## Courses

#### Algebraic Structures Pdf 201.1.7031 4.0 Credits

##### 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 as symmetries. Examples: cyclic, dihedral, symmetric, matrix groups.
• Homomorphism. Subgroups and normal subgroups. Quotient groups. Lagrange’s theorem. The isomorphism theorems. Direct products of groups.
• Actions of groups on sets. Cayley’s theorem.
• Group automorphisms.
• Sylow’s theorems. Application: classification of groups of small order.
• Composition series and Jordan–Hoelder theorem. Solvable groups.
• Classification of finite abelian groups, finitely-generated abelian groups.
• Symmetric group and alternating group. The alternating group is simple.
• Rings, maximal and prime ideals, integral domain, quotient ring. Homomorphism theorems.
• Multilinear algebra: Quotient spaces. Tensor products of vector spaces. Action of $S_n$ on tensor powers. Exterior and symmetric algebras. Multilinear forms and determinant.
• Optional topics: group of symmetries of platonic solids, free groups, semidirect products, representation theory of finite groups.

#### Infinitesimal Calculus 3(!) Pdf 201.1.0031 6.0 Credits

##### 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

#### Ordinary Differential Equations(!) Pdf 201.1.0061 5.0 Credits

##### 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

#### Fundamentals of Measure Theory(*) Pdf 201.1.0081 4.0 Credits

##### 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.

#### Logic Pdf 201.1.6061 4.0 Credits

##### Prof. 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.

#### Graph Theory Pdf 201.1.6081 4.0 Credits

##### 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.

#### Probability Pdf 201.1.8001 4.0 Credits

##### Prof. Ariel Yadin יום ב 11:00 - 09:00 ב- [90] חדר 238 יום ד 11:00 - 09:00 בגולדברגר [28] חדר 203

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

#### Theory of Numbers Pdf 201.1.6031 4.0 Credits

##### 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 4.0 Credits

##### 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.

#### Lie Groups Pdf 201.2.4141

##### 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.

##### Prof. Eitan Sayag Mon 14:00–16:00 Wed 12:00–13:00 Wed 14:00–15:00

The field of real numbers $\mathbb{R}$ is defined as the completion of the field of rational numbers $\mathbb{Q}$ with respect to the norm $\|·\|$. However, there are other norms on $\mathbb{Q}$, each corresponding to a prime number $p$, and the completion of $\mathbb{Q}$ with respect to any such norm leads to the field of $p$-adic numbers, denoted by $\mathbb{Q}_p$. This is a topological complete field, and thus it makes sense to develop an analysis on it.

Many features of the $p$-adic analysis are very different from familiar ones in real analysis. For example, “the first year calculus dream” of many students comes true: A series converges if and only if its general term goes to $0$. The overall picture of the $p$-adic analysis makes impression of a surprising and beautiful one, and easier than its real counterpart. Nowadays, the $p$-adic analysis has endless applications in geometry and number theory.

In this course we will study the field of $p$-adic numbers from different points of view, stressing similarities to and deviations from the real numbers. If time permits the culmination of the course will be Tate’s thesis (1950), that uses $p$-adic analysis to prove the meromorphic continuation of the zeta-function of Riemann and its functional equation.

1. Arithmetic of $\mathbb{Q}_p$: sums and products, square roots, finding roots of polynomials.
2. Algebraic number theory of $\mathbb{Q}_p$: finite extensions, algebraic closure of $\mathbb{Q}_p$, completion of the algebraic closure, local class field theory will be mentioned.
3. Topology of $\mathbb{Q}_p$: elementary topological properties, Euclidean models of $\mathbb{Z}_p$.
4. Analysis on $\mathbb{Q}_p$: convergence of sequences and series, radius of convergence, elementary functions $\ln_p$, $\exp_p$, the space of locally constant functions.
5. Harmonic analysis on $\mathbb{Q}_p$: characters of $\mathbb{Q}_p$, Haar measure, integration of locally constant functions, Fourier transform.
6. The ring of adeles as an object unifying $\mathbb{Q}_p$ for all $p$: topological properties, integration and Fourier transform, Poisson summation formula.
7. Tate’s thesis.

Prerequisites: topology, algebraic structures

#### Derived Categories III Pdf 201.2.0363

##### Prof. Amnon Yekutieli

Topics:

1. Review of material from past semesters (the courses “Derived Categories I and II”).

2. Derived categories in commutative algebra: dualizing complexes, Grothendieck’s local duality, MGM Equivalence, rigid dualizing complexes.

3. Derived categories in algebraic geometry: direct and inverse image functors, global Grothendieck duality, applications to birational geometry (survey), $l$-adic cohomology and Poincare-Verdier duality (survey), perverse sheaves (survey).

4. Derived categories in non-commutative ring theory: dualizing complexes, tilting complexes, derived Morita theory.

5. Derived algebraic geometry: nonabelian derived categories (survey), infinity categories (survey), derived algebraic stacks (survey), applications (survey).

#### Basic Concepts in Modern Analysis(#) Pdf 201.2.0351

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

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.

#### 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.