2015–16–A

• Basic concepts of topology of metric spaces: open and closed sets, connectedness, compactness, completeness.
• Normed spaces and inner product spaces. All norms on $\mathbb{R}^n$ are equivalent.
• Theorem on existence of a unique fixed point for a contraction mapping on a complete metric space.
• Differentiability of a map between Euclidean spaces. Partial derivatives. Gradient. Chain rule. Multivariable Taylor expansion.
• Open mapping theorem and implicit function theorem. Lagrange multipliers. Maxima and minima problems.
• Riemann integral. Subsets of zero measure and the Lebesgue integrability criterion. Jordan content.
• Fubini theorem. Jacobian and the change of variables formula.
• Path integrals. Closed and exact forms. Green’s theorem.
• Time permitting, surface integrals, Stokes’s theorem, Gauss’ theorem

Ordinary differential equations of first order, existence and uniqueness theorems, linear equations of order n and the Wronskian, vector fields and autonomous equations, systems of linear differential equations, nonlinear systems of differential equations and stability near equilibrium

Algebras and sigma-algebras of subsets, the extension theorem and construction of Lebesgue’s measure on the line, general measure spaces, measurable functions and their distribution functions, integration theory, convergence theorems (Egorov’s, relations between convergence in measure and a.e. convergence), the spaces $L_1$ and $L_2$ and their completeness, signed measures, the Radon-Nikodym theorem, measures in product spaces and Fubini’s theorem.

• Cesaro means: Convolutions, positive summability kernels and Fejer’s theorem.
• Applications of Fejer’s theorem: the Weierstrass approximation theorem for polynomials, Weyl’s equidistribution theorem, construction of a nowhere differentiable function (time permitting).
• Pointwise and uniform convergence and divergence of partial sums: the Dirichlet kernel and its properties, construction of a continuous function with divergent Fourier series, the Dini test.
• $L^2$ approximations. Parseval’s formula. Absolute convergence of Fourier series of $C^1$ functions. Time permitting, the isoperimetric problem or other applications.
• Applications to partial differential equations. The heat and wave equation on the circle and on the interval. The Poisson kernel and the Laplace equation on the disk.
• Fourier series of linear functionals on $C^n(\mathbb{T})$. The notion of a distribution on the circle.
• Time permitting: positive definite sequences and Herglotz’s theorem.
• The Fourier transform: convolutions, the inversion formula, Plancherel’s theorem, Hermite functions. Time permitting: tempered distributions, further applications to differential equations.
• Fourier analysis on finite cyclic groups, and the Fast Fourier Transform algorithm.

• An axiom system for predicate calculus and the completeness theorem.
• Introduction to model theory: The compactness Theorem, Skolem–Löwenheim Theorems, elementary substructures.
• Decidability and undecidability of theories, Gödel first Incompleteness Theorem.

Graphs and sub-graphs, trees, connectivity, Euler tours, Hamilton cycles, matching, vertex and edge colorings, planar graphs, introduction to Ramsey theory, directed graphs, probabilistic methods and linear algebra tools in Graph Theory.

• Groups as symmetries. Examples: cyclic, dihedral, symmetric and matrix groups.
• Homomorphism. Subgroups and normal subgroups. Quotient groups. Lagrange’s theorem. The isomorphism theorems. Direct products of groups.
• Actions of groups on sets. Cayley’s theorem.
• Group automorphisms.
• Sylow’s theorems. Application: classification of groups of small order.
• Composition series and Jordan–Hoelder theorem. Solvable groups.
• Classification of finite abelian groups, finitely-generated abelian groups.
• Symmetric group and alternating group. The alternating group is simple.
• Rings, maximal and prime ideals, integral domain, quotient ring. Homomorphism theorems.
• Multilinear algebra: Quotient spaces. Tensor products of vector spaces. Action of $S_n$ on tensor powers. Exterior and symmetric algebras. Multilinear forms and determinant.
• Optional topics: group of symmetries of platonic solids, free groups, semidirect products, representation theory of finite groups.

An introduction to the basic notions of probability theory:

sample spaces limits of events conditional probability independent events sigma algebras, continuous spaces, Lebesgue measure random variables and distributions independence expectation variance and covariance convergence of random variables: almost-sure, in Lp, in probability law of large numbers convergence in law central limit theorem