Real analysis
1. Review of metric spaces. Open and closed sets. Convergence. Complete spaces and compact spaces. Continuous and uniformly continuous functions.
2. Construction of a measure on a \( \sigma \)-algebra of subsets. Outer measure. Measurable sets and sets of measure zero.
3. The Lebesgue measure on \( \mathbb R^n \) . Borel sets and sets of measure zero. The \(\sigma\)-algebra of Lebesgue measurable sets. Approximation by open and closed sets. Measurable functions. Egoroff’s theorem, Lusin’s theorem.
4. Integration. Construction of the Lebesgue integral. Basic properties. Fatou’s lemma. Monotone convergence theorem. Dominated convergence theorem. Product measures and Fubini’s theorem on \( \mathbb R^n \) .
5. The Radon -
Nikodym theorem. The Lebesgue differentiation theorem. Functions of bounded variation on the real line. Absolutely continuous functions.
6. \( L^p \) spaces. Measure spaces and measurable functions. The spaces \( L^1 \) and \(L^2\) , including completeness. The inner product in \( L^2 \) . Convolutions and their continuity and smoothing properties.
7. The Fourier transform on \(L^2\) (\( \mathbb R^n ) \) . Definition and basic properties. Inversion formula and Plancherel’s theorem. Applications to PDE’s with constant coefficients.
Complex analysis
1. Basics. Geometric description of complex numbers. The complex plane and the Riemann sphere. Conformal mappings. Linear transformations as conformal maps. Representation by complex numbers.
2. Holomorphic functions. Definition. Exponential and trigonometric functions. Conformality and the Cauchy-Riemann equations. Relation to harmonic functions. Power series: uniform convergence, Weierstrass’ Mtest, continuity, integrability, differentiability. Integration along curves, primitives.
3. Cauchy’s Theorem and Applications. Goursat’s theorem. Cauchy’s theorem on a disk. Evaluation of integrals. Morera’s theorem. Cauchy integral formulas. Cauchy estimates and Liouville’s theorem. Fundamental theorem of algebra. Isolated zeros and analytic continuation. Sequences of holomorphic functions. Schwarz reflection principle.
4. Meromorphic functions. Zeros and poles. Laurent series. The residue formula for some domains. Jordan and “small arc” lemmas. Computation of integrals by residue calculus. Riemann’s theorem on removable singularities. Essential singularities and Casorati-Weierstrass theorem. The argument principle and applications (Rouché’s theorem, open mapping theorem, maximum modulus theorem).
5. Plane topology. Simply and multiply connected domains. Jordan curve theorem (without proof). Homotopies and the winding number. General form of Cauchy’s theorem. Roots and logarithm (including branches and cuts). Additional examples of computation of integrals using residues.
6. Conformal maps. Elementary conformal maps. Schwarz lemma and automorphisms of the disk and upperhalf plane. Fractional linear transformations (cross ratio, behavior of lines and circles). Normal familes. Montel’s theorem and the Riemann mapping theorem.
Functional analysis
1.Normed spaces. Banach spaces. Linear operators. Examples.
2. Spaces of bounded linear operators. The uniform boundedness principle and the open mapping theorem.
3. Bounded linear functionals. Dual spaces. The Hahn-Banach extension theorem. Separation of convex sets.
4. Spaces of continuous functions. Ascoli’s theorem, Stone-Weierstrass’ theorem. The space \( C^{0,\gamma } \) of Holder continuous functions and the space \(C^k \) of \(k \)-times differentiable functions.
5. Hilbert spaces. Perpendicular projections. Orthonormal bases. Selfadjoint operators.
6. Compact operators on a Hilbert space. Fredholm’s alternative. Spectrum and eigenfunctions of a compact, self-adjoint operator. Applications to Sturm-Liouville boundary value problems.
ODEs and PDEs
1.Existence and uniqueness theorems for solutions of ODE; explicit
solutions of simple equations.
2.self-adjoint boundary value
problems on finite intervals; critical points, phase space, stability
analysis
3.First order partial differential equations, linear and quasi-linear
PDE.Heat equation, Dirichlet problem, fundamental solutions.
Green functions and existence of solutions of
Dirichlet problem, harmonic functions, maximal principle and
applications, existence of solutions of Neumann’s problem
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Logic
Propositional formulas (predicate formulas) and representations, conjunctions; propositional formulas (predicate formulas) of tautology, implication, and equivalence. The paradigm of the proposition formula, the toe paradigm of the predicate formula; constraint and free variables; reasoning theory.
Set theory
The concept and representation of sets, the operations of sets; sequenced pairs and Cartesian products; binary relations, five properties of relations, compound relations and inverse relations; closure operations of relations; equivalence relations, division; compatibility relations, coverage; Partially ordered relations, partially ordered sets,
The potential of the set
Abstruct algebra
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(\(n\), \( \mathbb F\)) and its subgroups), \(p\)-groups,
solvable and nilpotent groups, group extensions, semi-direct products, free groups,
amalgamated products and 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
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,Galois theory of number fields,
transcendence degree, function fields.
Module
Modules and algebra Free and projective; tensor products; irreducible modules and
Schur’s lemma; se misimple, simple and primitive rings; density and Wederburn theorems;
the structure of finitely generated modules over principal ideal domains
Lattice
Definition of lattices, sublattices, lattice homomorphism , product algebra; distributive lattices, bounded lattices, complementary lattice, complementary distributive lattice and properties
Graph theory
Graphs and related concepts, graph isomorphism; roads and loops, connectivity; matrix representation of graphs; Euler graphs and Hamilton graphs; trees, spanning trees.
Number theory
Integers, Euclidean algorithm, unique decomposition;Euler theorem, Fermat theorem; congruence and the Chinese Remainder theorem; Quadratic reciprocity ; Indeterminate Equations. Polynomials, Euclidean algorithm, uniqueness decomposition, zeros; The fundamental theorem of algebra.
