## Courses

#### Introduction to Computational Algebraic GeometryPdf 0281

##### Prof. Amnon Besser

יום ד 14:00 - 12:00 בבנין 90 (מקיף ז’) [90] חדר 225
יום ה 12:00 - 10:00 בבנין 90 (מקיף ז’) [90] חדר 134

• Rings and modules, Polynomial rings in several variables over a field
• Monomial orders and the division algorithm in several variables
• Grobner bases and the Buchberger algorithm, Elimination and equation solving
• Applications of Grobner bases:
• integer programming
• graph coloring
• robotics
• coding theory
• combinatorics and more
• The Hilbert function and the Hilbert series, Speeding up the Buchberger algorithm, The f4 and f5 algorithms

#### Differential GeometryPdf 201.1.0051

##### Dr. Michael Brandenbursky4.0 Credits

יום א 14:00 - 12:00 בקרייטמן-זלוטובסקי(חדש) [34] חדר 2
יום ג 12:00 - 10:00 בבנין 90 (מקיף ז’) [90] חדר 234

1. Geometry of Curves. Parametrizations, arc length, curvature, torsion, Frenet equations, global properties of curves in the plane.
2. Extrinsic Geometry of Surfaces. Parametrizations, tangent plane, differentials, first and second fundamental forms, curves in surfaces, normal and geodesic curvature of curves.
3. Differential equations without coordinates. Vector and line fields and flows, frame fields, Frobenius theorem. Geometry of fixed point and singular points in ODEs.
4. Intrinsic and Extrinsic Geometry of Surfaces. Frames and frame fields, covariant derivatives and connections, Riemannian metric, Gaussian curvature, Fundamental Forms and the equations of Gauss and Codazzi-Mainardi.
5. Geometry of geodesics. Exponential map, geodesic polar coordinates, properties of geodesics, Jacobi fields, convex neighborhoods.
6. Global results about surfaces. The Gauss-Bonnet Theorem, Hopf-Rinow theorem, Hopf-Poincaret theorem.

#### Introduction to TopologyPdf 201.1.0091

##### Dr. Izhar Oppenheim4.0 Credits

יום א 11:00 - 09:00 בבנין 90 (מקיף ז’) [90] חדר 223
יום ה 16:00 - 14:00 בבנין 90 (מקיף ז’) [90] חדר 141

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 AnalysisPdf 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.

#### Theory of Functions of a Complex VariablePdf 201.1.0251

יום א 16:00 - 14:00 בבנין 90 (מקיף ז’) [90] חדר 224
יום ד 10:00 - 08:00 בבנין 90 (מקיף ז’) [90] חדר 224

• 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 Mittag-Leffler. Entire functions. Normal families.
• Riemann Mapping Theorem. Harmonic functions, Dirichlet problem.

#### Introduction to Singularity TheoryPdf 201.1.0361

##### Prof. Dmitry Kerner

יום א 12:00 - 10:00 בגוטמן [32] חדר 113
יום ה 17:00 - 16:00 בקרייטמן-זלוטובסקי(חדש) [34] חדר 16

See on the web page

#### Classical Set TheoryPdf 201.1.0371

##### Prof. Menachem Kojman

יום ג 14:00 - 12:00 בגוטמן [32] חדר 114
יום ד 16:00 - 14:00 בצוקר, גולדשטיין-גורן [72] חדר 119

The course covers central ideas and central methods in classical set theory, without the axiomatic development that is required for proving independence results. The course is aimed as 2nd and 3rd year students and will equip its participants with a broad variety of set theoretic proof techniques that can be used in different branches of modern mathematics.

##### Sylabus
• The notion of cardinality. Computation of cardinalities of various known sets.
• Sets of real numbers. The Cantor-Bendixsohn derivative. The structure of closed subsets of Euclidean spaces.
• What is Cantor’s Continuum Hypothesis.
• Ordinals. Which ordinals are order-embeddable into the real line. Existence theorems ordinals. Hartogs’ theorem.
• Transfinite recursion. Applications.
• Various formulations of Zermelo’s axiom of Choice. Applications in algebra and geometry.
• Cardinals as initial ordinals. Hausdorff’s cofinality function. Regular and singular cardinals.
• Hausdorff’s formula. Konig’s lemma. Constraints of cardinal arithmetic.
• Ideal and filters. Ultrafilters and their applications.
• The filter of closed and unbounded subsets of a regular uncountable cardinal. Fodor’s pressing down lemma and applications in combinatorics.
• Partition calculus of infinite cardinals and ordinals. Ramsey’s theorem. The Erdos-Rado theorem. Dushnik-Miller theorem. Applications.
• Combinatorics of singular cardinals. Silver’s theorem.
• Negative partition theorems. Todorcevic’s theorem.
• Other topics
##### Bibliography.
1. Winfried Just and Martin Wese. Discovering modern set theory I, II. Graduate Studies in Mathematics, vol. 8, The AMS, 1996.
2. Azriel Levy. Basic Set Theory. Dover, 2002.
3. Ralf Schindler. Set Theory. Springer 2014.

#### Field Theory and Galois Theory(*)Pdf 201.1.7041

##### Prof. Ilya Tyomkin4.0 Credits

• 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

#### Derived categories IVPdf 201.2.0364

##### Prof. Amnon Yekutieli

יום רביעי 12:00 - Wednesday 14:00

Topics:

1. Rigidity, residues and duality over commutative rings. We will study rigid residue complexes. We will prove their uniqueness and existence, the trace and localization functoriality, and the ind-rigid trace homomorphism.

2. Derived categories in geometry. This topic concerns geometry in the wide sense. We will prove existence of K-flat and K-injective resolutions, and talk about derived direct and inverse image functors.

3. Rigidity, residues and duality over schemes. The goal is to present an accessible approach to global Grothendieck duality for proper maps of schemes. This approach is based on rigid residue complexes and the ind-rigid trace. We will indicate a generalization of this approach to DM stacks.

4. Derived categories in noncommutative ring theory. Subtopics: dualizing complexes, tilting complexes, the derived Picard group, derived Morita theory, survey of noncommutative and derived algebraic geometry.

#### Introduction to Von Neumann AlgebrasPdf 201.2.0061

##### Dr. Daniel Markiewicz

יום א 16:00 - 14:00 בבנין 90 (מקיף ז’) [90] חדר 225
יום ג 18:00 - 16:00 בבנין 90 (מקיף ז’) [90] חדר 134

Basics of $C^*$-Algebra theory. The spectral theorem for bounded normal operators and the Borel functional calculus. Basic theory of von Neumann algebras. Density theorems, topologies and normal maps, traces, comparison of projections, type classification, examples of factors. Additional topics, for example, noncommutative dynamics, subfactors, group actions, and free probability.

#### Noncommutative algebra(#)Pdf 201.2.5121

##### Dr. Moshe Kamensky4.0 Credits

יום ד 08:00-10:00 בניין 34 חדר 7
יום ה 12:00-14:00 בניין 32 חדר 209

The course provides an introduction to the theory of non-commutative rings, and related structures. We will take as our motivating goal understanding the representation theory of finite groups. This will lead us to study the structure of semisimple rings, which is well understood due to a number of theorems by Wedderburn.

After drawing conclusions for representation theory, we will study in more details the building blocks in Wedderburn’s theory, namely, division rings and rings of matrices over them. This has applications in geometry and number theory, which we will try to outline.

Further topics will vary depending on the time constraints and the taste of the audience, but may include further study of group representations, more general classes of rings (non semisimple, non Artinian) and localisation. Whenever possible, we will try to include applications and relations to other fields.

Background required: A reasonable understand of linear algebra and basic Galois theory. Familiarity with the basic notions in commutative algebra and category theory is desirable, but not required (missing material will be reviewed as needed).

#### 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.