Courses
Fall 2018
Undergraduate Courses
Game Theory Pdf 201.1.8131
Prof Daniel Berend
יום א 12:00  11:00 בקרייטמןזלוטובסקי(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:
 Combinatorial games.
 Twoperson zerosum games.
 Linear programming.
 Generalsum games.
 Equilibrium points.
 Randomturn games.
 Stable marriages.
 Voting.
Infinitesimal Calculus 3 Pdf 201.1.0031
Dr Inna EntovaAizenbud
יום א 16:00  14:00 בצוקר, גולדשטייןגורן [72] חדר 213
יום ג 13:00  12:00 בקרייטמןזלוטובסקי(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
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
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 sigmaalgebras 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 RadonNikodym theorem, measures in product spaces and Fubini’s theorem.
Theory of Numbers Pdf 201.1.6031
Dr Ishai DanCohen
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
Algebraic Structures Pdf 201.1.7031
Prof Dmitry Kerner
יום א 18:00  16:00 בקרייטמןזלוטובסקי(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.
Logic Pdf 201.1.6061
Dr Moshe Kamensky 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.
Probability Pdf 201.1.8001
Prof Ariel Yadin
יום א 11:00  09:00 בצוקר, גולדשטייןגורן [72] חדר 488
יום ג 18:00  16:00 בצוקר, גולדשטייןגורן [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: almostsure, in Lp, in probability law of large numbers convergence in law central limit theorem
Graduate Courses
Basic Concept in Toplogy and Geometry (#) Pdf 201.2.5221
Dr Michael Brandenbursky
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 deRham cohomology.
 Additional topics as time permits.
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. BanachSteinhaus theorem; open mapping theorem and closed graph theorem. HahnBanach theorem. Duality. Measures on locally compact spaces; the dual of . Weak and weak topologies; BanachAlaoglu theorem. Convexity and the KreinMilman theorem. The StoneWeierstrass theorem. Compact operators on Hilbert space. Introduction to Banach algebras and Gelfand theory. Additional topics as time permits.
Commutative Algebra Pdf 201.2.2011
Prof Amnon Yekutieli יום ד 12:0014:00
Course Topics
 Modules: free modules, exact sequences, tensor products, Hom modules, flatness.
 Prime ideals and localization: local rings, Nakayama’s Lemma, the spectrum of a ring, dimension and connectedness.
 Noetherian rings: the Hilbert basis theorem, the ArtinRees lemma, completion, grading.
 Dimension theory: the Hilbert nullstellensatz, Noether normalization, transcendence degree.
Spring 2018
Undergraduate Courses
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 StoneCech 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 MittagLeffler. Entire functions. Normal families.
 Riemann Mapping Theorem. Harmonic functions, Dirichlet problem.
Coding Theory investigates errordetection and errorcorrection. 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 abovementioned communication technologies.
Topics
 The main problem of Coding Theory
 Bounds on codes
 Finite fields
 Linear codes
 Perfect codes
 Cyclic codes
 Sphere packing
 Asymptotic bounds
Bibliography:
R. Hill, A First Course in Coding Theory, Clarendon Press, Oxford 1986
Graphs and subgraphs, 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 CardanoTartaglia 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.
Graduate Courses
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 nonrelated 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 LiptonTarjan 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 ksets. An application of incidences to additive number theory.
 Coloring and hiting problems for geometric hypergraphs : VCdimension, Transversals and Epsilonnets. Weak epsnets for convex sets. Conflictfree colorings .
 Arrangements : Davenport Schinzel sequences and sub structures in arrangements. Geometric permutations.
 Geometric Ramsey and Turan type theorems: Application of Dilworth theorem, ErdosSzekeres theorem for convex sets, quasiplanar graphs.
Introduction to Model Theory Pdf 201.2.0091
Dr Moshe Kamensky ימי רביעי, 8:00–10:00
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
Prerequisites
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.
Course Topics:
 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.

groups.

Extension of groups: The first and second cohomology and applications.
Vector Bundles in Geometry and Analysis Pdf 201.2.5051
Prof Eitan Sayag Wed 17:0019:00, Math 201
 vector bundles and Kgroups of topological spaces
 Bott’s Periodicity theorem and applications to division algebras
 Index of Fredholm operators and Ktheory
 If time permits: smooth manifolds, DeRham cohomology, Chern classes, Elliptic operators, formulation of AtiyahSinger index theorem, relations to GaussBonnet 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.
Notes
 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.