Mar 5-Jun 22, 2018 Exam Period Ends: October 12, 2018
Coding Theory investigates error-detection and error-correction. Such errors can occur in various communication channels: satellite communication, cellular telephones, CDs and DVDs, barcode reading at the supermarket, etc. A mathematical analysis of the notions of error detection and correction leads to deep combinatorial problems, which can be sometimes solved using techniques ranging from linear algebra and ring theory to algebraic geometry and number theory. These techniques are in fact used in the above-mentioned communication technologies.
- The main problem of Coding Theory
- Bounds on codes
- Finite fields
- Linear codes
- Perfect codes
- Cyclic codes
- Sphere packing
- Asymptotic bounds
R. Hill, A First Course in Coding Theory, Clarendon Press, Oxford 1986
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.
- Fields: basic properties and examples, the characteristic, prime fields
- Polynomials: irreducibility, the Eisenstein criterion, Gauss’s lemma
- Extensions of fields: the tower property, algebraic and transcendental extensions, adjoining an element to a field
- Ruler and compass constructions
- Algebraic closures: existence and uniqueness
- Splitting fields
- Galois extensions: automorphisms, normality, separability, fixed fields, Galois groups, the fundamental theorem of Galois theory.
- Cyclic extensions
- Solving polynomial equations by radicals: the Galois group of a polynomial, the discriminant, the Cardano-Tartaglia method, solvable groups, Galois theorem
- Roots of unity: cyclotomic fields, the cyclotomic polynomials and their irreducibility
- Finite fields: existence and uniqueness, Galois groups over finite fields, primitive elements
- 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 spaces and continuous functions (product topology, quotient topology, metric topology). Connectedness and Compactness. Countabilty Axioms and Separation Axioms (the Urysohn lemma, the Urysohn metrization theorem, Partition of unity). The Tychonoff theorem and the Stone-Cech compactification. Metrization theorems and paracompactness.
- Complex numbers. Analytic functions, Cauchy–Riemann equations.
- Conformal mappings, Mobius transformations.
- Integration. Cauchy Theorem. Cauchy integral formula. Zeroes, poles, Taylor series, Laurent series. Residue calculus.
- The theorems of Weierstrass and of Mittag-Leffler. Entire functions. Normal families.
- Riemann Mapping Theorem. Harmonic functions, Dirichlet problem.
- Categories and functors: natural transformations, equivalence, adjoint functors, additive functors, exactness.
- Derived functors: projective, injective and flat modules; resolutions, the functors and ; examples and applications.
- Nonabelian cohomology and its applications.
Permutation representation and the Sylow theorems.
Representations of groups on groups, solvable groups, nilpotent groups, semidirect and central products.
Permutation groups, the symmetric and alternating groups.
The generalized Fitting subgroup of a finite group.
Extension of groups: The first and second cohomology and applications.
- vector bundles and K-groups of topological spaces
- Bott’s Periodicity theorem and applications to division algebras
- Index of Fredholm operators and K-theory
- If time permits: smooth manifolds, DeRham cohomology, Chern classes, Elliptic operators, formulation of Atiyah-Singer index theorem, relations to Gauss-Bonnet theorem
- Affine algebraic sets and varieties.
- Local properties of plane curves.
- Projective varieties and projective plane curves.
- Riemann–Roch theorem.
This course deals with random walks, harmonic functions, the relations between these notions, and their applications to geometry and algebra (mainly to finitely generated groups).
The modern point of view will be presented, following recent texts by: Gromov, Kleiner, Ozawa, Shalom & Tao, among others.
The course is intended for 3rd year undergraduate as well as M.Sc and Ph.D. students both in computer science and mathematics. We will touch main topics in the area of discrete geometry. Some of the topics are motivated by the analysis of algorithms in computational geometry, wireless and sensor networks. Some other beautiful and elegant tools are proved to be powerful in seemingly non-related areas such as additive number theory or hard Erdos problems. The course does not require any special background except for basic linear algebra, and a little of probability and combinatorics. During the course many open research problems will be presented.
- Fundamental theorems and basic definitions: Convex sets, convex combinations, separation , Helly’s theorem, fractional Helly, Radon’s theorem, Caratheodory’s theorem, centerpoint theorem. Tverberg’s theorem. Planar graphs. Koebe’s Theorem. A geometric proof of the Lipton-Tarjan separator theorem for planar graphs.
- Geometric graphs: the crossing lemma. Application of crossing lemma to Erdos problems: Incidences between points and lines and points and unit circles. Repeated distance problem, distinct distances problem. Selection lemmas for points inside discs, points inside simplexes. Counting k-sets. An application of incidences to additive number theory.
- Coloring and hiting problems for geometric hypergraphs : VC-dimension, Transversals and Epsilon-nets. Weak eps-nets for convex sets. Conflict-free colorings .
- Arrangements : Davenport Schinzel sequences and sub structures in arrangements. Geometric permutations.
- Geometric Ramsey and Turan type theorems: Application of Dilworth theorem, Erdos-Szekeres theorem for convex sets, quasi-planar graphs.
We will present some basic notions and constructions from model theory, motivated by concrete questions about structures and their theories. Notions we expect to cover include:
- Types and spaces of types
- Homogeneous and saturated models
- Quantifier elimination and model companions
- Elimination of imaginaries
- Definable groups and fields
Students should be familiar with the following concepts from logic: Languages, structures, formulas, theories, the compactness theorem. In addition, some familiarity with field theory, topology and probability will be beneficial.
- 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.