2016–17–A

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 $\mathbb{Z}/m$
  • Quadratic residues
  • 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

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.

p-adic analysis

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

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(#)

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.

2016–17–B

  • 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.
  • 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

Introduction to Topology
Pdf 201.1.0091 4.0 Credits

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 Stone-Cech compactification. Metrization theorems and paracompactness.

Theory of Functions of a Complex Variable
Pdf 201.1.0251 4.0 Credits

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

Differential Geometry
Pdf 201.1.0051 4.0 Credits

Dr. Michael Brandenbursky

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 Singularity Theory

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

  1. An introductory sketch and some motivating examples. Degenerate critical points of functions. Singular (nonsmooth) points of curves.
  2. Holomorphic functions of several variables. Weierstrass preparation theorem. Local Rings and germs of functions/sets.
  3. Isolated critical points of holomorphic functions. Unfolding and morsication. Finitely determined function germs.
  4. Classification of simple singularities. Basic singularity invariants. Plane curve singularities. Decomposition into branches and Puiseux expansion.
  5. Time permitting we will concentrate on some of the following topics: a. Blowups and resolution of plane curve singularities; b. Basic topological invariants of plane curve singularities (Milnor fibration); c. Versal deformation and the discriminant.

Classical Set Theory

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.

Introduction to Von Neumann Algebras

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 4.0 Credits

Dr. Moshe Kamensky

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

  1. Basic Algebraic Structures: rings, modules, algebras, the center, idempotents, group rings

  2. Division Rings: the Hamiltonian quaternions, generalized quaternion algebras, division algebras over $\mathbb{F}_q$, $\mathbb{C}$, $\mathbb{R}$, $\mathbb{Q}$ (theorems of Frobenius and Wedderburn), cyclic algebras, the Brauer–Cartan–Hua theorem

  3. Simplicity and semi-simplicity: simplicity of algebraic structures, semi-simple modules, semi-simple rings, Maschke’s theorem

  4. The Wedderburn–Artin Theory: homomorphisms and direct sums, Schur’s lemma, the Wedderburn–Artin structure theorem, Artinian rings

  5. Introduction to Group Representations: representations and characters, applications of the Wedderburn–Artin theory, orthogonality relations, dimensions of irreducible representations, Burnside’s theorem

  6. Tensor Products: tensor products of modules and algebras, scalar extensions, the Schur index, simplicity and center of tensor products, the Brauer group, the Skolem–Noether theorem, the double centralizer theorem, maximal fields in algebras, reduced norm and trace, crossed products

Derived categories IV

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.

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.