### Oct 30, 2016-Jan 27, 2017 Exam Period Ends: March 10, 2017

#### Algebraic Structures Pdf 201.1.7031

##### Dr Ishai Dan-Cohen יום ב 13:00 - 11:00 בקרייטמן-זלוטובסקי(חדש) [34] חדר 14 יום ה 12:00 - 10:00 בבנין 90 (מקיף ז’) [90] חדר 134
• 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.

#### Fourier Analysis Pdf 201.1.0231

• 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.

#### Fundamentals of Measure Theory Pdf 201.1.0081

##### Prof Tom Meyerovitch יום א 14:00 - 12:00 בבנין 90 (מקיף ז’) [90] חדר 237 יום ה 14:00 - 12:00 בבנין 90 (מקיף ז’) [90] חדר 243

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 $L_1$ and $L_2$ and their completeness, signed measures, the Radon-Nikodym theorem, measures in product spaces and Fubini’s theorem.

#### Graph Theory Pdf 201.1.6081

##### Prof Shakhar Smorodinsky יום ב 16:00 - 14:00 בצוקר, גולדשטיין-גורן [72] חדר 115 יום ג 12:00 - 10:00 בגולדברגר [28] חדר 301

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.

#### Infinitesimal Calculus 3 Pdf 201.1.0031

##### Prof Uri Onn יום ד 13:00 - 11:00 בגוטמן [32] חדר 114 יום א 11:00 - 09:00 בגוטמן [32] חדר 114 יום ב 14:00 - 13:00 בגוטמן [32] חדר 206
• Basic concepts of topology of metric spaces: open and closed sets, connectedness, compactness, completeness.
• Normed spaces and inner product spaces. All norms on $\mathbb{R}_n$ 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

#### Logic Pdf 201.1.6061

##### Dr Assaf Hasson יום ב 16:00 - 14:00 בבנין 90 (מקיף ז’) [90] חדר 136 יום ג 12:00 - 10:00 בגוטמן [32] חדר 114
• 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.

#### Ordinary Differential Equations Pdf 201.1.0061

##### Prof Victor Vinnikov יום א 16:00 - 14:00 בגוטמן [32] חדר 114 יום ג 14:00 - 12:00 בצוקר, גולדשטיין-גורן [72] חדר 123

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

#### Partial Differential Equations Pdf 201.1.0101

##### Prof Yitzchak Rubinstein יום ב 11:00 - 09:00 בגוטמן [32] חדר 210 יום ה 14:00 - 12:00 בבנין 90 (מקיף ז’) [90] חדר 227
1. Linear first order partial differential equations; characteristics - particles trajectories in a continuum; the Cauchy problem, propagation of singularities; complete integral and general solution.

2. System of two linear first order partial differential equations; classification; normal and canonical form; solution of the Cauchy problem for a hyperbolic system.

3. Classification of second order partial differential equations with a linear main part; canonical form; characteristics; propagation of singularities; Cauchy-Kovalevskaya theorem; physical phenomena leading to equations of various types.

4. One dimensional wave equation - example of a hyperbolic equation; initial and boundary conditions; Cauchy and boundary value problems; propagating waves method; D’Alambert’s formula; boundary value problems on a semi-axis and a segment; propagation of singularities; separation of variables; non-homogeneous problems; Duhamel’s principle.

5. One dimensional heat equation - example of a parabolic equation; typical problems - the Cauchy and boundary value problem; moments; solutions of heat equation on the axis, similarity variable and solution, fundamental solution and it’s properties, solution of the Cauchy problem; boundary value problems on a semi-axis, on a segment, separation of variables; non-homogeneous problems, Duhamel’s principle; Green’s functions; maximum principle and comparison theorems.

6. The Laplace’s equation - example of an elliptic equation; harmonic, sub- and super- harmonic functions and their properties, mean value theorem, maximum principle, Hopf’s lemma; Hadamard’s example and typical boundary value problems for elliptic equations; comparison theorems for linear and quasi-linear elliptic equations; fundamental solutions an their physical meaning; Green’s functions, method of images, inversion; separation of variables.

#### Probability Pdf 201.1.8001

##### Dr Ariel Yadin יום ב 11:00 - 09:00 בגוטמן [32] חדר 206 יום ד 11:00 - 09:00 בגולדברגר [28] חדר 203

Sample spaces and finite probability spaces with symmetric simple events, general probabilty spaces and the fields of events, the Borel filed and probabilities on it defined by densities, conditional probabilities and independent events, random variables and their distribution functions (discrete, absolutely continuous, mixed), the expectation of a random variable (for discrete, absolutely continuous and general distribution), the variance of a random variable, random vectors and the covariance, independent random variables, the central limit theorem for i.i.d. random variables, joint densities (discrete or continuous) with computation of the covariance and the marginal distributions, the weak law of large numbers.

#### Theory of Numbers Pdf 201.1.6031

##### Prof Eitan Sayag יום ב 18:00 - 16:00 בגוטמן [32] חדר 108 יום ד 18:00 - 16:00 בבנין 90 (מקיף ז’) [90] חדר 239

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 $\mathbb{Z}/m$
• Continued fractions
• Algebraic numbers and algebraic integers

#### Basic Concept in Toplogy and Geometry Pdf 201.2.5221

##### Prof Michael Levin יום א 16:00 - 14:00 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.

#### Basic concepts in Modern Analysis Pdf 201.2.0351

##### Dr Daniel Markiewicz יום א 16:00 - 14:00 יום ג 14:00 - 12:00

This course will cover the fundamentals of Functional Analysis, including Hilbert spaces, Banach spaces, and operators between such spaces.

#### Derived categories III Pdf 2012

##### Prof Amnon Yekutieli

Topics for the third course. Here is a tentative list of topics – the actual choice of topics will be influenced by the participants. Some of the material will be taken from textbooks, and some from research papers. There will be a few guest lectures.

1. Derived categories in commutative algebra: dualizing complexes, affine Grothendieck duality, local duality, rigid dualizing complexes.
2. Derived categories in algebraic geometry: derived direct and inverse image functors, rigid residue complexes, global Grothendieck duality, applications to birational geometry (survey), perverse sheaves (survey), l-adic cohomology and Poicaré-Verdier duality (survey).
3. Derived categories in noncommutative ring theory: dualizing complexes, tilting complexes, the derived Picard group, derived Morita theory.
4. Derived algebraic geometry (survey): nonlinear derived categories, infinity categories, derived algebraic stacks, applications to mathematical physics.

The lectures will be in English.

#### Lie Groups Pdf 201.2.4141

##### Prof Uri Onn יום ב 18:00 - 15:00 בגרוסמן/ דייכמן [58] חדר 101
1. Review of differentiable manifolds, definition of a Lie group. Quotients in the category of Lie groups, homogeneous manifolds, haar measure, connected components.
2. Algebraic groups, matrix groups, the classical groups.
3. Lie algebras and connection to Lie groups.
4. Nilpotent, solvable and semisimple Lie algebras and Lie groups, Lie theorem, Engel theorem, Levi decomposition.
5. Cartan-Killing form.
6. Representation of a Lie algebra over the complex numbers.
7. Root and weights, root systems, Dynkin diagrams, classication of complex semisimple Lie algebras.