• 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
• 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
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.

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.

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.

#### Spring 2018

Lecturer: Prof Michael Levin

The main purpose of the course is to introduce students to the basic concepts and theorems of Functional Analysis, and to develop an “abstract geometrical” approach to different mathematical problems.

1. Metric spaces. Properties of complete metric spaces. Mappings of metric spaces. Fixed points. Application to ODE. Compact sets in metric spaces.
2. Normed spaces. Holder’s inequality. The spaces: $C[0,1]$, $l_p$, $c_0$, $c$, $L_p$. Banach spaces. The completion theorem. Completeness of the above spaces.
3. Linear operators. The norm of a linear operator. Isomorphism and isometry. Any two $n$-dimensional normed spaces are isomorphic. The open mapping theorem and the closed graph theorem. Uniform boundedness principle. The completeness of the space $L(X, Y)$ of all linear bounded operators acting from a normed space $X$ into a Banach space $Y$.
4. Linear functionals. The Hahn–Banach extension theorem. The dual space. Dual spaces for the examples above. Adjoint operators.
5. Quotient space. Completeness of the quotient space. The dual space for a quotient space. Reflexive spaces.
6. Hilbert spaces. The Cauchy–Schwarz inequality. Orthogonal projectors. Orthonormal basis. Any two separable infinite-dimensional Hilbert spaces are isometric. Closed operators in a Hilbert space.
• 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.

#### Spring 2018

Lecturer: Dr Izhar Oppenheim

• 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.

#### Spring 2018

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.
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.

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.

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.

Coding Theory investigates error-detection and error-correction. Such errors can occur in various communication channels: satellite communication, cellular telephones, CDs and DVDs, barcode reading at the supermarket, etc. A mathematical analysis of the notions of error detection and correction leads to deep combinatorial problems, which can be sometimes solved using techniques ranging from linear algebra and ring theory to algebraic geometry and number theory. These techniques are in fact used in the above-mentioned communication technologies.

##### Topics
1. The main problem of Coding Theory
2. Bounds on codes
3. Finite fields
4. Linear codes
5. Perfect codes
6. Cyclic codes
7. Sphere packing
8. Asymptotic bounds
##### Bibliography:

R. Hill, A First Course in Coding Theory, Clarendon Press, Oxford 1986

#### Spring 2018

Lecturer: Prof Ido Efrat

• Review of Logic
• Crieteria for completeness of axiom systems
• Review of Turing machines computability
• The recursion theorem and its uses
• Formal systems and arithmetization of syntax
• representation of recursive function in arithmetic
• Godel’s fixed-point theorem. First incompleteness theorem
• Loeb’s theorem and Godel’s second incompleteness theorem

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
• 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.
1. Introduction.
2. Combinatorial objects: representation, identification, symmetry, constructive and analytical enumeration. The natural numbers, integers, rational and irrational numbers. The arithmetic operations, powers and radicals. The basic algebraic formulas including the binomial formula.
3. Permutation groups and combinatorial enumeration: Permutation groups. Introduction to Polya enumeration theory. Graph isomorphism problem.
4. Finite fields and applications: Finite fields. Coding theory. Incidence structures and block designs.
5. Symmetrical graphs: Cayley graphs, strongly regular graphs. $n$-dimensional cubes and distance transitive graphs.
6. Examples of applications: Design of statistical experiments. Cryptography. Recreational mathematics.

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.

#### Spring 2018

• Groups, the factor group and the homomorphism theorems, Sylow’s theorems and permutation actions of groups.
• Rings, Integral Domains and Fields. Ideals: maximal and prime. Unique Factorization Domains, Principle Ideal Domains, Euclidean Domains.
• Modules, structure theorems for finitely generated modules over a PID, application to finitely generated abelian groups and to the Jordan Canonical Form.
• 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

#### Spring 2018

Lecturer: Dr Ishai Dan-Cohen

algebra and category theory.

2. Multilinear algebra, tensor product of modules, the exterior algebra.
3. Categories: Examples of categories, Functors, Natural transformations, Universal constructions: product and coproduct, limits, adjoint functors, Applications.
4. Selected topics in commutative and non-commutative algebra, Applications.

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 Poisson Process and its applications. Elements of queuing theory. Queuing birth-and-death processes in equilibrium. Non-homogeneous Poisson Processes.
• Symmetric Brownian Motion. Brownian Motion with drift. First-Passage times and arcsine laws. Geometric Brownian motion. Application to option pricing by the Arbitrage Theorem (Black and Scholes).
• Discrete-time Markov Chains. Classification of states. Random walks in the line, plane and space. Limit theorems. Calculation of limiting proportions of sojourn times and expectations of recurrence times.

The course will present Game Theory mostly from a mathematical point of view. Topics to be covered:

1. Combinatorial games.
2. Two-person zero-sum games.
3. Linear programming.
4. General-sum games.
5. Equilibrium points.
6. Random-turn games.
7. Stable marriages.
8. Voting.

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.

This course will cover a number of fundamentals of model theory including:

• Quantifer Elimination
• Applications to algebra including algebraically closed fields and real closed fields.
• Types and saturated models.

Given time, the course may also touch upon the following topics:

• Vaught’s conjecture and Morley’s analysis of countable models
• $\omega$-stable theories and Morley rank
• Fraisse’s amalgamation theorem.
##### Prerequisites

Students should be familiar with the following concepts: Languages, structures, formulas, theories, Godel’s completeness theorem and the compactness theorem.

#### Spring 2018

We will present some basic notions and constructions from model theory, motivated by concrete questions about structures and their theories. Notions we expect to cover include:

• Types and spaces of types
• Homogeneous and saturated models
• Quantifier elimination and model companions
• Elimination of imaginaries
• Definable groups and fields
##### Prerequisites

Students should be familiar with the following concepts from logic: Languages, structures, formulas, theories, the compactness theorem. In addition, some familiarity with field theory, topology and probability will be beneficial.

Lecturer: Dr Moshe Kamensky

Hours: ימי רביעי, 8:00--10:00

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

• Fundamental theorems and basic definitions: Convex sets, separation , Helly’s theorem, fractional Helly, Radon’s theorem, Caratheodory’s theorem, centerpoint theorem. Tverberg’s theorem. Planar graphs. Koebe’s Theorem.
• Geometric graphs: the crossing lemma. Application of crossing lemma to Erdos problems: Geometric Incidences, Repeated distance problem, distinct distances problem. Selection lemmas. 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. $(p,q)$-Theorem, Conflict-free colorings.
• Arrangements : Davenport Schinzel sequences and sub structures in arrangements.
• Geometric Ramsey and Turan type theorems: Application of Dilworth theorem, Erdos-Szekeres theorem for convex sets, quasi-planar graphs.

#### Spring 2018

The course is intended for 3rd year undergraduate as well as M.Sc and Ph.D. 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.

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.

Info Core for completion of MSc degree
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.
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).

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).

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.

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
Info Core for completion of MSc degree

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

#### Spring 2018

##### Random walks and harmonic functions

This course deals with random walks, harmonic functions, the relations between these notions, and their applications to geometry and algebra (mainly to finitely generated groups).

The modern point of view will be presented, following recent texts by: Gromov, Kleiner, Ozawa, Shalom & Tao, among others.

##### Course Topics
1. Modules: free modules, exact sequences, tensor products, Hom modules, flatness.
2. Prime ideals and localization: local rings, Nakayama’s Lemma, the spectrum of a ring, dimension and connectedness.
3. Noetherian rings: the Hilbert basis theorem, the Artin-Rees lemma, completion, grading.
4. Dimension theory: the Hilbert nullstellensatz, Noether normalization, transcendence degree.
• 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

Course Topics:

1. Categories and functors: natural transformations, equivalence, adjoint functors, additive functors, exactness.
2. Derived functors: projective, injective and flat modules; resolutions, the functors $Ext$ and $Tor$; examples and applications.
3. Nonabelian cohomology and its applications.

#### Spring 2018

• תת-חבורות, חבורות מנה, קשר בין תת-חבורות של חבורה ושל חבורה מנה
• תת-חבורות של SYLOW, משפטי SYLOW
• חבורות פתירות ונילפוטנטיות, חבורות-$p$
• חבורות חופשיות ותכונותיהן
• אוטומורפיזמים ואיזומורפיזמים של חבורות, חבורות אוטומורפיזמים.

#### Spring 2018

• Permutation representation and the Sylow theorems.

• Representations of groups on groups, solvable groups, nilpotent groups, semidirect and central products.

• Permutation groups, the symmetric and alternating groups.

• The generalized Fitting subgroup of a finite group.

• $p$-groups.

• Extension of groups: The first and second cohomology and applications.

Lecturer: Prof Yoav Segev

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.
• 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
Info Core for completion of MSc degree
• vector bundles and K-groups of topological spaces
• Bott’s Periodicity theorem and applications to division algebras
• Index of Fredholm operators and K-theory
• If time permits: smooth manifolds, DeRham cohomology, Chern classes, Elliptic operators, formulation of Atiyah-Singer index theorem, relations to Gauss-Bonnet theorem

#### Spring 2018

Lecturer: Prof Eitan Sayag

Hours: Wed 17:00-19:00, Math 201

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

• 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.
Info Core for completion of MSc degree

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. Banach algebras. Spectrum of a Banach algebra element. Gelfand theory of commutative Banach algebras. The spectral theorem for normal operators (in the continuous functional calculus form).

1. Affine algebraic sets and varieties.
2. Local properties of plane curves.
3. Projective varieties and projective plane curves.
4. Riemann–Roch theorem.

#### Spring 2018

Lecturer: Prof Dmitry Kerner

#### 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.