algebra. II. Second edition. W. H. Freeman and Company, New York, 1989. xviii+686 pp.S. Lang, Algebra, Addison-Wesley冯克勤，李尚志，查建国，章璞，《近世代数引论》刘绍学，《近世代数基础》J. P. Serre, Linear representations of finite groupsJ. P. Serre: Complex semisimple Lie algebra and their representationsJ. Humphreys: Introduction to Lie algebra and representation theory, GTM 009.W. Fulton, Representation theory, a First Course, GTM 129. Analysis and differential equations (second draft)Calculus and mathematical analysisDerivatives, chain rule; maxima and minima, Lagrange multipliers; line and surface integrals of scalar and vector functions; Gauss’, Green’s and Stokes’ theorems. Sequences and series, Cauchy sequences, uniform convergence and its relation to derivatives and integrals; power series, radius of convergence, convergence of improper integrals. Inverse and implicit function theorems and applications; the derivative as a linear map; existence and uniqueness theorems for solutions of ordinary differential equations, explicit solutions of simple equations.; elementary Fourier series. Complex analysisAnalytic function, Cauchy's Integral Formula and Residues, Power Series Expansions, Entire Function, Normal Families, The Riemann Mapping Theorem, Harmonic Function, The Dirichlet Problem Simply Periodic Function and Elliptic Functions, The Weierstrass Theory Analytic Continuation, Algebraic Functions, Picard's TheoremPoint set topology of Rn Countable and uncountable sets, the axiom of choice, Zorn's lemma.Metric spaces. Completeness; separability; compactness; Baire category; uniformcontinuity; connectedness; continuous mappings of co。