Đề cương các môn học của Cao học Quốc tế

Đề cương các môn học của Cao học Quốc tế

(1)  Linear algebra and commutative algebra (*)

• Modules, free modules, exact sequences and tensor product

• Ideal and modules over a PID

• Noetherian rings

• Primary decomposition

(2)  Linear representation of finite groups (*)

• Characters, Schur lemma

• Regular representation, induced representation

• Frobenius reciprocity

(3)  Homological algebra (**)

• Abelian categories, derived functors

• Cohomology of groups

Geometry

(1)  Algebraic topology (**)

• Singular homology and cohomology

• Fundamental groups and covering

(2)  Differential geometry (**)

• Varieties, tangent and cotangent bundles

• Vector bundles, connections

• De Rham complex

• Characteristic classes

(3)  Morse Theory (**)

(1)  Galois theory (*)

• Galois groups

• Finite fields, Frobenius substitution

• Quadratic extension, cyclotomic extension

(2)  p-adic numbers (**)

• Hensel lemma

• Congruence, Bernouilli numbers

(3)  Algebraic numbers (**)

• Algebraic extension, ramification,

• Inert and decomposition classs

• Ideal class groups

• Chebotarev theorem

(4)  Adeles, Tate thesis, zeta function (**)

(1)  Complex analysis (*)

• Holomorphic and meromorphic functions, Hartogs phenomenon

• d-bar equation, Mittag-Leffler problem

• Weierstrass theorem, analytic sets

• Automorphisms, Kobayashi metrics

(2)  Functional analysis (*)

• Banach and Hilbert spaces

• Spectral theory

(3)  Ordinary differential equations (*)

• Dynamical systems

• Bifurcation theory

• Application in enviorment, biology,…

(4)Distribution (**)

• Distributions, Fourier transform

• Sobolev spaces

(5)  Partial differential equations (**)

• Cauchy problem of classical equations (transport, waves, heat, Schr¨odinger,..)

• Problems with elliptic limits

(6) Numerical analysis (**)

(1)  Notion of probability (*)

• Space of probability

• Random variables

• Independence

• Gaussian distributions

(2)  Convergence (*)

• Borel-Cantelli theorem

• Law of the large numbers

• Central limit theorem

(3)  Conditional expectations and martingales techniques (**)

• Definition and Properties.

• Condition and independence

• Martingales techniques

(1)  Non-linear analysis (*)

• Convex analysis

• Non-smooth analysis

• Fixed-point theory

(2)  Optimization (*)

• Optimal conditions

• Duality

• Algorithms

• Optimal control

(3)  Game theory (**)

• Information and rationality

• Zero-sum games

• Equilibrium et stability

(1)  Discrete mathematics (*)

• Graph theory

• Basic combinatorial methods

(2)  Algorithms (*)

• Deterministic Algorithms

• Randomized algorithms

(3)  Information Theory (**)

• Coding theory

• Complexity and Cryptography

(1) English