Faculty of Mathematics
The real numbers as a complete ordered field, infinite sequences of
real numbers, real valued functions of a single real variable:
limits and continuity, continuity on a closed interval, monotonic
functions, inverse functions, differentiability and the fundamental
theorem of differential calculus, Taylor's theorem, L'Hopital's rule,
curve tracing, elementary, functions, methods of
integration, definite integrals, integrable functions, fundamental
theorems of integral calculus, improper integrals.
Sequences and numerical infinite series, power series.
The n-dimensional Euclidean space Rn, real valued functions on Rn:
limits, continuity and differentiability, the chain rule and the
directional derivative, the gradient and its properties, implicit
functions and inverse mappings, extremal problems and Lagrange
multipliers, multiple integration: definition, applications
and techniques, the Jacobian and change of variables.
Vector analysis: line integrals and surface integrals,
Green's, Stokes' and Gauss' formulas.
Fields, complex numbers. Determinants, systems of linear equations
and matrices. The rank of a matrix. Vector spaces, basis and
dimension. Linear transformations, the kernel and the image of a
linear transformation. Matrix representation of a linear operator
and change of basis. Inverse matrix. Similar matrices.
Eigenvalues, eigenvectors, diagonizable matrices, the Cayley-
Hamilton theorem.
The final grade will be determined by weekly quizzes, one
midterm exam and a final exam.
The real numbers. Infinite sequences of real numbers. Real
functions of one variable: limits, continuity, continuity on
a closed interval, monotonic functions and inverse functions.
Differentiability and the main theorem of the differential
calculus. The Taylor theorem, the L'Hopital rule and study of
the behavior of a function.
The antiderivative and methods of integration. The definite
integral and its properties. Integrable functions. The
principle theorems of the integral calculus. Improper integrals.
Vectors in R2 and R3. Scalar product, vector product and mixed
product. The equations of planes and lines. Conditions for
colinearity and coplanarity.
In the course 104110 there is a greater emphasis on theory and
applications than in 104003
Functions of several variables, basic differential calculus
of such functions. Multiple integrals, line integrals,
surface integrals, vector calculus.
Numerical series, sequences and series of functions, power
series. Additional topics in differential calculus: Taylor's
formula, local and global extrema, implicit functions,
transformations in Rn.
Fields. Complex numbers. Matrices. Linear equations - solution
by elimination. Invertible matrices. Determinants. Linear
spaces: basis and dimensions. Linear transformations:
representation by matrices, similarity, characteristic values,
diagonalization. Cayley-Hamilton theorem. Minimal polynomial.
Jordan Normal Form. Inner product spaces. Gram-Schmidt process.
Unitary and Hermitian transformations. Quadratic forms.
Symmetric matrices.
Random experiments, probability spaces (S,F,P), axioms and basic
theorems, conditional probability, Bayes' theorem, independent events,
sequences of random experiments, random variables (discrete,
continuous and mixed), important distributions (discrete and
continuous), functions of random variables, random vectors,
important distributions (multivariate), functions of random vectors,
convolution, expectation and moments, generating functions,
characteristic functions and their applications, sequences of random
variables, convergence and limit theorems: applications.
First order equations: separable, homogeneous and exact equations:
integrating factors, linear, Bernoulli's, Riccati's, Lagrange's and
Clairaut's equations, existence and uniqueness theorems for equations
and systems, linear equations and linear sytems, Sturm's separation
and comparison theorems, solutions by power series, regular singular
points, Bessel's equation, Sturm-Liouville systems.
Fourier series: orthonormal series of piecewise continuous functions.
Gram-Schmidt orthonormalization. Bessel's inequality. Completeness and
closure. The Weierstrass approximation theorems. Convergence of
Fourier series.
Partial Differential equations: introduction, the Cauchy problem,
characteristic lines and canonical forms, the wave equation, the
Sturm-Liouville problem, the heat equation, the Laplace equation,
maximum and minimum principles for harmonic functions, the Poisson
integral.
Introduction: sets and numbers. Linear algebra: matrices and
determinants, systems of linear equations, vector algebra and
analytic geometry: scalar and vector products of vectors,
triple products. Differential calculus for functions of one variable:
elementary functions, limits and continuity.
Differentiation, Rolle's theorem, l'Hopital's rule, Taylor's formula,
extremal values. Integral calculus for functions of one variable:
definite integrals and antiderivatives, integration methods,
generalized integrals.
Differential calculus for functions of two variables, partial
derivatives, differentials, Taylor's formula for functions of two
variables, extremal values, Lagrange multipliers.
Series: sequences, numerical series, convergence , absolute
convergence, power series.
Integral calculus for functions of two variables: line integrals,
double integrals, applications.
Ordinary differential equations: separable equations, first order
linear equations, homegeneous equations, exact equations, integration
factors, second order linear equations with constant coefficients,
applications.
Differential and integral calculus of one real variable,
with emphasis on motivation through examples from science and
medicine. Elements of analytic geometry.
Functions of several variables, partial derivatives, multiple
and line integrals. Series. Overview of power series and
Fourier series. Differential equations. Emphasis on models
in science and medicine.
The extended Euclidean space: ideal points, lines and plane,
harmonic and cross ratios, plane projective geometry: axioms,
duality, Desargues' theorem, projective mappings between lines, the
fundamental theorem, Pappus' theorem, separation, construction of
coordinates on a line and in the plane, analytic definition of a
projective plane over a field, projective mappings of a plane, conic
sections.
Axiomatic systems: consistency, models, axiomatic systems for
Euclidean and hyperbolic geometries: incidence, order, plane
separation, distance and angle measurements, congruence of segments,
angles, triangles, axioms of parallels, the hyperbolic plane:
Saccheri quadrilaterals, parallels and ultraparallels, sums of angles,
area, models of the Euclidean and hyperbolic planes, isometries in the
Euclidean plane: properties, invariants, products of isometries,
homothety and similarity.
Inviscid flows (potential and vorticity flow): flow at low Reynolds
numbers (Stokes and Oseen), flow at high Reynolds numbers (boundary
layers), bending of beams, torsion, concentration of stresses, waves
in elastic media.
The project is intended for research-oriented students. The student
will have to study a subject and/or participate in the research of a
faculty member. A summary of the work will be submitted after
completion of the project.
Weierstrass' theorem, Jackson's theorems, interpolation, best
approximation in Chebyshev norm, least squares approximation.
Complex numbers, analytic functions, conformal mappings, complex
integration, Cauchy's theorem, Cauchy's formula, the maximum
principles, the argument principle, the calculus of residues,
harmonic functions, Taylor and Laurent series.
This course deals with classical topics of numerical analysis. The
main topics to be covered are: interpolation, a function of a single
variable numerical differentiation, numerical integration, and
iterative methods for solving nonlinear algebraic equations.
This course deals with numerical methods associated with linear
algebra. The main topics to be covered are: direct and iterative
methods for solving systems of linear equations,
least-square techniques, and
methods for computing eigenvalues and eigenvectors of matrices.
First order differential equations: linear, separable, exact,
integrating factors, homogeneous equations, existence and uniqueness
theorem (without proof), linear differential equations of order n,
systems of linear differential equations, solution of differential
equations by power series, Bessel's equation.
Eigenvalues, eigenvectors, diagonalization of
a linear operator, polynomials of matrices and linear operators:
characteristic polynomials, Cayley-Hamilton theorem, minimal
polynomials, the Jordan canonical form, inner product spaces, the
Cauchy-Schwarz inequality, norms, orthonormal bases, the Gram-Schmidt
process, linear functionals, adjoint operators, hermitian operators,
unitary operators, normal operators, the spectral decomposition
theorem, bilinear forms, quadratic forms, hermitian forms,
Sylvester's law of inertia.
Properties of integers, equivalence relations, groups, sub-groups,
cyclic groups, normal sub-groups, Lagrange's theorem, quotient groups,
the homomorphisms theorems, rings and fields: definition and
examples, polynomial rings, the Euclidean algorithm and the g.c.m.,
zero divisors, integral domains, ideals, quotient rings, and the
homomorphism theorem, unique factorization in rings of polynomials
over a field.
Introduction: sets and elements, propositional calculus, quantifiers,
arguments, algebras of sets, sets and generalized operations, power
sets, ordered pairs, Cartesian products, relations, equivalence and
order relations, functions, images and inverse images, composition of
functions, special classes of functions, families of sets, the number
system: from natural to complex, equivalence and domination between
sets, countable and uncountable sets, finite and infinite sets,
cardinals, partially ordered sets, Zorn's lemma, similarity.
Metric and topological spaces. Complete metric spaces. Baire's
category theorem. Lindeloff's theorem. Compactness. Connectedness.
Product spaces. Tichonoff's theorem. Axioms of separation.
Urysohn's lemma. Tietze extension theorem.
Hilbert spaces, orthonormal bases, bounded linear operators,
selfadjoint operators, applications to differential and
integral equations, compact operators, the spectrum of an operator,
eigenvalues and eigenvectors, the spectral theorem, spectral theory
of integral operators, applications to integral equations, the
Hilbert-Schmidt theorem, Mercer's theorem, Sturm-Liouville systems,
functions of compact operators, Banach spaces, the dual space, the
Hahn-Banach theorem, the Banach-Steinhaus theorem, the open mapping
theorem and the closed graph theorem, and their applications.
Lebesgue measure, measurable functions, integrable functions and
convergence theorems, the relation between the Riemann and Lebesgue
integrals, monotone functions and functions of bounded variation,
differentiation of monotone functions, absolutely continuous
functions, the Riemann-Stieltjes integral, the theorems of Fubini
and Tonelli.
Fields, complex numbers, vector spaces, subspaces, bases,
dimension, linear equations, matrices, the Gauss elimination process:
determinants, linear transformations, kernels, images, Hom (V,w),
Hom (V,V), determinants, eigenvalues and diagonalization.
The matrix representation of a linear transformation, Sylvester's
theorem, polynomials over a field, the unique factorization theorem,
eigenvalues and eigenvectors, characteristic and minimal polynomials,
the Hamilton-Cayley theorem, the Gram-Schmidt process, the adjoint of
a linear operator, normal transformations, the spectral theorem,
bilinear and quadratic forms, the inertia theorem, classification of
quadrics, invariant subspaces, the primary decomposition theorem,
the Jordan form.
Curves in E3 , curvature, radius of curvature, torsion,
Frenet-Serret formulas, surfaces in E3 , the space E3 , curves on
surfaces, the first linear element, the second fundamental form for
a surface, geodesics and geodesic curvature, Gaussian curvature, the
equations of Gauss and of Codazzi, surfaces of revolution, developable
surfaces, W-surfaces.
Permutations and combinations, generating functions, regression
formulas, inclusion-exclusion, Polya's counting theory, Ramsey's
theorem, distinct representatives, Hall's theorem, Latin rectangles,
orthogonal Latin squares, projective planes,
Sperner's lemma.
Solving problems from various fields of mathematics.
Solving problems from various fields of mathematics.
Tensors and dyadics (mainly Cartesian), transformation laws, continua,
description: strain, rate of strain, stress
(isotropic, deviatoric), conservation (balance) laws, continuity,
transport (mass, momentum, angular momentum, energy), kinematics,
velocity, acceleration, hydrodynamic derivatives, Eulerian and
Lagrangian descriptions, vorticity, circulation.
Dynamics, laws of motion.
Thermostatics, the first and second laws of thermodynamics, entropy,
ideal gas, energy balance.
Constitutive equations: ideal fluid and viscous fluid, stress-strain
relations. The Euler, Bernoulli and Navier-Stokes equations, initial
and boundary conditions.
Linear elasticity (Hooke's law).
Energy and virtual work.
Use of in-depth discussion and understanding of a problem in order to
formulate valid mathematical models for problems arising in
engineering, physics, chemistry, economics, etc. Use of mathematical
methods for the solution of the mathematical problems. The
interpretation of the solution vis-a-vis the original problem.
Examples of optimization problems from the fields of physics,
engineering and economics, linear programming, duality,simplex method,
one variable minimization, curve fitting: many variable minimization:
the method of steepest descent, Newton's method, restricted gradient,
penalty methods, feasible directions, duality methods.
Convex sets, cones, separation theorems, polyhedral sets, linear
inequalities, duality, introduction to game theory, nonlinear
programming, and theorems, quadratic programming,
convex functions and programming, introduction to
geometric programming, network flows and optimal control.
The real number system, functions, limits of sequences and of
functions, continuity and uniform continuity, the derivative,
Taylor's formula, investigation of functions in Cartesian
coordinates, polar coordinates and in parametric representation.
Primitive functions, the Riemann integral, improper integrals,
infinite series, sequences and series of real functions, real
functions of several real variables, partial derivatives, double
integrals, line integrals (on plane curves) and Green's formula.
Transformations from Rm to Rn: continuity, differentials,the Jacobian,
inverse transformations, implicit functions, functional dependence,
Taylor's formula and extrema of functions of several variables,
Lagrange multipliers, the Riemann integral in several variable, change
of variables in multiple integrals, line integrals, arc length,
vector analysis: gradient, divergence, rotor, orthogonal curvilinear
coordinates, surfaces and surface integration: Gauss' and Stokes'
theorems.
Selected topics in continuum mechanics, optimization, etc.,
emphasizing methods of applied mathematics.
Many of the problems to be studied will be chosen from those designed
to develop heuristic problem solving skills.
Solution of various non-algorithmic problems, most of which have
played an important role in the development of certain fields in
mathematics or in the creation of new fields. The course will
emphasize general approaches to families of similar problems, relate
the problems to material in other courses and encourage further
independent study.
Ordinary differential equations: introduction, initial
conditions, linear differential equations, equations with
variable coefficients, the Wronskian, nonhomogeneous linear equations,
power series solutions of second order linear equations (regular and
singular points), Bessel's equation and Bessel functions.
Partial differential equations: introduction, classification of
second order equations and their canonical forms, solutions of
initial and boundary value problems, the heat equation, Laplace's
equation, the method of separation of variables, Sturm-Liouville
problems, Fourier series and the Fourier transform, application
to solution of partial differential equations.
Inner product spaces, orthonormal systems, the Gram-Schmidt method,
Fourier series, convergence in norm and pointwise
convergence of Fourier series, the Fourier transform and its
properties, convolutions, applications to partial differential
equations, the Laplace transform and its properties, applications to
ordinary differential equations and integral equations, Hilbert
spaces, L2 spaces and Fourier series, introduction to operators on
Hilbert space.
Complex numbers, analytic functions, the Cauchy-Riemann equations,
harmonic functions, elementary functions of a complex variable, the
integral: Cauchy's integral theorem, Cauchy's integral formulas and
applications, Taylor and Laurent series, singularities, the
residue theorem and its applications, mappings by elementary
functions, conformal mappings and applications.
General discussion of partial differential equations and additional
conditions, solution of first order equations, the Cauchy problem and
characteristics, classification of second order equations and
canonical forms, well-posed problems, initial and boundary
conditions, the wave equation, the telegraph equation, d'Alembert's
representation, the method of separation of variables, Sturm-Liouville
problems, the heat equation, Laplace's equation, Poisson's equation.
Probability spaces, events, axioms of probability. Combinatorics,
dependent and independent events, conditional probability, Bayes'
theorem. Random variables - continuous and discrete. Distribution
functions and density functions. Expectation, variance and higher
moments. Classical probability distributions. Chebyshev's inequality.
Sums of independent random variables, the weak law of large numbers
and an application to Weierstrass' approximation theorem. Random
vectors, conditional expectation and the curve of regression.
Generating functions. The central limit theorem. Random walks.
Problems from abstract algebra, linear algebra and combinatorics,
where computer aided solutions can be obtained, will be presented.
The theoretical background of the problems will be discussed and
computers will be used to solve them.
Problems from analysis (real analysis, complex functions,
ordinary and partial differential equations),
where computer aided solutions can be obtained, will be presented. The
theoretical background of the problems will be discussed and computers
will be used to solve them.
Generalized Fourier series and the Sturm-Liouville problem, the
equations of Bessel and Legendre, applications to P.D.E. selfadjoint
and non-selfadjoint problems, solutions of P.D.E. by transform
methods (Fourier and Laplace), Green's functions and applications.
Rings, subrings, zero divisors. Integral domain, field of
fractions, ideals, maximal and prime ideals, quotient rings,
isomorphism theorems. Ring of polynomials. Euclidean ring.
Modules, submodules, factor modules. Structure theorem for
modules finitely generated over Euclidean rings. Jordan
form of a matrix, rational canonical form, cyclic invariant
subspaces. Basis theorem for finitely generated Abelian
groups. Invariants and matrix of relations for Abelian groups.
Groups - group action on a set, orbits stabilizers, Sylow
theorems. Composition series, solvable groups. Prime fields,
algebraic and transcendent extensions, finite extensions,
geometric constructions, Kronecker theorem. Splitting field
of a polynomial, algebraically closed field - existence
and uniqueness. Separability, normal extensions, automorphisms,
fundamental theorem of the Galois theory. Primitive element
theorem, cyclotomic extensions, solving equations by radicals,
Abel's theorem.
The course deals with basic notions of topology (setpoint and
algebraic) which are relevant to high school mathematics:
metric spaces, continuity, compactness and local connectedness,
product topology, Tichonoff theorem, two-dimensional surfaces,
models, Euler characteristic, topological invariants.
Hilbert spaces, convex sets and projections, orthonormal
systems, examples, normal spaces. Bounded operators, adjoint
space, Hahn-Banach theorem in Hilbert space, Riesz theorem.
Fourier series, Fourier and Laplace transforms, integral
operators, min-max, Hilbert-Schmidt-Mercer theorems.
Functions of compact operators, Dirichlet problems. Applications.
Selected topics for the following fields: graph theory (planar
graphs, Euler formula, connectivity), topology of surfaces
(triangulation, genus), geometry of surfaces (geodesics,
minimal surfaces, spherical triangles), group theory (abstract
groups, Lagrange theorem, generators).
Methods of projection: principal properties of orthogonal projections,
projections of point, line and plane, parallel, intersecting and skew
lines, perpendicularity and piercing, true size. The solid:
representation of solids and spatial geometrical shapes, axonometry,
sketching projections and axonometry of solids, developments and
intersections used in technique and in structure.
The perspective system, homology, main principles of perspective
representation on vertical and tilted planes, different methods used
by architects, representation of curved surfaces, shades and
shadows in perspective representation, reflection.
Orthographic projections, views of solids, technical sections,
dimensioning and tolerances, representation of machine parts
including casting, welding, machining, piping and ducts.
Hardware and languages for computer graphics, computerized drafting:
representation of 2-D configurations: translation, rotation,
reflection, representation of 3-D curves, "ruled" surfaces, solids:
orthographic, axonometric and perspective projections: technical
sections, intersections, developments, fundamentals of interactive
computer graphics, commercial graphics packages.
Fourier series, convergence and summability of Fourier series, the
conjugate function and Hardy spaces, interpolation and the
Hausdorff-Young theorem, Fourier transforms.
Free groups, subgroups of free groups: Nielsen-Schreier theory,
Presentation of groups by generators and relators, Magnus
representations, Fox calculus, free products with amalgamations,
Reidemeister-Schreier rewriting process, HNN extensions, one
relator groups.
Homology of complexes. Cohomology groups. Ext and Tor for
rings and algebras. Homological dimension for rings and
algebras.
Chess-like games, Zermelo's theorem. Two player zero-sum games,
the minimax theorem. Normal form games, Nash's theorem.
Characteristic function games, the core concept, the Shapley-
Bondareva theorem, the Shapley value, bargaining problems, the
Nash solution.
Algorithms and complexity, the ellipsoid algorithm, maximal flows,
matchings and weighted matchings, spanning trees and matroids, the
greedy algorithm, integer linear programming, algorithms of
intersecting planes.
The universal enveloping algebra, free Lie algebras,
Campbell-Hausdorff's formula. Solvable and nilpotent
algebras. Semi-simple algebras. Representations of Lie
algebras, Classification of simple Lie algebras in
characteristic 0.
Various subjects of non-linear analysis will be given.
1. Convergence concepts - convergence in probability, almost
sure convergence, weak convergence for probability measures and
sub-probability measures.
2. Kolmogorov's three series theorem, the law of large numbers,
applications.
3. Markov chains.
4. Ergodic theory.
Tensor fields, differential forms, Riemannian manifolds, connections,
parallel translation, curvature and torsion, Geodesics: the Hopf-Rinow
theorem, Jacobi fields, the Hadamard-Cartan theorem, submanifolds.
Difference methods for parabolic and hyperbolic equations and their
stability. Characteristics for hyperbolic equations and systems.
Relaxation and alternating direction methods for elliptic equations.
Monotone matrices, method of finite elements.
Introduction to theory of quasiregular mapping definitions.
Basic properties. Topological and metric properties.
Abstract measure and integration, regular Borel measures, Lebesgue
spaces and function spaces, the Riesz representation theorems,
absolute continuity and the Radon-Nikodym theorem, product measures,
Fubini's theorem.
Simplexes and simplicial complexes, simplicial approximation,
homotopy groups, simplicial homology and cohomology groups.
Applications: theorems of Hopf, Lefschetz and Brouwer, Cech homology
and cohomology groups, singular homology and cohomology.
Study of selected topics in topology. The aim of the seminar is to
introduce the student to reading, understanding and lecturing about
mathematical results.
Study of selected topics in topology. The aim of the seminar is to
introduce the student to reading, understanding and lecturing about
mathematical results.
Bases in Banach spaces, spectral theory of operators in Banach spaces.
Geometric properties of Banach spaces: operators in Banach spaces.
Initial value problems, existence and dependence on parameters,
linear equations with constant and periodic coefficients, stability
of linear and nonlinear equations.
The Routh-Hurwitz problem, Cauchy indices, Lyapunov's theorem,
Hankel matrices, the problem of moments.
Generalized sequences of Moore-Smith and filters, methods of
topologizing a set, Borel sets, open mappings, closed mappings and
homeomorphisms, identification topology, weak topologies, covering
characterization of normal spaces, completely regular spaces,
covering axioms, mappings of metric spaces into affine spaces,
topologies induced by families of pseudometrics, metrization theorems
of K. Morita, A.H. Stone and A. Arhangelskii, compactifications,
K-spaces, function spaces, homotopy, mappings into spheres,
topology of Euclidean n-dimensional spaces, the notion of dimension.
Existence and uniqueness theorems, dependence on initial values and
parameters: self-adjoint problems on finite intervals, oscillation
and comparison theorems, singular self-adjoint problems, two
dimensional autonomous systems and the Poincare-Bendixon theory.
Locally convex spaces, separation theorems, dual spaces, weak
topologies, weak compactness, reflexivity, the Krein-Milman theorem
and its applications, spectral theory in Banach algebras and its
applications.
Linear spaces and operators, matrices, determinants and permanents,
expansion theorems, the compound and induced matrices, the Kronecker
product, unitary spaces, the characteristic polynomial, pencils of
quadratic forms and the characterization of eigenvalues by extremal
problems, field of values, inclusion domains for eigenvalues,
Gershgorin's theorem, matrix and vector norms, nonnegative matrices,
The Perron-Frobenius theorem, doubly stochastic matrices, the
Konig-Frobenius theorem and Birkhoff's theorem, the proof of the van
der Waerden conjecture for permanents.
Extrema of integrals, the Euler-Lagrange equations, applications
in geometry and mechanics, second variation, Hamilton-Jacobi theory,
problems in optimal control.
Representation of functions by infinite series and products, entire
functions, analytic continuation, normal families, the Riemann
mapping theorem, the Schwarz-Christoffel transformation, harmonic
functions, the Dirichlet problem, harmonic measure, elliptic
functions, the monodromy theorem, Picard's theorems.
Connectivity, Euler paths and Hamiltonian circuits, trees, bipartite
graphs, matchings and covers, Hall's theorem, theorems of Konig,
Menger, Dilworth, Ramsey, directed and planar graphs, Kuratowski's
theorem, chromatic numbers, groups and graphs, flows in networks
and the maximum flow - minimum cut theorem.
Arithmetic functions, some examples, congruences, primitive
roots, quadratic reciprocity, the Riemann zeta function, Dirichlet
L-functions, primes in arithmetic progressions, diophantine
equations, Gauss' theory of binary quadratic forms.
Linear integral equations of the second kind with degenerate and
continuous kernels, kernels with weak singularities, Volterra
equations, equations of the first kind, Hilbert-Schmidt kernels:
Fredholm theorems, spectral theory of integral operators, integral
equations with symmetric kernels, connections with ordinary and
partial differential equations, nonlinear singular equations.
Planar curves: Frenet equations, rotation number, convex curves,
surfaces: fundamental forms, the Gauss and Codazzi equations,
intrinsic geometry, covariant differentiation, geodesics,
two-dimensional Riemannian geometry, the tangent bundle, compact
surfaces, the Gauss-Bonnet formula.
Linear and nonlinear approximation in various function classes,
splines, gamma-polynomials, rational approximation.
Selected from topics ranging through all phases of the development of
algebraic number theory, from Pythagoras's theorem, through Fermat's
last theorem until modern results.
Selected from topics ranging through all phases of the development
of algebraic number theory, from Pythagoras's theorem, through
Fermat's last theorem until modern results.
Study of selected topics in the theory of functions. The seminar is
intended to improve the ability of the students to independently read
mathematical material, understand it and lecture about it in a
clear manner.
Study of selected topics in the theory of functions. The seminar is
intended to improve the ability of the students to independently read
mathematical material, understand it and lecture about it in a clear
manner.
Solvable and nilpotent groups, finite simple groups: alternating
groups and the classical linear groups, free groups, generators
and relations, extensions of groups.
Elliptic equations: second order equations, the maximum principle,
existence and uniqueness theorems, higher order equations, weak
solutions and regularity theorems.
Hyperbolic equations: Well-posed problems, the energy integral,
existence and uniqueness theorems.
The course deals with numerical solutions of ordinary differential
equations. The purpose is to treat initial as well as boundary
value problems for both scalar equations and systems. We shall
mainly discuss finite difference methods. An introduction to finite
element methods will be given.
The course treats numerical solutions of partial differential
equations. We shall study boundary value problems of
elliptic type, and initial value problems of hyperbolic and
parabolic type. In each of these areas we shall discuss
both scalar equations and systems, for the linear as well
as the nonlinear cases.
A selection of topics from combinatorial geometry, including
classical results about polygons in the plane and other results
obtained by probabilistic methods.
The seminar will cover selected topics in nonlinear analysis
and its applications.
1. Discrete and continuous time martingales.
2. Brownian motion and its properties - continuity,
nondifferentiability, distribution of zeros, law of the iterated
logarithm, reflection principle.
3. Markov processes and semigroups. Application to stochastic
representations of solutions of partial differential equations.
4. Stochastic integrals.
5. Stochastic differential equations.
Differential calculus in Banach spaces, the implicit function
theorem. Degree theory. Fixed point theorems, bifurcation theory.
The Lyapunov-Schmidt method, theorems of Krasnoselski and
Rabinowitz. Critical points, existence theorems for critical points.
Elementary theory of monotone operators.
Riemann surfaces, covering surfaces, functions and differentials
on Riemann surfaces. Uniformization theory, groups of Mobius
transformations.
Conjugacy classes, Young subgroups, Young tableaux.
Regular representation of the symmetric groups, permutation
representations induced from Young subgroups, recursion
formulae for the representation of the symmetric groups,
the rules of Young, Littlewood and Richardson, Specht modules,
Weil modules, the representation theory of GL(n,F).
Selected topics in functional analysis and its applications.
Formulation of the problem of statistical estimation,
different kinds of estimators of parameters, the Rao-Cramer
inequality, the theory of regression and designs of experiments.
The variance reduction problem in the Monte Carlo method.
Nonparametric estimation.
Selected topics in linear and nonlinear operator theory
and their applications.
Basic notions:flows and semiflows, omega and alpha-limit
sets, attractors. Main examples: autonomous and time dependent
o.d.e.'s. Stability and periodicity: almost periodic functions,
recurrent motion, Poisson and Lyapunov stability.
The structure of the omega-limit sets. Applications:
LaSalle's invariance principle, limiting equations.
Invariant measures and ergodic theory.
teh mathematical basis of saving policies and loan schemes. the risks
of ceasing contributions to a plan. the mathematical basis of pure
endowment funds and whole life plans. endowment insurance plans.
pension schemes. evaluating expenses in terms of interest. premiums
reserves, and paid up values of a plan.
1. Affine algebraic groups, the associated Lie-algebras.
Homogeneous spaces, solvable subgroups, Borel subgroups,
centralizer of tori, the structure of reductive groups.
2. The irreducible reproduction of semisimple groups over a
field with characteristics and bilinear forms.
3. Introduction to the modular reproductive theory.
Differentiable manifolds, differentiable function and mappings,
rank emersions submanifolds, vector field on manifolds, tangent
covectors, tensor fields, extention algebras, partitions of unity
applications. Exterior differentiation.
Selected topics in ergodic theory and its applications.
Selected topics in optimization and convexity and their applications.
Selected topics in representation theory and its applications.
Selected topics in geometry and its applications.
Treatment of extremal problems by convexity methods.
The duality method and the relaxation method. Discussion
of various examples (connected with P.D.E. problems, flow problems,
etc.) where the above methods can be applied. Description of
the homogenization problem and examples.
Univalent functions, classes S. The area theorem.
Distortion theorems. Grunsky inequalities. De Branges theorem.
Subordination. Symmetrization. Poincare extension. Jorgensen
theorem.
Linearity of expectation - Ramsey numbers and tournaments. The
deletion and second moment methods. Large deviation inegualities and
the chromatic number. Random graphs. The local lemma. Semi-random
methods. Rodl theorem.
The solution of the a-problem in pseudoconvex domains: L-estimates
for the a-operator: existence and regularity theorems. Local
properties of holomorphic functions: the ring of germs of holomorphic
function, the Weiersrass preparation theorem. Sheaves and resolvents:
cohomology. Coherent analytic sheaves on Stein manifolds.
Advanced topics in algebraic groups which will be determined
whenever the course is given.
Advanced topics in number theory which will be determined whenever
the course is given.
Advanced topics in number theory which will be determined
whenever the courss is given.
Advanced topics in combinatorics which will be determined
whenever the course is given.
Advanced topics in analysis wich will be determined whenever
the course is given.
1. Enveloping algebra: filtration, isomorphism with
symmetric algebra, representation.
2. Two-sided ideals: prime and primitive spectrum, relation to
ideals of Lie algebras.
3. Center of enveloping algebra.
Advanced topics in algebra which will be determined whenever
the course is given.
Advanced topics in algebra which will be determined whenever the
course is given.
Advanced topics in topology which will be determined whenever
the course is given.
Advanced topics in algebraic geometry which will be determined
whenever the course is given.
Advanced topics in probalility which will be determined
whenever the course is given.
Selected topics in logic.
Topics in analysis which will be determind whenever the course is
given.
Banach spaces, conjugate spaces, Baire and open mapping theorems,
Auerbach theorem, Riesz lemma, Hahn-Banach theorem, reflexivity, weak
convergence closed graph theorem, the dual spaces and c(k)*, Alaoglu
and Krein-Milman theorems, metrizability of weak topologies, Banach
algebras, spectrum ideal, holomorphic caculus c(x) algebra, Abelian-
Banach algebras, Gelfand mapping.
The syllabus will be updated whenever the course is given.
Equations of classical mechanics, Lagrangian and Hamiltonian,
symplectic geometry, canonical transformations and generating
functions method of Hamilton-Jacobi, integrable systems.
Stochastigation and elements of KAM theory.
Uniform and non-uniform hyperbolicity, Anosov systems,
a homeomorphism, ergodic properties, Gibbs measure, metric and
topological entropy, symbolic dynamics, Lyapunov exponents, geometry
of strange attractors, interval mapping, renormalization and scaling.
Selected topics in continuum mechanics, optimization, etc.,
emphasizing methods of applied mathematics.
(a) Matrices and tensors, matrix determinants: adjoint and inverse,
linear transformations, orthogonal matrices, examples from the theory
of elasticity, homogeneous deformations.
(b) The Diffusion Equation I: the equation in various coordinates,
separation of variables in 1, 2 and 3 dimensions, special functions
(Bessel and Legendre), Green's functions.
(c) The Laplace and Fourier transform: the transforms, applications
to O.D.E., applications to the diffusion equation.
(d) Introduction to the theory of variations, Euler-Lagrange
equations, variations with constraints, the second variation,
Hamilton's principle.
(e) The Diffusion Equation II: numerical methods, nonlinear
equations, explicit and implicit methods, Crunk Nicholson,
stability and convergence, finite elements, examples.
Volterra equations. Fredholm equations of the second kind. Degenerate
kernels. The classical theory of Fredholm. Symmetric kernels.
Eigenfunctions and applications. Equations with a weak singularity.
Singular integral equations. Cauchy kernels. Applications to
boundary value problems.
Finite difference methods for solving partial differential equations.
Stability and convergence of solutions. Laxand Richmeyer's theory.
Elliptic equations. The equations of Laplace and Poisson. Iterative
methods: Jacobi, Gauss-Seidel, S.O.R. A.D.I. hyperbolic equations,
the wave equation. Explicit and implicit methods. Method of
characteristics.
Linear vibrations, Floquet theory, two-dimensional autonomous
systems, free and forced oscillations of systems with nonlinear
restoring force, self-sustained oscillations.
Examples from mechanics and electrical networks, including the
equations of Van der Pol, Rayleigh and Hill. Limit cycles.
Lienard's theorem. Introduction to autonomous and non-autonomous
systems. Selected topics.
Introduction: equations of flow, Stokes' equations, boundary
conditions, exact solutions, general solutions, reciprocal theory,
variational principles, paradoxes, motion of a particle: flow
around a sphere and cylinder, the Oseen approximation, matched
solutions, particles of arbitrary shape, conformal mappings, integral
representations, singular solutions, applications to biology and
engineering.
Longitudinal waves: linear sound waves, acoustic energy and
intensity, simple sound sources, nonlinear effects in sound waves
(shock waves) and in long waves in channels (hydraulic jumps),
transverse waves in homogeneous and isotropic systems: water waves,
dispersion relation, initial value problem, asymptotic behavior,
group velocity and energy propagation, transverse waves in
nonhomogeneous and anisotropic systems: internal gravity waves,
effects of nonhomogeneity, wind shear and variable Brunt-Vaisala
frequency, steady streaming generated by wave attenuation.
Stability of infinitesimal disturbances: initial value problem and
normal modes, the dispersion relation in the complex plane,
convective and absolute instability: energy considerations, necessary
and sufficient condition for stability bounds on phase speed and
growth rate of finite disturbances: weakly nonlinear theory, energy
integral approach, viscous and nonlinear critical layers, stability
of non-parallel flows.
Advanced methods of numerical solution of partial differential
equations and applications in science and engineering, to be examined
from the standpoint of numerical analysis.
The moden theory of boundary values problems for linear partial
differential equations. Pseudo-differential operators.