Fall 2016 term
Oct 25, 2015-Jan 22, 2016 Exam Period Ends: March 4, 2016
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 and and their completeness, signed measures, the Radon-Nikodym theorem, measures in product spaces and Fubini’s theorem.
- 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.
- Groups, the factor group and the homomorphism theorems, Sylow’s theorems and permutation actions of groups.
- Rings, Integral Domains and Fields. Ideals: maximal and prime. Unique Factorization Domains, Principle Ideal Domains, Euclidean Domains.
- Modules, structure theorems for finitely generated modules over a PID, application to finitely generated abelian groups and to the Jordan Canonical Form.
- Basic concepts of topology of metric spaces: open and closed sets, connectedness, compactness, completeness.
- Normed spaces and inner product spaces. All norms on 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
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.
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
- 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.
- approximations. Parseval’s formula. Absolute convergence of Fourier series of 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 . 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.
- Topological manifolds. The fundamental group and covering spaces. Applications.
- Singular homology and applications.
- Smooth manifolds. Differential forms and Stokes’ theorem, definition of de-Rham cohomology.
- Additional topics as time permits.
- Some review of complex variables
- Hilbert spaces of analytic functions. General theory
- Bergman space
- Hardy spaces
- Fractional Hardy space and self similar systems
- Bargmann-Fock space and second quantization
- Banach spaces of analytic functions.
- Frechet spaces and Schwartz spaces
- Proof of Riemann’s mapping theorem
- Dual of the space of functions analytic in a given open set
- Gelfand triples and applications
Banach spaces and Hilbert spaces. Basic properties of Hilbert spaces. Topological vector spaces. Banach-Steinhaus theorem; open mapping theorem and closed graph theorem. Hahn-Banach theorem. Duality. Measures on locally compact spaces; the dual of . Weak and weak- topologies; Banach-Alaoglu theorem. Convexity and the Krein-Milman theorem. The Stone-Weierstrass theorem. Compact operators on Hilbert space. Introduction to Banach algebras and Gelfand theory. Additional topics as time permits.
- Review of abelian categories and additive functors.
- Differential graded rings, modules and categories.
- The derived category of a differential graded category.
- Derived functors. Resolutions of differential graded modules.
- Courses marked with (*) are required for admission to the M.Sc. program in Mathematics.
- The M.Sc. degree requires the successful completion of at least 2 courses marked (#). See the graduate program for details
- The graduate courses are open to strong undergraduate students who have a grade average of 85 or above and who have obtained permission from the instructors and the head of the teaching committee.
- Please see the detailed undergraduate and graduate programs for the for details on the requirments and possibilities for complete the degree.