- 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
- coding theory
- combinatorics and more
- The Hilbert function and the Hilbert series, Speeding up the Buchberger algorithm, The f4 and f5 algorithms
- Normed spaces and spaces with inner products. The projection theorem for finite dimensional subspaces. Orthogonal systems in infinite dimensional spaces. The Bessel inequality and the Parseval equality, closed orthogonal systems. The Haar system.
- The Fourier series (in real and complex form). Approximate identities, closedness of the trigonometric / exponential system. Uniform convergence of the Fourier series of piecewise continuously differentiable functions on closed intervals of continuity; the Gibbs phenomenon. Integrability and differentiability term by term.
- The Fourier transform. The convolution theorem. The Plancherel equality. The inversion theorem. Applications: low pass filters and Shannon’s theorem.
- The Laplace transform. Basic relations and connection with the Fourier transform. A table of the Laplace transforms. The convolution integral. Application of the Laplace transform for solution of ODEs.
- Introduction to the theory of distributions. Differentiation of distributions, the delta function and its derivatives. Fourier series, Fourier transforms, and Laplace transforms of distributions.
- 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 Codazzi-Mainardi.
- Geometry of geodesics. Exponential map, geodesic polar coordinates, properties of geodesics, Jacobi fields, convex neighborhoods.
- Global results about surfaces. The Gauss-Bonnet Theorem, Hopf-Rinow theorem, Hopf-Poincaret theorem.
- Complex numbers, open sets in the plane.
- Continuity of functions of a complex variable
- Derivative at a point and Cauchy–Riemann equations
- Analytic functions; example of power series and elementary functions
- Cauchy’s theorem and applications.
- Cauchy’s formula and power series expansions
- Morera’s theorem
- Existence of a logarithm and of a square root
- Liouville’s theorem and the fundamental theorem of algebra
- Laurent series and classification of isolated singular points. The residue theorem
- Harmonic functions
- Schwarz’ lemma and applications
- Some ideas on conformal mappings
- Computations of integrals
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 and 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.
- Second order linear equations with two variables: classification of the equations in the case of constant and variable coefficients, characteristics, canonical forms.
- Sturm-Liouville theory.
- String or wave equation. Initial and boundary value conditions (fixed and free boundary conditions). The d’Alembert method for an infinitely long string. Characteristics. Wave problems for half-infinite and finite strings. A solution of a problem for a finite string with fixed and free boundary conditions by the method of separation of variables. The uniqueness proof by the energy method. Well-posedness of the vibrating string problem.
- Laplace and Poisson equations. Maximum principle. Well-posedness of the Dirichlet problem. Laplace equation in a rectangle. Laplace equation in a circle and Poisson formula. An ill-posed problem - the Cauchy problem. Uniqueness of a solution of the Dirichlet problem. Green formula in the plane and its application to Neumann problems.
- Heat equation. The method of separation of variables for the one-dimensional heat equation. Maximum principle. Uniqueness for the one-dimensional heat equation. The Cauchy problem for heat equations. Green?s function in one dimension. If time permits: Green?s function in the two dimensional case.
- Non-homogeneous heat equations, Poisson equations in a circle and non-homogeneous wave equations.
- If time permits: free vibrations in circular membranes. Bessel equations.
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.
- Metric spaces. Properties of complete metric spaces. Mappings of metric spaces. Fixed points. Application to ODE. Compact sets in metric spaces.
- Normed spaces. Holder’s inequality. The spaces: , , , , . Banach spaces. The completion theorem. Completeness of the above spaces.
- Linear operators. The norm of a linear operator. Isomorphism and isometry. Any two -dimensional normed spaces are isomorphic. The open mapping theorem and the closed graph theorem. Uniform boundedness principle. The completeness of the space of all linear bounded operators acting from a normed space into a Banach space .
- Linear functionals. The Hahn–Banach extension theorem. The dual space. Dual spaces for the examples above. Adjoint operators.
- Quotient space. Completeness of the quotient space. The dual space for a quotient space. Reflexive spaces.
- 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.
- Partially ordered sets. Chains and antichains. Examples. Erdos–Szekeres’ theorem or a similar theorem. The construction of a poset over the quotient space of a quasi-ordered set.
- Comparison of sets. The definition of cardinality as as an equivalence class over equinumerousity. The Cantor-Bernstein theorem. Cantor’s theorem on the cardinality of the power-set.
- Countable sets. The square of the natural numbers. Finite sequences over a countable set. Construction of the ordered set of rational numbers. Uniqueness of the rational ordering.
- Ramsey’s theorem. Applications.
- The construction of the ordered real line as a quotient over Cauchy sequences of rationals.
- Konig’s lemma on countably infinite trees with finite levels. Applications. A countable graph is k-colorable iff every finite subgraph of it is k-colorable.
- Well ordering. Isomorphisms between well-ordered sets. The axiom of choice formulated as the well-ordering principle. Example. Applications. An arbitrary graph is k–colorable iff every finite subgraph is k-colorable.
- Zorn’s lemma. Applications. Existence of a basis in a vector space. Existence of a spanning tree in an arbitrary graph.
- Discussion of the axioms of set theory and the need for them. Russel’s paradox. Ordinals.
- Transfinite induction and recursion. Applications. Construction of a subset of the plane with exactly 2 point in every line.
- Infinite cardinals as initial ordinals. Basic cardinal arithmetic. Cardinalities of well known sets. Continuous real functions, all real runctions, the automorphisms of the real field (with and without order).
קבוצות: שייכות, איחוד, חיתוך, הפרש.
מכפלה קרטזית, מושג היחס, יחסי שקילות, יחס סדר חלקי, יחס סדר קווי. הגדרת פונקציה כקבוצת סדורים.
תחשיב הפסוקים: ו/או גרירה, שקילות וטבלאות האמת שלהם, ערך האמת של פסוקים בהשמה, שקילות לוגית וגרירה לוגית, טאוטולוגיות ופסוקים שקריים, הטאוטולוגיות החשובות: למשל, חוקי הפילוג, ונוסחאות דה-מורגן.
תחשיב הפרדיקטים: הגדרת שפת תחשיב הפרדיקטים ומשמעותה; הגדרת מבנים; נוסחאות ופסוקים; הסתפקות במבנה ובהשמה, אמיתיות לוגית, גרירה לוגית, שקילות לוגית; השקילויות החשובות, סדר הכמתים, הכנסת השלילה פנימה.
תורת הקבוצות: התאמות חד-חד-ערכיות, הרכבת פונקציות והפונקציה ההפוכה; יחסי שקילות; הגדרת העוצמה, שיוויון עוצמות ואי-שיוויון עוצמות; משפט קנטור ברנשטיין (ללא הוכחה), המשפט שכל שתי עוצמות נתנות להשוואה (ללא הוכחה); משפט קנטור על עוצמת קבוצות החזקה , .
State space and transfer functions, stability-state space and input output, observability and controllability, realization theory, Hankel Matrices, statefeedback ans stabilization, Ricatti equations.
- 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.
- Complex numbers: Cartesian coordinates, polar coordinates. Functions of a complex variable. Basic properties of analytic functions, the exponential function, trigonometric functions. Definition of contour integral. The Cauchy Integral Formula. Residues and poles. Evaluation of impoper real integrals with the use of residues.
- Inner product functional spaces. Orthogonal and orthonormal systems. Generalized Fourier series. Theorem on orthogonal projection. Bessel’s inequality and Parseval’s equality.
- Trigonometric Fourier series. Complex form of Fourier series. Fourier series expansion defined over various intervals. Pointwise and uniform convergence of Fourier series. Completness of trigonometric system and Parseval’s equality. Differentiation and integration of Fourier series.
- The Fourier integral as a limit of Fourier series. The Fourier transform: definition and basic properties. The inverse Fourier transform. The convolution theorem, Parseval’s theorem for the Fourier transform. A relation between Fourier and Laplace transforms. Application of Fourier transform to partial differential equations and image processing.
- Distributions (generalized functions). The Heaviside step function, the impulse delta-function. Derivative of distribution. Convergence of sequences in the space of distributions. The Fourier transform of distributions.
- 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 aim of the course is to train students in creative problem solving in various areas of mathematics, in a manner simulating, to some extent, mathematical research (rather than ordinary courses, where homework assignments are, usually closely ? and quite obviously - related to the topics reviewed in class). The course will focus on problems whose solutions call for tools and ideas from several fields of mathematics, and will reveal ? on a miniature scale - the beauty of mathematics as an integral field of knowledge, where between seemingly unrelated subjects deep and surprising connections may arise. The problems will focus on topics in classical and modern mathematics that, due mostly to their interdisciplinary nature, are not ? as a rule ? covered in the core classes offered by the department (such topics as the axiom of choice and its applications, the Banach-Tarski paradox, transcendental number theory etc.). Classes will be divided between lectures, given by the instructor, filling in background material required to address the problems in question, and between students’ presentation of their solutions to the work sheets distributed previously. In addition to all of the above, the course will help students improve their skills in searching the mathematical literature and in the art of writing and presenting proofs.
Introduction to partial differential equations
- The first order equations: a linear equation, a quasilinear equation, resolving the initial value problem by the method of characteristic curves.
- Classification of the second order equations: elliptic, hyperbolic and parabolic equations, exam- ples of Laplace, Wave and Heat equations.
- Elliptic equations: Laplace and Poisson?s equations, Dirichlet and Neumann boundary value problems, Poisson’s kernel, Green’s functions, properties of harmonic functions, Maximum principle.
- Analytical methods for resolving partial differential equations: Sturm-Liouville problem and the method of separation of variables for bounded domains, applications for Laplace, Wave and Heat equations including non-homogenous problems. Applications of Fourier and Laplace transforms for resolving problems in unbounded domains.
- Heat equation: initial value problem in unbounded domain, basic formula for the solution, initial-boundary value problems in bounded domains, Maximum principle.
- Wave equation: Dalamber formula, non-homogenous equation, Wave equation in higher dimensions. If time permits: Legendre polynomials and spherical functions.
- Pinchover Y.; Rubinstein J. Introduction to partial differential equations (in Hebrew), Department of mathematics, Technion, 2011
- John F. Partial differential equations, Reprin t of the fourth edition. Applied Mathematical Scien ces, 1. Springer-Verlag, New York, 1991,
- Evans Lawrence C. Partial Differential Equations, Second Edition
- Gilbarg D.; Trudinger N. S. Elliptic partial differential equations of second order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001
- Zauderer E. Partial differential equations of applied mathematics, Second edition. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1989. xvi+891 pp. ISBN: 0-471-61298-7.
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.
- 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 introductory sketch and some motivating examples. Degenerate critical points of functions. Singular (nonsmooth) points of curves.
- Holomorphic functions of several variables. Weierstrass preparation theorem. Local Rings and germs of functions/sets.
- Isolated critical points of holomorphic functions. Unfolding and morsication. Finitely determined function germs.
- Classification of simple singularities. Basic singularity invariants. Plane curve singularities. Decomposition into branches and Puiseux expansion.
- 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.
- 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
- 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.
Goal: To teach the basic methods of numerical algebraic geometry that allow to find all solutions to a system of polynomial equations over the complex numbers and applications of these methods Topics: Polynomials in several variables, The zero set, Resultants, Elimination and Bezut theorems, The Newton method in several variables, Homotopy continuation and finding isolated solutions, Computation of higher dimensional components, Applications in robotics and elsewhere.
The process diffusion limited aggregation, or DLA, is a remarkable process stemming from physics. Although it is known for almost 40 years, it still is mysterious and difficult to mathematically understand. Classically it has been studied in the plane. For this project we will simulate the process in other geometries, Euclidean as well as non-Euclidean. The goal is to obtain large-scale simulations, which may lead to further understanding and new ideas. Further, we will use the simulations repeatedly to measure experimentally different quantities associated with DLA, such as growth rate, fractal dimension, etc.
- Introduction: Sets, subsets, permutations, functions, partitions. Indistinguishable elements, multisets, binary algebra of subsets. Rules of sum and product, convolutions, counting pairs. Binomial and multinomial coefficients. Stirling numbers of second kind, definition and a recurrenat formula.
- Graphs: General notions and examples. Isomorphism. Connectivity. Euler graphs. Trees. Cayley’s theorem. Bipartite graphs. Konig’s theorem, P. Hall’s theorem.
- The inclusion-exclusion method: The complete inclusion-exclusion theorem. An explicit formula for the Stirling numbers. Counting permutations under constraints, rook polynomials.
- Generating functions: General notion, combinatorial meaning of operations on generating functions. Theory of recurrence equations with constant coefficients: the general solution of the homogeneous equation, general and special cases of nonhomogeneity. Catalan numbers. Partitions of numbers, Ferrers diagrams. Exponential generating functions, counting words, set partitions, etc.
The goal of the workshop is to accompany first year mathematics majors, and to improve their skills in writing formal proofs. In the course of the workshop, the students will work in small groups on writing proofs, with an emphasis on topics related to the foundational first year courses.
The system of the real numbers (without Dedekind cuts). The supremum axiom. Convergent sequences, subsequences, monotonic sequences, upper and lower limits. Series: partial sums, convergent and divergent series, examples, nonnegative series, the root test, the quotient test, general series, Dirichlet, Leibnitz, absolute convergence implies convergence (without a proof). Limits of functions, continuity, the continuity of the elementary functions, extrema in compact intervals. The derivative of a function, Lagrange’s Mean Value Theorem, high order derivatives, L’hospital’s rules, Taylor’s Theorem, error estimates, lots of examples. The Riemann integral: only for piecewise continuous functions (finitely many points of discontinuity). Riemann sums and the definition of the integral, The Fundamental Theorem of Calculus, the existence of primitive functions (anti-derivatives). Integration techniques: integration by parts, substitutions, partial fractions (without proofs), improper integrals, applications of integrals, estimation of series with the aid of integrals, Hardy’s symbols O, o and Omega, approximation of momenta and the Stirling formula.
Pointwise convergence of sequences and of series of functions. Functions of several variables: limits, continuity, directional derivatives, the gradient, the orthogonality of the gradient to level surfaces, the chain rule, critical points (the necessity of the vanishing of the first order derivatives and examples of saddle points). Integration in 2 variables, repeated integrals and changing the integration order, the dependence of the boundaries of the integrals on the order of integration. Optimization with Lagrange multipliers, examples and less proofs. Depending on time: the Euler Lagrange equation (in Variational Calculus).
1) Probability space 2) Law of total probability 3) Conditional probability, independent events 4) Bayes’ law 5) Discrete random variables. Discrete distributions: uniform, Bernoulli, binomial, geometric, Poisson 6) Continuous random variable. Continuous distributions: uniform, exponential, normal 7) Discrete two-dimensional joint random variables 8) Independence of random variables 9) Expectation 10) Variance, covariance, correlation coefficient
Sample spaces and finite probability spaces with symmetric simple events, general probabilty spaces and the fields of events, the Borel filed and probabilities on it defined by densities, conditional probabilities and independent events, random variables and their distribution functions (discrete, absolutely continuous, mixed), the expectation of a random variable (for discrete, absolutely continuous and general distribution), the variance of a random variable, random vectors and the covariance, independent random variables, the central limit theorem for i.i.d. random variables, examples related to analysis of simple algorithms, joint densities (discrete or continuous) with computation of the covariance and the marginal distributions, the weak law of large numbers.1. A.M. Mood, F.A. Graybill And D.C.Boes. Introduction To The Theory Of Statistics 3rd Edition, Mcgraw-Hill, 1974. 2. A. Dvoretzky, Probability theory (in Hebrew), Academon, Jerusalem, 1968.3. B. Gnedenko, The theory of Probability, Chelsea 1967 (or Moscow 1982) in English; Russian origina titled ‘A course in probability theory”.
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.
- The main problem of Coding Theory
- Bounds on codes
- Finite fields
- Linear codes
- Perfect codes
- Cyclic codes
- Sphere packing
- Asymptotic bounds
R. Hill, A First Course in Coding Theory, Clarendon Press, Oxford 1986
- 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
- Additional advanced material
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
- The multiplicative group of
- Quadratic residues
- 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.
- 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.
- Permutation groups and combinatorial enumeration: Permutation groups. Introduction to Polya enumeration theory. Graph isomorphism problem.
- Finite fields and applications: Finite fields. Coding theory. Incidence structures and block designs.
- Symmetrical graphs: Cayley graphs, strongly regular graphs. -dimensional cubes and distance transitive graphs.
- Examples of applications: Design of statistical experiments. Cryptography. Recreational mathematics.
Prerequisites: 20119531 Linear Algebra
Operations over sets, logical notation, relations.
Enumeration of combinatorial objects: integer numbers, functions, main principles of combinatorics.
Elementary combinatorics: ordered and unordered sets and multisets, binomial and multinomial coefficients.
Principle of inclusion and exclusion, Euler function.
Graphs: representation and isomorphism of graphs, valency, paths and cycles.
Recursion and generating functions: recursive definitions, usual and exponential generating functions, linear recurrent relations with constant coefficients.
(Optional) Modular arithmetics: congruences of integer numbers, , invertible elements in .
- General background: sets and operations on them, Complex numbers: definition (via ordered pairs), addition and multiplication, inverses, adjoint, absolute value. Real and complex polynomials and their roots.
- Fields: Definition, properties, examples: Rationals, reals, complex numbers, integers mod p.
- Linear equations over fields, matrices and elementary row operations, rank of a matrix, solutions of homogeneous and non homogeneous systems of linear equations and the connections between them.
- Vector spaces over fields, subspaces, bases and dimensions, coordinates change of coordinate matrix, row rank as rank of a subspace, sums, direct sums of subspaces and the dimension theorem.
- Matrices multiplication, the algebra of square matrices, inverse determinants: properties, Cramer’s rule, adjoint and its use for finding the inverse.
- Linerar transformationsbasic propertieskernel and image of a linear trasformationrepresentaion of linear transformations by matrices and the effect of change of bases.linear functionals, dual bases
- Polynomials: algebras and ideals, the algebra of polynomials and its ideal structure, Lagrange interpolation, the prime factorization of a polynomial.
- Elementary canonical forms, characteristic values and vectors of linear transformations and matrices, characteristic polynomials and annihilating polynomials, invariant subspaces, direct sum decompositions, invariant direct sums. the primary decomposition theorem, diagonalization: necessary and sufficient conditions for diagonilaztion, computing diagonalizing matrices.
- Inner product spaces: inner products, inner product spaces, linear functionals and adjoints, unitary operators, Hermitian operators, normal operators and the spectral decomposition theorem, singular value decomposition theorem and applications
- Jordan forms (optional), cyclic subspaces and annihilators, cyclic decompositions, the Jordan form and its computation
- 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
Rings and ideals (revisited and expanded). Modules, exact sequences, tensor products. Noetherian rings and modules over them. Hilbert basis theorem. Finitely generated modules over PID. Hilbert Nullstellensatz. Affine varieties. Prime ideals and localization. Primary decomposition. Discrete valuation rings.
Rings and ideals (revisited and expanded). Modules, exact sequences, tensor products. Noetherian rings and modules over them. Hilbert basis theorem. Finitely generated modules over PID. Hilbert Nullstellensatz. Affine varieties. Prime ideals and localization. Primary decomposition. Discrete valuation rings.
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
(1) Parametric point estimation: sampling, statistics, estimators, sample mean, sample variance, method of moments, maximum likelihood.(2) Properties of point estimators: mean square error, loss and risk functions, unbiased estimators, lower bound for variance (Rao-Cramer inequality), efficient estimators.(3) Sufficient statistics: factorization criterion, sufficiency and completeness, exponential families of distributions, finding of minimal variance unbiased estimators.(4) Parametric interval estimation: confidence interval, sampling from normal distribution, confidence intervals for mean and variance, large sample confidence interval.(5) Test of hypotheses: simple hypothese versus simple alternative, most powerful test, composite hypotheses, uniformly most powerful test, sampling from normal distribution, tests for mean and variance.(6) Comparison of two populations: confidence interval for differences of two means, test on two means, tests concerning variances, goodness of fit tests, tests on independence.
- 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.
Probability theory: discrete and continuous variables, independent vs dependent variables, six basic discrete distributions: Bernoulli, binomial, uniform, geometric, negative binomial, Poissonian. Mean, variance, moments, probability generating function. Five basic continuous distributions: uniform, normal, exponential, gamma, beta. Moment generating function. Events, conditional probability, aging of molecules, entropy and related concepts, scores and support. Generating various probabilities. Many random variables. EST library. Covariance and correlation, iid, minimum and maximum of many random variables.Theoretical statistics; random sampling. Classical vs Bayesian approach. Distributions of the mean and variance of the sample; methods of calculating point estimates; point estimator for the mean; point estimator for the variance; biased vs unbiased, MSE, confidence intervals for the parameters of distribution. Basic ideas and definitions for the test of the hypothesis; errors of type I and II; P-values, tests for mean values, variances and proportions; test for the goodness of fit; test of independence; correlation coefficient; and its tests; linear regression; Likelihood ratios, information and support; maximum value as test statistic. Nonparametric: Mann-Whitney and permutation tests.Bayesian approach to hypothesis testing and estimation.ANOVA - analysis of variance: one-way and two-way.More theory on classical estimation: optimality aspects.BLAST.
The course will present Game Theory mostly from a mathematical point of view. Topics to be covered:
- Combinatorial games.
- Two-person zero-sum games.
- Linear programming.
- General-sum games.
- Equilibrium points.
- Random-turn games.
- Stable marriages.
First order differential equations.1. Separable equations.2. Exact equations. Integrating factors.3. Homogeneous equations.4. Linear equations. Equation Bernulli.5. The existence theoremSecond order equations.1. Reduction of order.2. Fundamental solutions of the homogeneous equations3. Linear independence. Liouville formula. Wronskian.4. Homogeneous equations with constant coefficients.5. The nonhomogeneous problem.6. The method of undetermined coefficients.7. The method of variation of parameters. 8. Euler equation.9. Series solutions of second order linear equatHigher order linear equations.1. The Laplace transform2. Definition of the Laplace transform3. Solution of differential equations by method of Laplase transform.4. Step functions.5. The convolution integral.Systems of first order equations.1. Solution of linear systems by elimination.2. Linear homogeneous systems with constant coefficients.3. The matrix method. Eigenvalues and eigenvectors.4. Nonhomogeneous linear systems.
The course discusses elements of linear algebra over the real and complex numbers. 1. The complex numbers. Linear equations: elementary operations, row reduction, homogeneous and non-homogeneous linear equations. 2. Real and complex vector spaces. Examples, subspaces, linear dependence, bases, dimension. 3. Matrix algebra. Matrix addition and multiplication, elementary operations, LU decomposition, inverse matrix, the determinant, Cramer’s law. 4. Linear transformations: examples, kernel and image, matrix representation. 5. Inner products, orthonormal sequences, the Gram-Schmidt process. 6. Diagonalization: eigenvalues and eigenvectors, the characteristic polynomial. Time permitting: unitary and orthogonal matrices and diagonalization of Hermitian and symmetric matrices.
(1) Probability space.(2) Conditional probability, independent events, Bayes’s theorem, complete probabilities.(3) Random discrete variable, discrete distributions: uniform, binomial, geometric, hypergeometric, negative binomial, Poisson.(4) Random continuos variable, continuos distributions: uniform, exponential, normal.(5) Random discrete two dimensional variable, independence of variables.(6) Mean, variance, correlation coefficient.(7) Chebyshev inequalitiy, large numbers law.(8) Central Limit Theorem, normal approximation.
- Review of probability: a. Basic notions. b. Random variables, Transformation of random variables, Independence. c. Expectation, Variance, Co-variance. Conditional Expectation.
- Probability inequalities: Mean estimation, Hoeffding?s inequality.
- Convergence of random variables: a. Types of convergence. b. The law of large numbers. c. The central limit theorem.
- Statistical inference: a. Introduction. b. Parametric and non-parametric models. c. Point estimation, confidence interval and hypothesis testing.
- Parametric point estimation: a. Methods for finding estimators: method of moments; maximum likelihood; other methods. b. Properties of point estimators: bias; mean square error; consistency c. Properties of maximum likelihood estimators. d. Computing of maximum likelihood estimate
- Parametric interval estimation a. Introduction. b. Pivotal Quantity. c. Sampling from the normal distribution: confidence interval for mean, variance. d. Large-sample confidence intervals.
- Hypothesis testing concepts: parametric vs. nonparametric a. Introduction and main definitions. b. Sampling from the Normal distribution. c. p-values. d. Chi-square distribution and tests. e. Goodness-of-fit tests. f. Tests of independence. g. Empirical cumulative distribution function. Kolmogorov-Smirnov Goodness-of fit test.
- Regression. a. Simple linear regression. b. Least Squares and Maximum Likelihood. c. Properties of least Squares estimators. d. Prediction.
- Handling noisy data, outliers.
- Introduction to number theory. Intervals and segments. Concept of a function. Elementary functions. 2. Limit of a function.3. Continuity and discontinuity of functions.4. Derivative and differential. Basic derivatives. Differentiability and continuity. Linear approximation by differentials. High-order derivatives. The fundamental theorems of differentiation and their applications. L’Hopital’s theorem and its application to calculation of limits.5. Taylor’s polynom. Expansion of functions into Taylor’s and McLoran’s series. Expansions of some usage functions. Application of Taylor’s and McLoran’s polynoms a) to approximate calculations, and b) to calculation of limits.6. Investigation of a function. Extremal points. Necessary and sufficient conditions for extrema. Max. and min. of a function within a segment. Convexity and concavity, inflection point. Asymptotes. Graph construction.7. Primitive function and indefinite integral. Table integrals. Calculation of indefinite integrals by decomposition, by parts, by substitution. Integration of rational and trigonometric functions.8. Definite integrals. Reimann’s sum. The fundamental theorem. Formula of Newton-Leibnitz. Calculation of definite integrals. Integration by decomposition, by parts, by substitution.9. Use in definite integrals to calculation of areas, volumes and curve lengthes. Rectungular and polar coordinate systems.10. First-order ordinary differential equations. General definitions. Cauchy problem. Separated variables.
- Infinite series. Tests for convergence. Taylor series and Taylor polynomials. Absolute convergence. Alternating series. Conditional convergence. Power series for functions. Convergence of power series; differentiation and integration. 2. Vectors and parametric equations. Parametric equation in analytic geometry. Space coordinates. Vectors in space. The scalar product of two vectors. The vector product of two vectors in space. Equations of lines and planes. product of three vectors and more. Catalog of the quadratic surfaces. Cylindres.3. Vector functions and their derivatives. Vector functions. differentiation formulas. Velocity and acceleration. Tangential vectors. Curvature and normal vectors. Polar coordinates.4. Partial differentiation. Functions of two and more variables. The directional derivative. limits and continuity. Tangent plane and normal lines. The gradient. The chain rule for partial derivatives. The total differentiation. Maxima and minima of functions of several independent variables. Higher order derivatives.5. Multiple integrals. Double integrals. Area and volume by double integrals. Double integrals in polar coordinates. Physical applications. triple integrals. Integration in cylindrical and spherical coordinates. Surface area. Change of variable in multiple integrals.6. Vector analysis. Vector fields. Line integrals. Independence of path. Green’s theorem. Surface integrals. The divergence theorem. Stokes’ theorem.
Ordinary differential equations: explicit solutions of first order equations. 2nd order equations. Higher order ordinary differential equations. Systems of ordinary differential equations. 2. Fourier series: Review of series of functions, Fourier expansions and properties of Fourier series, convergence of Fourier series, Gibbs phenomenon. Application to the heat conduction equation. 3. Additional applications as time permits.
- Real numbers and real line, elementary functions and graphs, some functions arising in economics. The limit of a function, calculating limits using the limit laws, continuity, the number e.2. The derivative of a function, differential rules, higher derivatives, L’Hospital rules.3. Extreme values of functions, monotonic functions, point of inflection, concavity, curve sketching, applications to economics.4. Indefinite integrals, techniques of integration, definite and improper integrals, areas between curves, applications to economics.5. Functions of several variables, economics examples, partial derivatives, linearization, the chain rile, implicit and homogeneous functions, maximum and minimum, Lagrange multipliers.6. Introduction to linear algebra, matrices, linear systems.
- Ordinary differential equations: explicit solutions of first -order equations. 2nd order equations. Higher order ordinary differential equations. Systems of ordinary differential equations.
- Fourier series: Review of series of functions, Fourier expansions and properties of Fourier series, convergence of Fourier series, Gibbs phenomenon. Application to periodic ODE’s.
- The Laplace transform and applications to ODE’s.
- Introduction: the real and complex numbers, polynomials.
- Systems of linear equations and Gauss elimination.
- Vector spaces: examples (Euclidean 2-space and 3-space, function spaces, matrix spaces), basic concepts, basis and dimension of a vector space. Application to systems of linear equations.
- Inverse matrices, the determinant, scalar products.
- Linear transformations: kernel and image, the matrix representation of a transformation, change of basis.
- Eigenvalues, eigenvectors and diagonalization.
- The real numbers, inequalities in real numbers, the complex numbers, the Cartesian representation, the polar representation, the exponential representation, the Theorem of de Moivre, root computations.
- Systems of linear equations over the real or complex numbers, the solution set and its parametric representation, echelon form and the reduced echelon form of a matrix, backwards substitution, forward substitution and their complexity, the Gauss elimination algorithm and its complexity, the reduction algorithm and its complexity.
- Vector spaces, sub-spaces of vector spaces, linear combinations of vectors, the span of a set of vectors, linear dependence and linear independence, the dimension of a vector space, row spaces and column spaces of matrices, the rank of a matrix.
- Linear mappings between vector spaces, invertible mappings and isomorphisms, the matrix representation of finite dimensional linear mappings, inversion of a square matrix, composition of mappings, multiplication of matrices, the algebra of matrices, the kernel and the image of a linear mapping and the computation of bases, changing of a basis, the dimension theorem for linear mappings.
- Inner product spaces, orthogonality, the norm of a vector, orthonormal sets of vectors, the Cauchy-Schwarz inequality, the orthogonal complement of a sub-space, orthogonal sequences of vectors, the Gram-Schmidt algorithm, orthogonal transformations and orthogonal matrices.
- The determinant of a square matrix, minors and cofactors, Laplace expansions of the determinant, the adjoint matrix and Laplace theorem, conjugation of a square matrix, similarity transformations and their invariants (the determinant and the trace).
- Eigenvalues, eigenvectors, eigenspaces, diagonalization and similarity, the characteristic polynomial, the algebraic and the geometric multiplicities of an eigenvalue, the spectral theorem for Hermitian matrices.
Basic concepts, direction fields. First order differential equations. Separable and exact equations, integrating factors. Methods for finding explicit solutions, Bernoulli equations. Euler approximations. Examples, polulation growth. Second order differential equations. Equations with constant coefficients, the solution space, the Wronskian. Nonhomogeneous equations. Variation of parameters. Systems of two first order equations with constant coefficients. Examples and applications.
- Descriptive statistics: organizing, processing and displaying data. 2. Sampling distributions: Normal distribution, the student t-distribution, Chi-Square distribution and Fisher’s F-distribution 3. Estimation: A point estimate and Confidence Interval of population parameters: Mean variance and proportion. Tolerance interval. 4. Testing hypothesis about a population’s parameters: Mean, variance and proportion. 5. Evaluating the properties of a statistical test: errors, significance level and power of a test. 6. Testing hypothesis about equality of variances, equality of means and equality of proportions of two populations. 7. Testing for independence of factors: Normal and Chi-Square methods. 8. Testing for goodness of fit of data to a probabilistic model: Chi-Square test. 9. Linear regression: Inference about the utility of a linear regression model. Covariance and correlation coefficient. Confidence and prediction intervals. 10. Weibull distribution: estimating the distribution’s parameters
Topics: 1. Limits and Continuity of functions, applications 2. Differentiability of functions, applications 3. Differentiation techniques 4. Differentiation of Implicit functions, applications 5. Investigation of functions. 6. Multivariable functions, Partial derivatives, applications 7. The Definite Integral 8. The Indefinite Integral 9. Applications of Integrals 10. Integration techniques 11. Taylor polynomials 12. Simple Differential Equations.
First order equations: separable equations, exact equations, linear equations, Bernulli equations. Existence and uniqueness. Second order equations. Reduction of order. Linear homogeneous equations, fundamental solutions and Wronskian. Inhomogeneous equations, variation of parameters. Equations with constant coefficients and the method of undetermined coefficients. Linear equations of higher order. Euler equations. Systems of differential equations. Laplace transform and its properties. Solution of initial value problems with discontinuous forcing functions. Impulse functions. Convolution.
First-Order PDE Cauchy Problem Method of Characteristics The Wave Equation: Vibrations of an Elastic String D’Alembert’s Solution Fourier Series Fourier Sine Series Initial-Boundary Value Problems The Wave Equation .Separation of Variables Fourier Series Solution of the Heat Equation The Heat Equation. Duhamel’s Principle. Laplace’s Equation. Dirichlet Problem for a Disc
Ordinary Differential EquationsBasic concepts: ordinary differential equations, differential equations of the first order, general solution, initial value problems, partial solutions. Linear differential equations with separable variables, exact equations, integration factor, homogeneous equations. Existence and Uniqueness theorem (without proof). System of differential equation of first order, solution by matrixes. Linear differential equations of second order, non- homogeneous equations, Wronskian. Linear differential equations of n-th order.Integral TransformsLaplace transform, properties of the Laplace transform. Convolution of two functions and convolution theorem. Heavyside (unit step) function, ?-function (Dirac), particularly continuous functions, their Laplace transform. Solution of non-homogeneous differential equations by Laplace transform.Fourier transform, properties of the Fourier transform. Convolution of two functions and convolution theorem. Cosines and Sine Fourier transform. Solution of integral equations by Fourier transform..
Fields. Fields of rational, real and complex numbers. Finite fields. Calculations with complex numbers. Systems of linear equations. Gauss elimination method. Matrices. Canonical form of a matrix. Vector spaces . Homogeneous and non homogeneous systems. Vector spaces. Vector spaces. Vector subspace generated by a system of vectors. Vector subspace of solutions of a system of linear homogeneous equations. Linear dependence. Mutual disposition of subspaces. Basis and dimension of a vector space. Rank of a matrix. Intersection and sum of vector subspaces. Matrices and determinants. Operations with matrices. Invertible matrices. Change of a basis. Determinants. Polynomials over fields. Divisibility. Decomposition into prime polynomials over R and over C. Linear transformations and matrices. Linear transformations and matrices. Kernel and image. Linear operators and matrices. Algebra of linear operators. Invertible linear operators. Eigenvectors and eigenvalues of matrices and linear operators. Diagonalization of matrices and linear operators. Scalar multiplication. Orthogonalization process of Gram-Shmidt. Orthogonal diagonalization of symmetric matrices.
Complex numbers.Systems of linear equations. Solving linear systems: row reduction and echelon forms. Homogenous and inhomogenous systems.Rank of matrix.Vector spaces. Linearly independent and linearly dependent sets of vectors. Linear combinations of vectors.Inner (dot) product, length, and orthogonality. The Gram - Schmidt process.Matrices: vector space of matrices, linear matrix operations, matrix multiplication, inverse matrix. An algorithm for finding inverse matrix by means of elementary row operations.Rank of matrix and its invertibility. Solving systems of linear equations by means of inverse matrix.Determinants. Condition detA=0 and its meaning. Tranposed matrix.Eigenvectors and eigenvalues. The characteristic polynomial and characteristic equation. Finding of eigenvectors and eigenvalues.Diagonalization and diagonalizable matrices. Symmetric matrices.
Analytic Geometry: planes and lines, quadric surfaces, cylinders.Vector functions: derivatives and integrals.Partial derivatives: functions of two or more arguments, chain rules, gradient, directional derivatives, tangent planes, higher order derivatives, linear approximation, differential of the first and higher order, maxima, minima and saddle points, Lagrange multipliers.Multiple integrals: double integrals, area, changing to polar coordinates, triple integrals in rectangular coordinates, physical applications.Vector analysis: vector and scalar fields, surface integrals, line integrals and work, Green’s theorem, the divergence theorem, Stokes’s theorem.Infinite series: tests for convergence of series with nonnegative terms, absolute convergence, Alternating series, conditional convergence, arbitrary series.Power series: power series for functions, Taylor’s theorem with remainder: sine, cosine and e , logarithm, arctangent, convergence of power series, integration, differentiation.
- Basic notions: equations of the first order, general solution, initial value problem, particular solution. Linear equations, separable equations, exact equations, homogeneous equations, integrating factor. The existence and uniqueness theorem (without proof). The Riccatti equations, the Bernoulli equations. Linear systems of the first order differential equations. Solution via the matrix calculus. The second order linear equations. Non-linear equations and the Wronskian. The Euler equations. Linear equations of the first order. 2. The Laplace transformation, properties of the Laplace transformation, solutions of the linear non-homogeneous equations via the Laplace transformation, the Heaviside functions, the delta functions.3. The Fourier transformation, properties of the Fourier transformation. Cosine and sine Fourier transformation. Solution of the integral equations via the Fourier transformation.
- Classification of linear Partial Differential Equations of order 2, canonical form.
- Fourier series (definition, Fourier theorem, odd and even periodic extensions, derivative, uniform convergence).
- Examples: Heat equation (Dirichlet’s and Newman’s problems), Wave equation (mixed type problem), Potential equation on a rectangle.
- Superposition of solutions, non-homogeneous equation.
- Infinite and semi-infinite Heat equation: Fourier integral, Green’s function. Duhamel’s principle.
- Infinite and semi-infinite Wave equation: D’Alembert’s solution.
- Potential equation on the disc: Poisson’s formula and solution as series.
The aim of the course is to learn basis of Calculus of functions of two and more variables. It includes: a) short study of vector algebra and analytic geometry in plane and space; b) differential calculus of two and more variables and its applications to local and global extremum problems, directional derivatives, Teylor’s formula, etc.;c) Integral calculus (line, double and triple integrals) and its applications; d) vector field theory and in particular its applications for studying potential vector fields.
- Lines and planes. Cross product. Vector valued functions of a single variable, curves in the plane, tangents, motion on a curve.
- Functions of several variables: open and closed sets, limits, continuity, differentiability, directional derivatives, partial derivatives, the gradient, scalar and vector fields, the chain rule, the Jacobian. Implicit differentiation and the implicit function theorem. Extremum problems in the plane and in space: the Hessian and the second derivatives test, Lagrange multipliers.
- Line integrals in the plane and in space, definition and basic properties, work, independence from the path, connection to the gradient, conservative vector field, construction of potential functions. Applications to ODEs: exact equations and integrating factors. Line integral of second kind and arclength.
- Double and triple integrals: definition and basic properties, Fubini theorem. Change of variable and the Jacobian, polar coordinates in the plane and cylindrical and spherical coordinates in space. Green’s theorem in the plane.
- Parametric representation of surfaces in space, normals, the area of a parametrized surface, surface integrals including reparametrizations
- Curl and divergence of vector fields. The theorems of Gauss and Stokes.
- Fields: definitions, the field of complex numbers.
- Linear equations: elementary operations, row reduction, homogeneous and inhomogeneous systems, representations of the solutions.
- Vector spaces: examples, subspaces, linear dependence, bases, dimension.
- Matrix algebra: matrix addition and multiplication, elemetary operations, the inverse of a matrix, the determinant, Cramer’s rule.
- Linear transformations: examples, kernel and image, matrix representation.
- Diagonalization: eigenvectors and eigenvalues, the characteristic polynomial, applications.
- Bilinear forms.
- Finite dimensional inner product spaces.
- Operators on finite dimensional inner product spaces: the adjoint, self adjoint operators, normal operators, diagonalization of normal operators.
The course provides a basic introduction to “naive” set theory, propositional logic and predicate logic. The course studies many important concepts which serve as the building blocks for any mathematical theory An emphasis is given to clear proof writing and correct usage of the mathematical language.A. Basics of Set Theory1. The notion of a set. Set operations: union, intersection, difference, complementation, and the power set.2. Cartesian products and binary relations. Operations on relations.3. Functions: domain and range, one to one and onto. Composition.4. Equivalence relations and set partitions.5. Order relations: partial, linear and well-founded.6. Induction theorems: mathematical, complete and well-founded.B. Set Cardinality1. The notion of cardinality. Finite, infinite and countable sets.2. Cantor’s theorem.3. The cardinality of the set of real numbers and other sets.C. Propositional Calculus1. The language of propositional calculus. Logical connectives.2. Logical implication and logical equivalence of propositional formulas.3. Disjunctive normal form.4. Complete sets of logical connectives.D. Predicate Calculus (First Order Logic)1. The language of predicate calculus: terms, formulas, sentences.2. Structures, assignments for a given structure.3. Logical implication and logical equivalence of first order formulas.4. Elementary equivalence of structures, definable sets in a given structure.
Sets. Set operations and the laws of set theory. Power set. Cartesian product of sets.The rules of sum and product. Permutations, combination, distributions. The Binomial Theorem. The well-ordering principle: mathematical induction. The principle of inclusion and exclusion. The pigeonhole principle. Recurrence relations. Generating functions.Relations and functions. Properties of relations. Equivalence relations and their properties. Partial order. Functions and their properties. Injective, surjective functions. Function composition and inverse functions.Graph, subgraph, complements. Graph isomorphism. Euler`s formula. Planar graph. Euler trails and circuits. Trees.Propositional logic. Syntax of propositional logic. Logical equivalence. The laws of logic. Logical implication. Equivalence and disjunctive normal form. Predicate logic. Syntax of predicate logic. Models. Equivalence of formulas. Normal form.Algebraic structures. Rings, groups, fields. The integer modulo n. Boolean algebra and its structure.
- Real numbers. Supremum and Infimum of a set. 2. Convergent sequences, subsequences, Cauchy sequences. The Bolzano-Weierstrass theorem. Limit superior and limit inferior. 3. Series. Partial sums, convergent and divergent series, Cauchy criterion. Series of non-negative terms. The root and the ratio tests. Conditional and absolute convergence. The Leibnitz test for series with alternating signs. Rearrangements of series (without proof) 4. The limit of a function. Continuous functions. Continuity of the elementary functions. Properties of functions continuous on a closed interval: boundedness and attainment of extrema. Uniform continuity, Cantor?s theorem. 5. The derivative of a function. Mean value theorems. Derivatives of higher order. L’Hospital’s rule. Taylor’s theorem. Lagrange remainder formula.
- The Riemann integral: Riemann sums, the fundamental theorem of calculus and the indefinite integral. Methods for computing integrals: integration by parts, substitution, partial fractions. Improper integrals and application to series. 2. Uniform and pointwise convergence. Cauchy criterion and the Weierstrass M-test. Power series. Taylor series. 3. First order ODE’s: initial value problem, local uniqueness and existence theorem. Explicit solutions: linear, separable and homogeneous equations, Bernoulli equations. 4. Systems of ODE’s. Uniqueness and existence (without proof). Homogeneous systems of linear ODE’s with constant coefficients. 5. Higher order ODE’s: uniqueness and existence theorem (without proof), basic theory. The method of undetermined coefficients for inhomogeneous second order linear equations with constant coefficients. The harmonic oscillator and/or RLC circuits. If time permits: variation of parameters, Wronskian theory.
In this course the basic concepts of one-dimensional analysis (a limit, a derivative, an integral) are introduced and explored in different applications: graphing functions, approximations, calculating areas etc. 1. Limit of a function, continuity. 2. Derivative, basic derivative formulas. 3. Derivative of an inverse function; derivative of a composite function, the chain rule; derivative of an implicit function. 4. Derivatives of high order. 5. The mean value problem theorem. Indeterminate forms and l’Hopital’s rule. 6. Rise and fall of a function; local minimal and maximal values of a function. 7. Concavity and points of inflection. Asymptotes. Graphing functions. 8. Linear approximations and differentials. Teylor’s theorem and approximations of an arbitrary order. 9. Indefinite integrals: definition and properties. 10. Integration methods: the substitution method, integration by parts. 11. Definite integrals. The fundamental theorem of integral calculus (Newton-Leibniz’s theorem). 12. Calculating areas. Bibliography Thomas & Finney, Calculus and Analytic Geometry, 8th Edition, Addison-Wesley(World Student Series).
- Infinite series of nonnegative terms and general series. Absolute and conditional convergence. Power series.
- Vector algebra. Dot product, cross product and box product.
- Analytic geometry of a line and a plane. Parametric equations for a line. Canonic equations for a plane. Points, lines and planes in space.
- Vector-valued functions. Derivative. Parametrized curves. Tangent lines. Velocity and acceleration. Integration of the equation of motion.
- Surfaces in space. Quadric rotation surfaces. Cylindrical and spherical coordinates.
- Scalar functions of several variables. Scalar field. Level surfaces. Limit and continuity. Partial derivatives. Directional derivative. Gradient vector. Differential. Tangent plane and normal line. Chain rules. Implicit function and its derivative. Taylor and MacLaurin formulas. Local extreme values. Absolute maxima and minima on closed bounded regions.
- Vector-valued functions of several variables. Vector field. Field curves. Divergence and curl.
- Line and path integrals. Work, circulation. Conservative fields. Potential function.
- Double integral and its applications. Green’s theorem.
- Parametrized surfaces. Tangent plane and normal line. Surface integrals. Flux. Stokes’s theorem.
- Triple integral and its applications. Divergence theorem.
1) The system of the real numbers: the natural numbers, well order, the inegers, the rational numbers, arithmetical operations and the field axioms, the order on the rational numbers, the Archimedian axiom, non completeness of the rationals, irrational numbers and the completeness of the reals, bounded sets, upper and lower bounds, maximum and minimum, supremum and infimum, rational and irrational powers, the basic inequalities (Bernoulli, Cauchy-Schwarz, the mean values, Holder, Minkowskii), the rational roots of a polynomial equation over the rational numbers.2) Sequences and their limits, the arithmetic of limits, divergence and tending to infinity, inequalities between sequences and between their limits, the Sandwitch Theorem, monotonic sequences, recursive sequences, Cantor’s Lemma, sub sequences, the Theorem of Bolzano-Weierstrass, the exponent, Cauchy’s criterion for the convergence of a sequence, upper limit, lower limit.3) Functions of a single variable, arithmetic operations on functions, monotone functions, the elementary functions.4) The limit of a function, the sequential definition and non sequential definition, the arithmetic of limits, one sided limits, Cauchy’s criterion, bounded functions, the order of magnitude of a function (big Oh and small Oh).5) Continuous functions, classification of discontinuities, the mean value theorem and its applications, continuity and being monotone.6) The derivative of a function, the graph of a function and the tangent line to the graph, the slope of the tangent line, the velocity, differentiability and continuity, the arithmetic of the differential operator in particular the Liebnitz rule, composition of functions, the chain rule, higher order derivatives, a theorem of Fermat, Roll’s Theorem, the maen value theorem of Lagrange, the mean value theorem of Cauchy, L’hospital rules, Taylor’s Theorem.7) Local and global maximum and minimum, inflection points, convex and concave functions, asymptotic lines, sketching
- Analytic geometry in space. Vector algebra in R3. Scalar, cross and triple product and their geometric meaning. Lines, planes and quadric surfaces in space including the standard equations for cones, ellipsoids, paraboloids and hyperboloids.
- Functions of several variables.Graphs and level curves and surfaces. Limits and continuity. Properties of the continuous functions on a closed bounded domain. Partial derivatives. The plane tangent to graph of the function. Differentiability, the total differential and the linear approximation. Differentiability implies continuity. The chain rule. The gradient vector and the directional derivative. Tangent plane and the normal line to a surface at a point. 201.1.9761
- Maxima and minima for functions of several variables. Higher-order partial derivatives and differentials. Taylor’s formula. Local extrema and saddle points. Necessary conditions for local maxima and minima of a differentiable function. Sufficient conditions for local maxima and minima via the Hessian. Global extrema in closed bounded sets. Lagrange Multipliers.
- Double integrals . Double integrals on rectangles. Connection with the volume. Properties and evaluation of double integrals in non-rectangular domains. Iterated integrals and change of order of integration. Change of variables formula for the double integral and the Jacobian. Double integrals in polar coordinates. Applications of the change of variables formula to the computation of area.
The aim of the course is to study main principles of probability theory. Such themes as probability spaces, random variables, probability distributions are given in details.Some applications are also considered.1. Probability space: sample space, probability function, finite symmetric probability space, combinatorial methods, and geometrical probabilities.2. Conditional probability, independent events, total probability formula, Bayes formula. 3. Discrete random variable, special distributions: uniform, binomial, geometric, negative binomial, hypergeometric and Poisson distribution. Poisson process.4. Continuous random variable, density function, cummulative distribution function. Special distributions: uniform, exponential, gamma and normal. Transformations of random variables. Distribution of maximum and minimum. Random variable of mixed type.5. Moments of random variable. Expectation and variance. Chebyshev inequality.6. Random vector, joint probability function, joint density function, marginal distributions. Conditional density, covariance and correlation coefficient.7. Central Limit Theorem. Normal approximation. Law of Large Numbers.
Logic and Set theory: propositional calculus, boolean operators and their truth tables, truth values of formulae, logical equivalence and logical inference, tautologies and contradictions, the important tautologies, e.g., the distributive laws and De Morgan formulae. Sets: inclusion, union, intersection, difference, cartesian product, relations, equivalence relations, partial orders, linear orders, and functions. Basic Combinatorics: induction, basic counting arguments, binomial coefficients, inclusion-exclusion, recursion and, generating functions. Graphs: general notions and examples, isomorphism, connectivity, Euler graphs, trees.
Introduction to computer science 201-1-1011 1. Basic Programming 2. Some order and arrays 3. Arrays and functions 4. Strings, Binary search and Selection sort 5. Insertion sort, handling matrices 6. Introduction to recursive programming 7. Recursive programming : binary search, selection sort, Merge sort 8. Last examples : recursive programming 9. Introduction to object oriented programming 10. Object oriented programming 11. Lecture canceled 12. Interfaces 13. Inheritance 14. Inheritance and abstract inheritance 15. Exceptions 16. Data structures 17. Stack application 18. Queue and linked lists 19. Lists - the recursice perspective 20. Trees
The Lab will be based on a LINUX platform, and use the C programming language. The emphasis is on low-level programming. Goals of this lab are to introduce issues in low level programming, as well as techniques on how to learn needed information on demand
- Variable types: Numbers and strings, basic operations
- Input and output
- Conditionals and loops
- Lists, dictionaries and other data types
- Object oriented programming
- Introduction to computational complexity
- Introduction to scalars and vectors: operations with scalars and vectors, scalar product, vector product, vector equations, indices, Einstein summation convention, kronecker delta, Levi-Civita symbol.
- Introduction to functions: types, continuity, limits, derivatives, integrals, integration methods, taylor series, remainder.
- Functions with scalar input and vector output: curves, tangent, normal, velocity, acceleration, curvature, torsion. Frenet-Serret basis and kinematics.
- Functions with vector input and scalar output: extrema, contours, gradient, directional derivative, tangent space.
- Multiple integrals
- Functions with vector input and vector output: conservative fields, rotational fields, divergence, curl, line integral, surface integral, Stokes and Gauss
- Differential equations: damped harmonic oscillator with driving
- Rotations: scalars, vectors, tensors
- Curvilinear coordinates
Number representation in different bases, binary codes and binary arithmetic. Combinational systems: Boolean algebra, switching function representations and minimization. Karnaugh maps, prime implicate table. Hazards. Combinational circuit design. Switching devices: logic gates (NAND, AND, NOR, OR, NOT, XOR); modules: HA, FA, HS, FS, Multipliers, Decoders, Multiplexers, Demultiplexers, PROM, PLA. Using modules to the implementation of combinational circuits. Sequential systems: Basic models, synchronous and asynchronous system structure. Bistable memory devices (Latches, Flip-Flops), transition table and state table. Master-Slave Flip-Flops, Edge-Triggered Flip-Flops. Design and implementation of sequential systems, synchronous and asynchronous. Race problem solution. Special modules: Registers, Shift Registers, Counters.
- Introduction: overall description of the multi-purpose digital computer operation, the central processing unit (CPU) organization, representation and processing of information (data and programs).
- Machine language: representation of operations and commands, selecting the command set, designing the command architecture, addressing modes.
- Control Unit: application methods by logic and micro-programming, interpretation and execution of command set.
- Arithmetic Unit: numbers representation, basic arithmetic operations in digital systems, addition, subtraction, multiplication and division of fix and floating point numbers.
- Memory Unit: memory types, addressing methods, memory organization, virtual memory, cache memory.
- Input/Output Unit: I/O addressing modes, I/O data transferring control by serial polling, interrupt and direct memory access (DMA).
Definitions of expander graphs, Cheeger-Buser inequality, Expander mixing lemma, Alon-Boppna Theorem, Existence of expander graphs (non constructive proof), Application of expander graphs to error-correcting codes, Groups – basic concepts (actions, Cayley graphs, normal subgroups, unitary representations), Margulis expanders, Kazhdan property (T) – definition, hereditary properties, relation to expander graphs, other constructions of expander graphs.
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.
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
- -stable theories and Morley rank
- Fraisse’s amalgamation theorem.
Students should be familiar with the following concepts: Languages, structures, formulas, theories, Godel’s completeness theorem and the compactness theorem.
- Sheaves on topological spaces
- Affine schemes
- Schemes and morphisms between them.
- Quasi-coherent sheaves
- Separated and proper morphisms.
- Vector bundles and the Picard group of a scheme.
- The functor of points and moduli spaces.
- Morphisms to projective space and blow-ups.
- Smooth morphisms and differential forms.
- Sheaf cohomology.
- Group schemes.
Algebraic Geometry - Schemes (1)
The course will be in English . It will continue in the Spring semester, as: Algebraic Geometry - Schemes (2)
Course Topics (tentative, for both semesters; more details in course web site):
- Sheaves on topological spaces.
- Ringed spaces.
- Affine Schemes.
- Maps of schemes.
- Morphisms to projective space and blow-ups.
- Calculating some invariants.
- The functor of points and moduli spaces.
- Algebraic differential calculus.
- Group schemes and their Lie algebras.
The field of real numbers is defined as the completion of the field of rational numbers with respect to the norm . However, there are other norms on , each corresponding to a prime number , and the completion of with respect to any such norm leads to the field of -adic numbers, denoted by . This is a topological complete field, and thus it makes sense to develop an analysis on it.
Many features of the -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 . The overall picture of the -adic analysis makes impression of a surprising and beautiful one, and easier than its real counterpart. Nowadays, the -adic analysis has endless applications in geometry and number theory.
In this course we will study the field of -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 -adic analysis to prove the meromorphic continuation of the zeta-function of Riemann and its functional equation.
- Arithmetic of : sums and products, square roots, finding roots of polynomials.
- Algebraic number theory of : finite extensions, algebraic closure of , completion of the algebraic closure, local class field theory will be mentioned.
- Topology of : elementary topological properties, Euclidean models of .
- Analysis on : convergence of sequences and series, radius of convergence, elementary functions , , the space of locally constant functions.
- Harmonic analysis on : characters of , Haar measure, integration of locally constant functions, Fourier transform.
- The ring of adeles as an object unifying for all : topological properties, integration and Fourier transform, Poisson summation formula.
- 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 -sets. An application of incidences to additive number theory.
- Coloring and hiting problems for geometric hypergraphs : -dimension, Transversals and Epsilon-nets. Weak eps-nets for convex sets. -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.
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 . 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.
- Review of abelian categories and additive functors.
- Differential graded rings, modules and categories.
- The derived category of a differential graded category.
- Derived functors. Resolutions of differential graded modules.
Commutative algebra via derived categories (regular and CM rings, Grothendieck’s Local Duality, MGM Equivalence, rigid dualizing complexes).
Geometric derived categories (of sheaves on spaces). Direct and inverse image functors, Grothendieck Duality, Poicar?-Verdier Duality, perverse sheaves).
Derived categories associated to nocommutative rings (dualizing complexes, tilting complexes and derived Morita theory).
Derived categories in modern algebraic geometry and modern string theory (a survey).
Review of material from past semesters (the courses “Derived Categories I and II”).
Derived categories in commutative algebra: dualizing complexes, Grothendieck’s local duality, MGM Equivalence, rigid dualizing complexes.
Derived categories in algebraic geometry: direct and inverse image functors, global Grothendieck duality, applications to birational geometry (survey), -adic cohomology and Poincare-Verdier duality (survey), perverse sheaves (survey).
Derived categories in non-commutative ring theory: dualizing complexes, tilting complexes, derived Morita theory.
Derived algebraic geometry: nonabelian derived categories (survey), infinity categories (survey), derived algebraic stacks (survey), applications (survey).
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.
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.
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.
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.
- Rings, ideals, and homomorphisms
- Modules, Cayley-Hamilton theorem, and Nakayama’s lemma
- Noetherian rings and modules, Hilbert basis theorem
- Integral extensions, Noether normalization lemma, and Nullstellensatz
- Affine varieties
- Localization of rings and modules
- Primary decomposition theorem
- Discrete valuation rings
- Selected topics
- Miles Reid, Undergraduate Commutative Algebra
- Miles Reid, Undergraduate Algebraic Geometry
- Altman, Kleiman, A Term of Commutative Algebra
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
- Modules: free modules, exact sequences, tensor products, Hom modules, flatness.
- Prime ideals and localization: local rings, Nakayama’s Lemma, the spectrum of a ring, dimension and connectedness.
- Noetherian rings: the Hilbert basis theorem, the Artin-Rees lemma, completion, grading.
- 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
- Categories and functors: natural transformations, equivalence, adjoint functors, additive functors, exactness.
- Derived functors: projective, injective and flat modules; resolutions, the functors and ; examples and applications.
- Nonabelian cohomology and its applications.
- תת-חבורות, חבורות מנה, קשר בין תת-חבורות של חבורה ושל חבורה מנה
- תת-חבורות של SYLOW, משפטי SYLOW
- חבורות פתירות ונילפוטנטיות, חבורות-
- חבורות חופשיות ותכונותיהן
- אוטומורפיזמים ואיזומורפיזמים של חבורות, חבורות אוטומורפיזמים.
- Review of differentiable manifolds, definition of a Lie group. Quotients in the category of Lie groups, homogeneous manifolds, haar measure, connected components.
- Algebraic groups, matrix groups, the classical groups.
- Lie algebras and connection to Lie groups.
- Nilpotent, solvable and semisimple Lie algebras and Lie groups, Lie theorem, Engel theorem, Levi decomposition.
- Cartan-Killing form.
- Representation of a Lie algebra over the complex numbers.
- 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
- Consistency of the negation of the continuum hypothesis
- Consistency of the negation of the Axiom of choice
- 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
Basic Algebraic Structures: rings, modules, algebras, the center, idempotents, group rings
Division Rings: the Hamiltonian quaternions, generalized quaternion algebras, division algebras over , , , (theorems of Frobenius and Wedderburn), cyclic algebras, the Brauer–Cartan–Hua theorem
Simplicity and semi-simplicity: simplicity of algebraic structures, semi-simple modules, semi-simple rings, Maschke’s theorem
The Wedderburn–Artin Theory: homomorphisms and direct sums, Schur’s lemma, the Wedderburn–Artin structure theorem, Artinian rings
Introduction to Group Representations: representations and characters, applications of the Wedderburn–Artin theory, orthogonality relations, dimensions of irreducible representations, Burnside’s theorem
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.
Examples: time flow for solutions of differential equations, symbolic dynamics, cellular automata, geodesic and horocyclic flows, interval exchange transformations, Chabauty space, profinite actions. Basic concepts: Factors and extensions, topological transitivity, minimality, equicontinutity, distality, proximality, weak mixing, almost 1-1 extenstions. Topological entropy. The Ellis semgroup, Ellis theorem on the existence of idempotents, Ellis-Auslander theorem, Hindeman?s theorem. Universal constructions. Stone Cech compactification. The universal ambit and the universal minimal flow. Universal proximal and strongly proximal flows. Furstenberg boundary.
- Lattices. Doubly periodic functions.
- Riemann surfaces: definition and examples (Riemann sphere, roots, logarithms).
- Riemann surfaces: Euler characteristic, genus, Hurwitz formula. Riemann–Roch theorem (statement)
- Complex tori and elliptic curves.
- Morphisms between elliptic curves. The group law. Arithmetic of elliptic curves (a glimpse: Mordell’s theorem, L-functions, the Birch and Swinnerton-Dyer conjecture).
- The group SL(2,Z). The space of all lattices. The trefoil knot.
- Modular forms for SL(2,Z): finite-dimensionality. Eisenstein series.
- Theta series. Four squares. Even unimodular lattices and sphere packing in dimension 8.
- Hecke operators. `Reminders’ on the Riemann zeta function. L-functions. Modularity of elliptic curves (statement).
- The Ramanujan conjecture. Expander graphs. The j-invariant. Moonshine.
- Modular curves and their compactification.
- Complex multiplication.
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 . 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).
- Sheaves on topological spaces
- Affine schemes
- Schemes and morphisms between them.
- Quasi-coherent sheaves
- Separated and proper morphisms.
- Vector bundles and the Picard group of a scheme.
- The functor of points and moduli spaces.
- Morphisms to projective space and blow-ups.
- Smooth morphisms and differential forms.
- Sheaf cohomology.
- Group schemes.
- Normal families and rational maps.
- The Fatou and Julia sets.
- Properties of the Julia set.
- The structure of the Fatou set.
- Periodic points.
- Forward invariant components.
- The no wandering domains theorem.
- Commuting and semiconjugate rational functions.
- Introduction to arithmetic dynamics.
- Affine algebraic sets and varieties.
- Local properties of plane curves.
- Projective varieties and projective plane curves.
- Riemann–Roch theorem.
- Real numbers (axiomatic theory). Supremum and Infimum of a set. Existence of an n-th root for any a > 0. 2. Convergent sequences, subsequences, Cauchy sequences. The Bolzano-Weierstrass theorem. Upper and lower limits. 3. Series. Partial sums, convergent and divergent series, Cauchy criterion. Series of non-negative terms. The root and the ratio tests. Series of arbitrary terms. Dirichlet, Leibnitz, and Abel theorems. Rearrangements of series. The Riemann Theorem. 4. The limit of a function. Continuous functions. Continuity of the elementary functions. Properties of functions continuous on a closed interval. Uniformly continuous functions. Cantor?s theorem. 5. The derivative of a function. Mean value theorems. Derivatives of higher order. L’Hospital’s rule. Taylor’s theorem.
- The Riemann integral: Riemann sums, the fundamental theorem of calculus. Methods for computing integrals (integration by parts, substitution, partial fractions). Improper integrals and application to series. Numerical integration. Stirling’s formula and additional applications time permitting.
- Uniform and pointwise convergence. Cauchy’s criterion and the Weierstrass M-test. Power series. Taylor series, analytic and non-analytic functions. Convolutions, approximate identities and the Weierstrass approximation theorem. Additional applications time permitting.
- A review of vectors in R^n and linear maps. The Euclidean norm and the Cauchy-Schwarz inequality. Basic topological notions in R^n. Continuous functions of several variables. Curves in R^n, arc-length. Partial and directional derivatives, differentiability and C^1 functions. The chain rule. The gradient. Implicit functions and Lagrange multipliers. Extrema in bounded domains.
- Basic concepts of topology of metric spaces: open and closed sets, connectedness, compactness, completeness.
- Normed spaces and inner product spaces. All norms on 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
- 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.