丘成桐大學生數學競賽

居然有這種東西了...
簡體網站 繁體網站

競賽內容

競賽將全面測試學生的知識與技能。測試將分成4個科目：
1 分析與微分方程
2 幾何與拓撲
3 代數、數論與組合
4 應用、計算和概率統計

競賽學術與出題委員會將提供每個科目的詳細大綱和參考書。每個科目將有五道題目。每
個參加者可選其中的三個科目。

競賽前15名優勝者將參加 2010年12月14日的口試。

但是大綱和參考書有點太誇張了...
大學生考這個!?

幾何與拓樸
Space curves and surfaces
Curves and Parametrization, Regular Surfaces;
Inverse Images of Regular Values.
Gauss Map and Fundamental Properties;
Isometries;
Conformal Maps;
Rigidity of the Sphere.

Topological space
Space, maps, compactness and connectedness, quotients;
Paths and Homotopy.
The Fundamental Group of the Circle.
Induced Homomorphisms.
Free Products of Groups.
The van Kampen Theorem.
Covering Spaces and Lifting Properties;
Simplex and complexes. Triangulations.
Surfaces and its classification.

Differential Manifolds
Differentiable Manifolds and Submanifolds,
Differentiable Functions and Mappings;
The Tangent Space,
Vector Field and  Covector Fields.
Tensors and Tensor Fields and differential forms.
The Riemannian Metrics as examples,
Orientation and Volume Element;
Exterior Differentiation and Frobenius's Theorem;
Integration on manifolds,
Manifolds with Boundary and Stokes' Theorem.

Homology and cohomology
Simplicial and Singular Homology.
Homotopy Invariance.
Exact Sequences and Excision.
Degree.
Cellular Homology.
Mayer-Vietoris Sequences.
Homology with Coefficients.
The Universal Coefficient Theorem.
Cohomology of Spaces.
The Cohomology Ring.
A Kunneth Formula.
Spaces with Polynomial Cohomology.
Orientations and Homology.
Cup Product and Duality.

Riemannian Manifolds
Differentiation  and connection,
Constant Vector Fields and Parallel Displacement
Riemann Curvatures and the Equations of Structure
Manifolds of Constant Curvature,
Spaces of Positive Curvature,
Spaces of Zero Curvature,
Spaces of Constant Negative Curvature

References:
Do Carmo , Differentia geometry of curves and surfaces.
Chen Qing and Chia Kuai Peng, Differential Geometry
Armstrong,  Basic Topology
Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry
Spivak, A comprehensive introduction to differential geometry
Hicks, Notes on differential geometry
Frenkel, Geometry of Physics
Milnor, Morse Theory
Hatcher, Algebraic Topology
Milnor, Topology from the differentiable viewpoint
Bott and Tu, Differential forms in algebraic topology
Guillemin, A. Pollack, Differential topology

代數、數論、組合
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 polynomials
Integers, 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,
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,
group presentations.

Ring
Basic 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 Wedderburn 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 representation
Irreducible 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 Algebra
Basic concepts,
semisimple Lie algebras,
root systems,
isomorphism and conjugacy theorems,
representation theory.

Combinatorics (TBA)

References:
Strang, Linear algebra,
Gelfand, Linear Algebra
《整數與多項式》馮克勤 餘紅兵著
Jacobson, Basic algebra. I & II
Lang, Algebra,
馮克勤，李尚志，查建國，章璞，《近世代數引論》
劉紹學，《近世代數基礎》
Serre, Linear representations of finite groups
Serre: Complex semisimple Lie algebra and their representations
Humphreys: Introduction to Lie algebra and representation theory,
Fulton, Representation theory, a First Course,

 作者  Tass (為文載道尊於勢)                                       看板  Tass
 標題  Re: [數學] 丘成桐大學生數學競賽
 時間  Thu Sep  9 13:12:51 2010
───────────────────────────────────────

分析與微分方程
Calculus and mathematical analysis
Derivatives, 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 ODEs,
explicit solutions of simple equations;
elementary Fourier series.

Complex analysis
Analytic 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 Theorem

Point set topology of R^n
Countable and uncountable sets,
the axiom of choice,
Zorn's lemma.
Metric spaces.
Completeness;
separability;
compactness;
Baire category;
uniform continuity;
connectedness;
continuous mappings of compact spaces.
Functions on topological spaces.
Equicontinuity and Ascoli's theorem;
the Stone-Weierstrass theorem;
topologies on function spaces;
compactness in function spaces.

Measure and integration
Measures;
Borel sets and cantor sets;
Lebesgue measures;
distributions;
product measures.
Measurable functions.
approximation by simple functions;
convergence in measure;
Construction and properties of the integral;
convergence theorems;
Radon-Nykodym theorem;
Fubini's theorem;
mean convergence.
Monotone functions;
functions of bounded variation and Borel measures;
Absolute continuity,
convex functions;
semicontinuity.

Banach and Hilbert spaces
Lp spaces;
C(X);
completeness and the Riesz-Fischer theorem;
orthonormal bases;
linear functionals;
Riesz representation theorem;
linear transformations and dual spaces;
interpolation of linear operators;
Hahn-Banach theorem;
open mapping theorem;
uniform boundedness (or Banach-Steinhaus) theorem;
closed graph theorem.
Basic properties of compact operators,
Riesz- Fredholm theory,
spectrum of compact operators.
Basic properties of Fourier series and the Fourier transform;
Poission summation formula;
convolution.

Basic partial differential equations
First order partial differential equations,
linear and quasi-linear PDE,
Wave equations:
    initial condition and boundary condition,
    well-poseness,
    Sturn-Liouville eigen-value problem,
    energy functional method,
    uniqueness and stability of solutions
Heat equations:
    initial conditions,
    maximal principle
    uniqueness and stability
Potential equations:
    Green functions
    existence of solutions of Dirichlet problem,
    harmonic functions,
    Hopf's maximal principle
    existence of solutions of Neumann’s problem,
    weak solutions,
    eigen-value problem of the Laplace operator
Generalized functions and fundamental solutions of PDE

References:
Rudin, Principles of mathematical analysis
Courant & John, Introduction to calculus and analysis.
Arnold,  Ordinary Differential Equations,
Ahlfors, An Introduction to the Theory of Analytic Functions of One Complex
    Variable
Kodaira, Complex Analysis
Rudin, Real and complex analysis
龔升，簡明複分析
Royden, Real Analysis, except chapters 8, 13, 15.
Stein & Shakarchi; Real Analysis: Measure Theory, Integration, and Hilbert
    Spaces,
周民強, 實變函數論,
夏道行等，《實變函數論與泛函分析》
Lax, Functional Analysis,
Basic Partial Differential Equations, Bleecker, Csordas
《數學物理方法》，柯朗、希爾伯特著。

作者  Tass (為文載道尊於勢)                                       看板  Tass
 標題  Re: [數學] 丘成桐大學生數學競賽
 時間  Thu Sep  9 13:24:27 2010
───────────────────────────────────────

計算數學、應用數學、機率和統計

Computational Mathematics
Interpolation and approximation
Polynomial interpolation and least square approximation;
trigonometric interpolation and approximation,
fast Fourier transform;
approximations by rational functions;
splines.

Nonlinear equation solvers
Convergence of iterative methods
    (bisection, secant method, Newton method, other iterative methods)
    for both scalar equations and systems;
finding roots of polynomials.

Linear systems and eigenvalue problems
Direct solvers
    (Gauss elimination, LU decomposition, pivoting, operation count,
     banded matrices, round-off error accumulation);
iterative solvers
    (Jacobi, Gauss-Seidel, successive over-relaxation, conjugate gradient
     method, multi-grid method, Krylov methods);
numerical solutions for eigenvalues and eigenvectors

Numerical solutions of ordinary differential equations
One step methods
    (Taylor series method and Runge-Kutta method);
stability, accuracy and convergence;
absolute stability, long time behavior;
multi-step methods

Numerical solutions of partial differential equations
Finite difference method;
stability,
accuracy and convergence,
Lax equivalence theorem;
finite element method,
boundary value problems

References:

De Boor and Conte, Elementary Numerical Analysis, an algorithmic approach
Golub and van Loan, Matrix Computations,
Hairer, Syvert and Wanner, Solving Ordinary Differential Equations
Gustafsson, Kreiss and Oliger, Time Dependent Problems and Difference Methods
Strang and Fix, An Analysis of the Finite Element Method

Applied Mathematics
ODE with constant coefficients;
Nonlinear ODE:
critical points,
phase space & stability analysis;
Hamiltonian, gradient, conservative ODE's.

Calculus of Variations:
Euler-Lagrange Equations;
Boundary Conditions,
parametric formulation;
optimal control and Hamiltonian,
Pontryagin maximum principle.

1st order partial differential equations (PDE) and method of characteristics;
Heat, wave, and Laplace’s equation;
Separation of variables and eigen-function expansions;
Stationary phase method;
Homogenization method for elliptic and linear hyperbolic PDEs;
Homogenization and front propagation of Hamilton-Jacobi equations;
Geometric optics for dispersive wave equations.

References:
Boyce and DiPrima, Elementary Differential Equations
Wan, Introduction to Calculus of Variations and Its Applications,
Whitham, Linear and Nonlinear Waves
Keener, Principles of Applied Mathematics
Benssousan, Lions, Papanicolaou, Asymptotic Analysis for Periodic Structures
Jikov, Kozlov, Oleinik, Homogenization of differential operators and integral functions
Xin, An Introduction to Fronts in Random Media

Probability
Random Variables;
Conditional Probability and Conditional Expectation;
Markov Chains;
The Exponential Distribution and the Poisson Process;
Continuous-Time Markov Chains;
Renewal Theory and Its Applications;
Queueing Theory;
Reliability Theory;
Brownian Motion and Stationary Processes;
Simulation.

Reference:
Ross, Introduction to Probability Models

Statistics
Distribution Theory and Basic Statistics
Families of continuous distributions:
    Chi-sq, t, F, gamma, beta;
Families of discrete distributions:
    Multinomial, Poisson, negative binomial;
Basic statistics:
    Mean, median, quantiles, order statistics
Likelihood principle
Likelihood function,
parametric inference,
sufficiency,
factorization theorem,
ancillary statistic,
conditional likelihood,
marginal likelihood.
Testing
Neyman-Pearson paradigm,
null and alternative hypotheses,
simple and composite hypotheses,
type I and type II errors,
power, most powerful test,
likelihood ratio test,
Neyman-Pearson Theorem,
monotone likelihood ratio,
uniformly most powerful test,
generalized likelihood ratio test.
Estimation
Parameter estimation,
method of moments,
maximum likelihood estimation,
unbiasedness,
quadratic and other criterion functions,
Rao-Blackwell Theorem,
Fisher information,
Cramer-Rao bound,
confidence interval,
duality between
confidence interval and hypothesis testing.
Bayesian Statistics
Prior,
posterior,
conjugate priors,
Bayesian loss
Nonparametric statistics
Permutation test,
permutation distribution,
rank statistics,
Wilcoxon-Mann-Whitney test,
log-rank test,
Kolmogorov-Smirnov-type tests.
Regression
Linear regression,
least squares method,
Gauss-Markov Theorem,
logistic
regression,
maximum likelihood
Large sample theory
Consistency,
asymptotic normality,
chi-sq approximation to likelihood ratio statistic,
large-sample based confidence interval,
asymptotic properties of empirical distribution.

References
Casella and Berger. Statistical Inference
茆詩松，程依明，濮曉龍，概率論與數理統計教程
陳家鼎，孫山澤，李東風，劉力平，數理統計學講義
鄭明，陳子毅，汪嘉岡，數理統計講義
陳希孺，倪國熙，數理統計學教程