## Courses

#### 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 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.

#### Theory of Numbers Pdf 201.1.6031

##### Dr. Ishai Dan-Cohen 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 $\mathbb{Z}/m$
• Continued fractions
• Algebraic numbers and algebraic integers

#### Game Theory Pdf 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.

#### Algebraic Structures Pdf 201.1.7031

##### Prof. Dmitry Kerner 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.

#### Infinitesimal Calculus 3 Pdf 201.1.0031

##### Dr. Inna Entova-Aizenbud 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 $\mathbb{R}^n$ 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. 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, 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

#### 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

#### 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 de-Rham 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. Banach-Steinhaus theorem; open mapping theorem and closed graph theorem. Hahn-Banach theorem. Duality. Measures on locally compact spaces; the dual of $C(X)$. 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.

#### Commutative Algebra Pdf 201.2.2011

##### 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.

#### 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.