2017–18–B

Mar 5—Jun 22, 2018

Courses

Intro to logic and sets
Pdf 201.1.0171 4.0 Credits

  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. Russel’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).

Introduction to Topology
Pdf 201.1.0091 4.0 Credits

Prof. Michael Levin

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.

Fourier Analysis
Pdf 201.1.0231 4.0 Credits

Prof. Izhar Oppenheim
  • 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.

Coding Theory
Pdf 201.1.4501

Prof. Ido Efrat

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.

Topics
  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
Bibliography:

R. Hill, A First Course in Coding Theory, Clarendon Press, Oxford 1986

Graph Theory
Pdf 201.1.6081 4.0 Credits

Dr. Yaar Solomon

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.

Field Theory and Galois Theory(*)
Pdf 201.1.7041 4.0 Credits

Dr. Ishai Dan-Cohen
  • 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

Introduction to Partial Differetial Equations
Pdf 201.1.0291 4.0 Credits

  1. The Fourier transform: convolutions, the inversion formula, Plancherel’s theorem, Hermite functions, tempered distributions. The Poisson summation formula. The Fourier transform in R^n.
  2. The Laplace transform. Connections with convolutions and the Fourier transform. Laguerre polynomials. Applications to ODE’s. Uniqueness, Lerch’s theorem.
  3. Classification of the second order PDE: elliptic, hyperbolic and parabolic equations, examples of Laplace, Wave and Heat equations.
  4. Elliptic equations: Laplace and Poisson equations, Dirichlet and Neumann boundary value problems, Poisson kernel, Green’s functions, properties of harmonic functions, Maximum principle
  5. 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.

Bibliography

  1. Stein E. and Shakarchi R., Fourier analysis, Princeton University Press, 2003.
  2. Korner T.W., Fourier analysis, Cambridge University Press, 1988.
  3. Katznelson Y., An Introduction to Harmonic Analysis, Dover publications. 4. John, Partial differential equations, Reprint of the fourth edition. Applied Mathematical Sciences, 1. Springer-Verlag, New York, 1991.
  4. Evans Lawrence C. Partial Differential Equations, Second Edition.
  5. 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.
  6. Zauderer E. Partial differential equations of applied mathematics, Second edition. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1989. xvi+891 pp. ISBN: 0-471-61298-7.

Vector Bundles in Geometry and Analysis
Pdf 201.2.5051 2.0 Credits

Prof. Eitan Sayag
  • 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

Random walks and harmonic functions
Pdf 201.2.0391

Prof. Ariel Yadin

In a random process, by definition, it is not possible to deterministically predict the next step. However, we will see in this course how to predict rigorously the long term behavior of processes. We will study in this course processes, known as Markov processes, in which the next step depends only on the current position. We will see that these processes are deeply related to electrical networks, and to notions from information theory such as entropy. We will develop techniques of discrete analysis, which are counterparts of classical analysis in the discrete setting. These notions are at the cutting edge of current research methods in these fields

Algebraic curves
Pdf 201.2.4451

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

Discrete Geometry
Pdf 201.2.0191

Prof. Shakhar Smorodinsky
  • Fundamental theorems and basic definitions: Convex sets, separation , Helly’s theorem, fractional Helly, Radon’s theorem, Caratheodory’s theorem, centerpoint theorem. Tverberg’s theorem. Planar graphs. Koebe’s Theorem.
  • Geometric graphs: the crossing lemma. Application of crossing lemma to Erdos problems: Geometric Incidences, Repeated distance problem, distinct distances problem. Selection lemmas. 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. $(p,q)$-Theorem, Conflict-free colorings.
  • Arrangements : Davenport Schinzel sequences and sub structures in arrangements.
  • Geometric Ramsey and Turan type theorems: Application of Dilworth theorem, Erdos-Szekeres theorem for convex sets, quasi-planar graphs.

Introduction to Model Theory
Pdf 201.2.0091

Dr. Moshe Kamensky

This course will cover a number of fundamentals of model theory including:

  • Quantifer Elimination
  • Applications to algebra including algebraically closed fields and real closed fields.
  • Types and saturated models.

Given time, the course may also touch upon the following topics:

  • Vaught’s conjecture and Morley’s analysis of countable models
  • $\omega$-stable theories and Morley rank
  • Fraisse’s amalgamation theorem.
Prerequisites

Students should be familiar with the following concepts: Languages, structures, formulas, theories, Godel’s completeness theorem and the compactness theorem.

Group Theory
Pdf 201.2.4041

Emeritus Prof Yoav Segev

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; p-groups; Extension of groups: The first and second cohomology and applications.

Homological Algebra
Pdf 201.2.2091 2.0 Credits

Prof. Amnon Yekutieli
  1. Recalling prior material. Rings (including noncommutative), ideals, modules and bimodules, exact sequences, infinite direct sums and products, tensor products of modules and rings.
  2. Categories and functors. Morphisms of functors, equivalences. Linear categories and linear functors. Exactness of functors.
  3. Special modules. Projective, injective and flat modules.
  4. Morita Theory. Equivalences of module categories realized as tensor products.
  5. Complexes of modules. Operations on complexes, homotopies, the long exact cohomology sequence.
  6. Resolutions. Projective, injective and flat resolutions – existence and uniqueness.
  7. Left and right derived functors. The general theory. Tor and Ext functors.
  8. Applications to commutative algebra. Some local and global theorems, involving $Tor$ and $Ext$ functors. Derived completion and torsion functors.
  9. Sheaf cohomology. A survey of the role of homological algebra in geometry.
  10. Nonabelian cohomology. A survey of classification theorems: Galois cohomology, vector bundles.

Bibliography

  1. R. Hartshorne, “Algebraic Geometry”, Springer-Verlag, New-York, 1977.
  2. P.J. Hilton and U. Stammbach, “A Course in Homological Algebra”, Springer, 1971.
  3. S. Maclane, “Homology”, Springer, 1994.
  4. J. Rotman, “An Introduction to Homological Algebra”, Academic Press, 1979.
  5. L.R. Rowen, “Ring Theory” (Student Edition), Academic Press, 1991.
  6. C. Weibel, “An introduction to homological algebra”, Cambridge Univ. Press, 1994.
  7. M. Kashiwara and P. Schapira, Sheaves on Manifolds, Springer, 1990.
  8. The Stacks Project, an online reference, J.A. de Jong (Editor). (9) A. Yekutieli, “Derived Categories”, Cambridge Univ. Press, 2019. Free prepublication version. (10) Course notes, to be uploaded every week to the course web page

Infinitesimal Calculus 1(!)
Pdf 201.1.0011 6.0 Credits

  1. Real numbers (axiomatic theory). Supremum and Infimum of a set. Existence of an n-th root for any a > 0. 2. Convergent sequences, subsequences, Cauchy sequences. The Bolzano-Weierstrass theorem. Upper and lower limits. 3. Series. Partial sums, convergent and divergent series, Cauchy criterion. Series of non-negative terms. The root and the ratio tests. Series of arbitrary terms. Dirichlet, Leibnitz, and Abel theorems. Rearrangements of series. The Riemann Theorem. 4. The limit of a function. Continuous functions. Continuity of the elementary functions. Properties of functions continuous on a closed interval. Uniformly continuous functions. Cantor?s theorem. 5. The derivative of a function. Mean value theorems. Derivatives of higher order. L’Hospital’s rule. Taylor’s theorem.

Infinitesimal Calculus 2(!)
Pdf 201.1.0021 6.0 Credits

  1. The Riemann integral: Riemann sums, the fundamental theorem of calculus. Methods for computing integrals (integration by parts, substitution, partial fractions). Improper integrals and application to series. Numerical integration. Stirling’s formula and additional applications time permitting.
  2. Uniform and pointwise convergence. Cauchy’s criterion and the Weierstrass M-test. Power series. Taylor series, analytic and non-analytic functions. Convolutions, approximate identities and the Weierstrass approximation theorem. Additional applications time permitting.
  3. A review of vectors in R^n and linear maps. The Euclidean norm and the Cauchy-Schwarz inequality. Basic topological notions in R^n. Continuous functions of several variables. Curves in R^n, arc-length. Partial and directional derivatives, differentiability and C^1 functions. The chain rule. The gradient. Implicit functions and Lagrange multipliers. Extrema in bounded domains.

Fourier Analysis for Electrical Engineering(!)
Pdf 201.1.0041 4.5 Credits

  1. Normed spaces and spaces with inner products. The projection theorem for finite dimensional subspaces. Orthogonal systems in infinite dimensional spaces. The Bessel inequality and the Parseval equality, closed orthogonal systems. The Haar system.
  2. The Fourier series (in real and complex form). Approximate identities, closedness of the trigonometric / exponential system. Uniform convergence of the Fourier series of piecewise continuously differentiable functions on closed intervals of continuity; the Gibbs phenomenon. Integrability and differentiability term by term.
  3. The Fourier transform. The convolution theorem. The Plancherel equality. The inversion theorem. Applications: low pass filters and Shannon’s theorem.
  4. The Laplace transform. Basic relations and connection with the Fourier transform. A table of the Laplace transforms. The convolution integral. Application of the Laplace transform for solution of ODEs.
  5. Introduction to the theory of distributions. Differentiation of distributions, the delta function and its derivatives. Fourier series, Fourier transforms, and Laplace transforms of distributions.

Topics in mathematics(!)
Pdf 201.1.0261

The aim of the course is to train students in creative problem solving in various areas of mathematics, in a manner simulating, to some extent, mathematical research (rather than ordinary courses, where homework assignments are, usually closely ? and quite obviously - related to the topics reviewed in class). The course will focus on problems whose solutions call for tools and ideas from several fields of mathematics, and will reveal ? on a miniature scale - the beauty of mathematics as an integral field of knowledge, where between seemingly unrelated subjects deep and surprising connections may arise. The problems will focus on topics in classical and modern mathematics that, due mostly to their interdisciplinary nature, are not ? as a rule ? covered in the core classes offered by the department (such topics as the axiom of choice and its applications, the Banach-Tarski paradox, transcendental number theory etc.). Classes will be divided between lectures, given by the instructor, filling in background material required to address the problems in question, and between students’ presentation of their solutions to the work sheets distributed previously. In addition to all of the above, the course will help students improve their skills in searching the mathematical literature and in the art of writing and presenting proofs.

Statistics for Bio-Informatics(!)
Pdf 201.1.8041 3.5 Credits

Dr. Ilya Frenkel

Probability theory: discrete and continuous variables, independent vs dependent variables, six basic discrete distributions: Bernoulli, binomial, uniform, geometric, negative binomial, Poissonian. Mean, variance, moments, probability generating function. Five basic continuous distributions: uniform, normal, exponential, gamma, beta. Moment generating function. Events, conditional probability, aging of molecules, entropy and related concepts, scores and support. Generating various probabilities. Many random variables. EST library. Covariance and correlation, iid, minimum and maximum of many random variables.Theoretical statistics; random sampling. Classical vs Bayesian approach. Distributions of the mean and variance of the sample; methods of calculating point estimates; point estimator for the mean; point estimator for the variance; biased vs unbiased, MSE, confidence intervals for the parameters of distribution. Basic ideas and definitions for the test of the hypothesis; errors of type I and II; P-values, tests for mean values, variances and proportions; test for the goodness of fit; test of independence; correlation coefficient; and its tests; linear regression; Likelihood ratios, information and support; maximum value as test statistic. Nonparametric: Mann-Whitney and permutation tests.Bayesian approach to hypothesis testing and estimation.ANOVA - analysis of variance: one-way and two-way.More theory on classical estimation: optimality aspects.BLAST.

Statistical methods for big data
Pdf 201.1.9131 3.5 Credits

Dr. Luba Sapir
  1. Review of probability: a. Basic notions. b. Random variables, Transformation of random variables, Independence. c. Expectation, Variance, Co-variance. Conditional Expectation.
  2. Probability inequalities: Mean estimation, Hoeffding?s inequality.
  3. Convergence of random variables: a. Types of convergence. b. The law of large numbers. c. The central limit theorem.
  4. Statistical inference: a. Introduction. b. Parametric and non-parametric models. c. Point estimation, confidence interval and hypothesis testing.
  5. Parametric point estimation: a. Methods for finding estimators: method of moments; maximum likelihood; other methods. b. Properties of point estimators: bias; mean square error; consistency c. Properties of maximum likelihood estimators. d. Computing of maximum likelihood estimate
  6. Parametric interval estimation a. Introduction. b. Pivotal Quantity. c. Sampling from the normal distribution: confidence interval for mean, variance. d. Large-sample confidence intervals.
  7. Hypothesis testing concepts: parametric vs. nonparametric a. Introduction and main definitions. b. Sampling from the Normal distribution. c. p-values. d. Chi-square distribution and tests. e. Goodness-of-fit tests. f. Tests of independence. g. Empirical cumulative distribution function. Kolmogorov-Smirnov Goodness-of fit test.
  8. Regression. a. Simple linear regression. b. Least Squares and Maximum Likelihood. c. Properties of least Squares estimators. d. Prediction.
  9. Handling noisy data, outliers.

Linear Algebra for Engineering
Pdf 201.1.9531 4.5 Credits

Dr. Noa Eidelstein

Fields. Fields of rational, real and complex numbers. Finite fields. Calculations with complex numbers. Systems of linear equations. Gauss elimination method. Matrices. Canonical form of a matrix. Vector spaces . Homogeneous and non homogeneous systems. Vector spaces. Vector spaces. Vector subspace generated by a system of vectors. Vector subspace of solutions of a system of linear homogeneous equations. Linear dependence. Mutual disposition of subspaces. Basis and dimension of a vector space. Rank of a matrix. Intersection and sum of vector subspaces. Matrices and determinants. Operations with matrices. Invertible matrices. Change of a basis. Determinants. Polynomials over fields. Divisibility. Decomposition into prime polynomials over R and over C. Linear transformations and matrices. Linear transformations and matrices. Kernel and image. Linear operators and matrices. Algebra of linear operators. Invertible linear operators. Eigenvectors and eigenvalues of matrices and linear operators. Diagonalization of matrices and linear operators. Scalar multiplication. Orthogonalization process of Gram-Shmidt. Orthogonal diagonalization of symmetric matrices.

Linear Algebra for physics students(!)
Pdf 201.1.9641 6.0 Credits

  • Fields: definitions, the field of complex numbers.
  • Linear equations: elementary operations, row reduction, homogeneous and inhomogeneous systems, representations of the solutions.
  • Vector spaces: examples, subspaces, linear dependence, bases, dimension.
  • Matrix algebra: matrix addition and multiplication, elemetary operations, the inverse of a matrix, the determinant, Cramer’s rule.
  • Linear transformations: examples, kernel and image, matrix representation.
  • Diagonalization: eigenvectors and eigenvalues, the characteristic polynomial, applications.
  • Bilinear forms.
  • Finite dimensional inner product spaces.
  • Operators on finite dimensional inner product spaces: the adjoint, self adjoint operators, normal operators, diagonalization of normal operators.

Fundamentals of Logic and Set Theory for Information Systems(!)
Pdf 201.1.9651 5.0 Credits

Prof. Gregory Mashevitsky

The course provides a basic introduction to “naive” set theory, propositional logic and predicate logic. The course studies many important concepts which serve as the building blocks for any mathematical theory An emphasis is given to clear proof writing and correct usage of the mathematical language.A. Basics of Set Theory1. The notion of a set. Set operations: union, intersection, difference, complementation, and the power set.2. Cartesian products and binary relations. Operations on relations.3. Functions: domain and range, one to one and onto. Composition.4. Equivalence relations and set partitions.5. Order relations: partial, linear and well-founded.6. Induction theorems: mathematical, complete and well-founded.B. Set Cardinality1. The notion of cardinality. Finite, infinite and countable sets.2. Cantor’s theorem.3. The cardinality of the set of real numbers and other sets.C. Propositional Calculus1. The language of propositional calculus. Logical connectives.2. Logical implication and logical equivalence of propositional formulas.3. Disjunctive normal form.4. Complete sets of logical connectives.D. Predicate Calculus (First Order Logic)1. The language of predicate calculus: terms, formulas, sentences.2. Structures, assignments for a given structure.3. Logical implication and logical equivalence of first order formulas.4. Elementary equivalence of structures, definable sets in a given structure.

Introduction to logic and Discrete Mathematics For Computer Engineering(!)
Pdf 201.1.9861 5.0 Credits

Prof. Shakhar Smorodinsky

Logic and Set theory: propositional calculus, boolean operators and their truth tables, truth values of formulae, logical equivalence and logical inference, tautologies and contradictions, the important tautologies, e.g., the distributive laws and De Morgan formulae. Sets: inclusion, union, intersection, difference, cartesian product, relations, equivalence relations, partial orders, linear orders, and functions. Basic Combinatorics: induction, basic counting arguments, binomial coefficients, inclusion-exclusion, recursion and, generating functions. Graphs: general notions and examples, isomorphism, connectivity, Euler graphs, trees.

Advanced Anlysis for Communication Engineering
Pdf 201.1.0241 3.5 Credits

  1. Complex numbers: Cartesian coordinates, polar coordinates. Functions of a complex variable. Basic properties of analytic functions, the exponential function, trigonometric functions. Definition of contour integral. The Cauchy Integral Formula. Residues and poles. Evaluation of impoper real integrals with the use of residues.
  2. Inner product functional spaces. Orthogonal and orthonormal systems. Generalized Fourier series. Theorem on orthogonal projection. Bessel’s inequality and Parseval’s equality.
  3. Trigonometric Fourier series. Complex form of Fourier series. Fourier series expansion defined over various intervals. Pointwise and uniform convergence of Fourier series. Completness of trigonometric system and Parseval’s equality. Differentiation and integration of Fourier series.
  4. The Fourier integral as a limit of Fourier series. The Fourier transform: definition and basic properties. The inverse Fourier transform. The convolution theorem, Parseval’s theorem for the Fourier transform. A relation between Fourier and Laplace transforms. Application of Fourier transform to partial differential equations and image processing.
  5. Distributions (generalized functions). The Heaviside step function, the impulse delta-function. Derivative of distribution. Convergence of sequences in the space of distributions. The Fourier transform of distributions.

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.