Spring 2017 term
Mar 13Jun 30, 2017 Exam Period Ends: October 27, 2017
Advanced Courses
Undergraduate Courses
Classical Set Theory Pdf 201.1.0371
Prof Menachem Kojman
יום ג 14:00  12:00 בגוטמן [32] חדר 114
יום ד 16:00  14:00 בצוקר, גולדשטייןגורן [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 CantorBendixsohn derivative. The structure of closed subsets of Euclidean spaces.
 What is Cantor’s Continuum Hypothesis.
 Ordinals. Which ordinals are orderembeddable 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 ErdosRado theorem. DushnikMiller theorem. Applications.
 Combinatorics of singular cardinals. Silver’s theorem.
 Negative partition theorems. Todorcevic’s theorem.
 Other topics
Bibliography.
 Winfried Just and Martin Wese. Discovering modern set theory I, II. Graduate Studies in Mathematics, vol. 8, The AMS, 1996.
 Azriel Levy. Basic Set Theory. Dover, 2002.
 Ralf Schindler. Set Theory. Springer 2014.
Differential Geometry Pdf 201.1.0051
Dr Michael Brandenbursky
יום א 14:00  12:00 בקרייטמןזלוטובסקי(חדש) [34] חדר 2
יום ג 12:00  10:00 בבנין 90 (מקיף ז’) [90] חדר 234
 Geometry of Curves. Parametrizations, arc length, curvature, torsion, Frenet equations, global properties of curves in the plane.
 Extrinsic Geometry of Surfaces. Parametrizations, tangent plane, differentials, first and second fundamental forms, curves in surfaces, normal and geodesic curvature of curves.
 Differential equations without coordinates. Vector and line fields and flows, frame fields, Frobenius theorem. Geometry of fixed point and singular points in ODEs.
 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 CodazziMainardi.
 Geometry of geodesics. Exponential map, geodesic polar coordinates, properties of geodesics, Jacobi fields, convex neighborhoods.
 Global results about surfaces. The GaussBonnet Theorem, HopfRinow theorem, HopfPoincaret theorem.
 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 CardanoTartaglia 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
 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.
 approximations. Parseval’s formula. Absolute convergence of Fourier series of 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 . 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.
Introduction to Computational Algebraic Geometry Pdf 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
More details: https://www.math.bgu.ac.il/~bessera/computeralgebra07adv.pdf
Introduction to Singularity Theory Pdf 201.1.0361
Dr Dmitry Kerner
יום א 12:00  10:00 בגוטמן [32] חדר 113
יום ה 17:00  16:00 בקרייטמןזלוטובסקי(חדש) [34] חדר 16
See on the web page
Introduction to Topology Pdf 201.1.0091
Dr Izhar Oppenheim
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 StoneCech compactification. Metrization theorems and paracompactness.
Theory of Functions of a Complex Variable Pdf 201.1.0251
Prof Arkady Poliakovsky
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 MittagLeffler. Entire functions. Normal families.
 Riemann Mapping Theorem. Harmonic functions, Dirichlet problem.
Graduate Courses
Derived categories IV Pdf advertdercatsIV.pdf 201.2.0364
Prof Amnon Yekutieli יום רביעי 12:00  Wednesday 14:00
Topics:

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 indrigid trace homomorphism.

Derived categories in geometry. This topic concerns geometry in the wide sense. We will prove existence of Kflat and Kinjective resolutions, and talk about derived direct and inverse image functors.

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 indrigid trace. We will indicate a generalization of this approach to DM stacks.

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 Algebras Pdf 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 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 Kamensky
יום ד 08:0010:00 בניין 34 חדר 7
יום ה 12:0014:00 בניין 32 חדר 209
The course provides an introduction to the theory of noncommutative 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).