2015–16–A

  • 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 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

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.

  • 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.
  • 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.

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.

  • 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.

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

  • Some review of complex variables
  • Hilbert spaces of analytic functions. General theory
  • Bergman space
  • Hardy spaces
  • Fractional Hardy space and self similar systems
  • Bargmann-Fock space and second quantization
  • Banach spaces of analytic functions.
  • Frechet spaces and Schwartz spaces
  • Proof of Riemann’s mapping theorem
  • Dual of the space of functions analytic in a given open set
  • Gelfand triples and applications
  • 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.

Derived categories I

Prof. Amnon Yekutieli

Wed 12:00-14:00, math (58) -101

  1. Review of abelian categories and additive functors.
  2. Differential graded rings, modules and categories.
  3. The derived category of a differential graded category.
  4. Derived functors. Resolutions of differential graded modules.

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.

2015–16–B

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.

  • 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.

The course is intended for 3rd year undergraduate as well as M.Sc students both in computer science and mathematics. We will touch main topics in the area of discrete geometry. Some of the topics are motivated by the analysis of algorithms in computational geometry, wireless and sensor networks. Some other beautiful and elegant tools are proved to be powerful in seemingly non-related areas such as additive number theory or hard Erdos problems. The course does not require any special background except for basic linear algebra, and a little of probability and combinatorics. During the course many open research problems will be presented.

Detailed Syllabus:

  • Fundamental theorems and basic definitions: Convex sets, convex combinations, separation , Helly’s theorem, fractional Helly, Radon’s theorem, Caratheodory’s theorem, centerpoint theorem. Tverberg’s theorem. Planar graphs. Koebe’s Theorem. A geometric proof of the Lipton-Tarjan separator theorem for planar graphs.
  • Geometric graphs: the crossing lemma. Application of crossing lemma to Erdos problems: Incidences between points and lines and points and unit circles. Repeated distance problem, distinct distances problem. Selection lemmas for points inside discs, points inside simplexes. Counting k-sets. An application of incidences to additive number theory.
  • Coloring and hiting problems for geometric hypergraphs : VC-dimension, Transversals and Epsilon-nets. Weak eps-nets for convex sets. Conflict-free colorings .
  • Arrangements : Davenport Schinzel sequences and sub structures in arrangements. Geometric permutations.
  • Geometric Ramsey and Turan type theorems: Application of Dilworth theorem, Erdos-Szekeres theorem for convex sets, quasi-planar graphs.

An introduction to applications of algebra and number theory in the field of cryptography. In particular, the use of elliptic curves in cryptography is studied in great detail.

  • Introduction to cryptography and in particular to public key systems, RSA, Diffie-Hellman, ElGamal.
  • Finite filelds, construction of all finite fields, efficient arithmetic in finite fields.
  • Elliptic curves, the group law of an elliptic curve, methods for counting the number of points of an elliptic curves over a finite field: Baby-step giant step, Schoof’s method.
  • Construction of elliptic curves based cryptographic systems.
  • Methods for prime decomposition, the elliptic curves method, the quadratic sieve method.
  • Safety of public key cryptographic methods.

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
  • 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
  • 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.

Probability
Pdf 201.1.8001 4.0 Credits

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

  • The reflection theorem
  • Mostowski collapse theorem
  • Absoluteness of formulas
  • The constructible universe
  • Forcing
  • Consistency of the negation of the continuum hypothesis
  • Consistency of the negation of the Axiom of choice
  1. Commutative algebra via derived categories (regular and CM rings, Grothendieck’s Local Duality, MGM Equivalence, rigid dualizing complexes).

  2. Geometric derived categories (of sheaves on spaces). Direct and inverse image functors, Grothendieck Duality, Poicar?-Verdier Duality, perverse sheaves).

  3. Derived categories associated to nocommutative rings (dualizing complexes, tilting complexes and derived Morita theory).

  4. Derived categories in modern algebraic geometry and modern string theory (a survey).

Random walks

Prof. Ariel Yadin

ב’ 12:00-14:00, ד’ 13:00-15:00

In a random process, by definition, it is not possible to deterministically predict the next step. However, we will see in this course how to predict rigorously the long term behavior of processes. We will study in this course processes, known as Markov processes, in which the next step depends only on the current position. We will see that these processes are deeply related to electrical networks, and to notions from information theory such as entropy. We will develop techniques of discrete analysis, which are counterparts of classical analysis in the discrete setting. These notions are at the cutting edge of current research methods in these fields

This is a first course in modern commutative algebra that provides the background for further study of commutative and homological algebra, algebraic geometry, etc.

Syllabus

  1. Rings, ideals, and homomorphisms
  2. Modules, Cayley-Hamilton theorem, and Nakayama’s lemma
  3. Noetherian rings and modules, Hilbert basis theorem
  4. Integral extensions, Noether normalization lemma, and Nullstellensatz
  5. Affine varieties
  6. Localization of rings and modules
  7. Primary decomposition theorem
  8. Discrete valuation rings
  9. Selected topics

Literature

  1. Miles Reid, Undergraduate Commutative Algebra
  2. Miles Reid, Undergraduate Algebraic Geometry
  3. Altman, Kleiman, A Term of Commutative Algebra

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.