2017–18–A term

Infinitesimal Calculus 3(!)Pdf 201.1.0031

Dr. Inna Entova-Aizenbud 6.0 Credits

יום א 16:00 - 14:00 בצוקר, גולדשטיין-גורן [72] חדר 213
יום ג 13:00 - 12:00 בקרייטמן-זלוטובסקי(חדש) [34] חדר 303
יום ה 16:00 - 14:00 בגוטמן [32] חדר 111

  • 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(!)Pdf 201.1.0061

Prof. Victor Vinnikov 5.0 Credits

יום ג 12:00 - 10:00 בבנין 90 (מקיף ז’) [90] חדר 234
יום ה 10:00 - 08:00 בגוטמן [32] חדר 309

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

Fundamentals of Measure Theory(*)Pdf 201.1.0081

Prof. Ilan Hirshberg 4.0 Credits

יום ב 16:00 - 14:00 בגולדברגר [28] חדר 304
יום ד 16:00 - 14:00 בבנין 90 (מקיף ז’) [90] חדר 144

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.

Theory of NumbersPdf 201.1.6031

Dr. Ishai Dan-Cohen 4.0 Credits

יום ב 16:00 - 12:00 בבנין 90 (מקיף ז’) [90] חדר 223
יום ה 12:00 - 10:00 בגוטמן [32] חדר 207

Number Theory studies the structure of the integers and the natural numbers. In addition to classical topics (prime numbers, congruences, quadratic residues, etc.) there is an emphasis on algorithmic questions and in particular on applications to cryptography.

  • Divisibility and prime numbers
  • Congruences
  • The multiplicative group of
  • Quadratic residues
  • Continued fractions
  • Algebraic numbers and algebraic integers

LogicPdf 201.1.6061

Dr. Moshe Kamensky 4.0 Credits

יום ב 12:00 – 10:00 בגוטמן [32] חדר 309 יום ד 12:00 – 10:00 בגולדברגר [28] חדר 103

  • 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.

Algebraic StructuresPdf 201.1.7031

Prof. Dmitry Kerner 4.0 Credits

יום א 18:00 - 16:00 בקרייטמן-זלוטובסקי(חדש) [34] חדר 116
יום ד 14:00 - 12:00 בבנין 90 (מקיף ז’) [90] חדר 237

  • 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.

ProbabilityPdf 201.1.8001

Prof. Ariel Yadin 4.0 Credits

יום א 11:00 - 09:00 בצוקר, גולדשטיין-גורן [72] חדר 488
יום ג 18:00 - 16:00 בצוקר, גולדשטיין-גורן [72] חדר 489

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

Game TheoryPdf 201.1.8131

Prof. Daniel Berend

יום א 12:00 - 11:00 בקרייטמן-זלוטובסקי(חדש) [34] חדר 205
יום ד 18:00 - 16:00 בבנין 90 (מקיף ז’) [90] חדר 144

The course will present Game Theory mostly from a mathematical point of view. Topics to be covered:

  1. Combinatorial games.
  2. Two-person zero-sum games.
  3. Linear programming.
  4. General-sum games.
  5. Equilibrium points.
  6. Random-turn games.
  7. Stable marriages.
  8. Voting.

Basic Concepts in Topology and Geometry(#)Pdf 201.2.5221

Dr. Michael Brandenbursky 4.0 Credits

יום א 12:00 - 10:00
יום ג 12:00 - 10:00

  • 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.
Course Topics
  1. Modules: free modules, exact sequences, tensor products, Hom modules, flatness.
  2. Prime ideals and localization: local rings, Nakayama’s Lemma, the spectrum of a ring, dimension and connectedness.
  3. Noetherian rings: the Hilbert basis theorem, the Artin-Rees lemma, completion, grading.
  4. Dimension theory: the Hilbert nullstellensatz, Noether normalization, transcendence degree.

Basic Concepts in Modern Analysis(#)Pdf 201.2.0351

Dr. Izhar Oppenheim

יום א 14:00 - 12:00 בבנין 90 (מקיף ז’) [90] חדר 225
יום ד 18:00 - 16:00 בגולדברגר [28] חדר 204

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.

2017–18–B term

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.

  1. Partially ordered sets. Chains and antichains. Examples. Erdos-Szekeres’ theorem or a similar theorem. The construction of a poset over the quotient space of a quasi-ordered set.
  2. Comparison of sets. The definition of cardinality as as an equivalence class over equinumerousity. The Cantor-Bernstein theorem. Cantor’s theorem on the cardinality of the power-set.
  3. Countable sets. The square of the natural numbers. Finite sequences over a countable set. Construction of the ordered set of rational numbers. Uniqueness of the rational ordering.
  4. Ramsey’s theorem. Applications.
  5. The construction of the ordered real line as a quotient over Cauchy sequences of rationals.
  6. Konig’s lemma on countably infinite trees with finite levels. Applications. A countable graph is k-colorable iff every finite subgraph of it is k-colorable.
  7. Well ordering. Isomorphisms between well-ordered sets. The axiom of choice formulated as the well-ordering principle. Example. Applications. An arbitrary graph is k–colorable iff every finite subgraph is k-colorable.
  8. Zorn’s lemma. Applications. Existence of a basis in a vector space. Existence of a spanning tree in an arbitrary graph.
  9. Discussion of the axioms of set theory and the need for them. Russle’s paradox. Ordinals.
  10. Transfinite induction and recursion. Applications. Construction of a subset of the plane with exactly 2 point in every line.
  11. Infinite cardinals as initial ordinals. Basic cardinal arithmetic. Cardinalities of well known sets. Continuous real functions, all real runctions, the automorphisms of the real field (with and without order).
  • 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.

Introduction to partial differential equations ? The first order equations: a linear equation, a quasilinear equation, resolving the initial value problem by the method of characteristic curves. ? Classification of the second order equations: elliptic, hyperbolic and parabolic equations, exam- ples of Laplace, Wave and Heat equations. ? Elliptic equations: Laplace and Poisson?s equations, Dirichlet and Neumann boundary value problems, Poisson?s kernel, Green?s functions, properties of harmonic functions, Maximum prin- ciple. ? Analytical methods for resolving partial differential equations: Sturm-Liouville problem and the method of separation of variables for bounded domains, applications for Laplace, Wave and Heat equations including non-homogenous problems. Applications of Fourier and Laplace transforms for resolving problems in unbounded domains. ? Heat equation: initial value problem in unbounded domain, basic formula for the solution, initial-boundary value problems in bounded domains, Maximum principle. ? Wave equation: Dalamber formula, non-homogenous equation, Wave equation in higher dimen- sions. ? If time permits: Legendre polynomials and spherical functions. Literature: ? Pinchover Y.; Rubinstein J. Introduction to partial differential equations (in Hebrew), Depart- ment of mathematics, Technion, 2011, ? John F. Partial differential equations, Reprin t of the fourth edition. Applied Mathematical Scien ces, 1. Springer-Verlag, New York, 1991, ? Evans Lawrence C. Partial Differential Equations, Second Edition, ? Gilbarg D.; Trudinger N. S. Elliptic partial differential equations of second order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Ver lag, Berlin, 2001, ? Zauderer E. Partial differential equations of applied mathematics, Second edition. Pure and Appli ed Mathematics (New York). A Wiley-Interscience Public ation. John Wiley & Sons, Inc., New York, 1989. xvi+891 pp. ISBN: 0-471-61298-7. 1

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.

  1. The main problem of Coding Theory
  2. Bounds on codes
  3. Finite fields
  4. Linear codes
  5. Perfect codes
  6. Cyclic codes
  7. Sphere packing
  8. 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

Course Topics:

  1. Categories and functors: natural transformations, equivalence, adjoint functors, additive functors, exactness.
  2. Derived functors: projective, injective and flat modules; resolutions, the functors and ; examples and applications.
  3. 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.

  • -groups.

  • Extension of groups: The first and second cohomology and applications.

  1. Affine algebraic sets and varieties.
  2. Local properties of plane curves.
  3. Projective varieties and projective plane curves.
  4. Riemann–Roch theorem.
  • 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

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.

Detailed Syllabus:

  • 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.