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Elective courses for 2nd stage studies in Mathematics (course group defined by Faculty of Mathematics, Informatics, and Mechanics)

Faculty: Faculty of Mathematics, Informatics, and Mechanics Courses displayed below are part of group defined by this faculty, but this faculty is not necessarily the one that organizes these courses. Read Help for more information on this subject.
Course group: Elective courses for 2nd stage studies in Mathematics
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2023Z - Winter semester 2023/24
2023L - Summer semester 2023/24
(there could be semester, trimester or one-year classes)
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2023Z 2023L
1000-135AF*
n/a
Classes
Winter semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

This is a fundamental course in functional analysis. The course gives a basic knowledge on Banach and Hilbert spaces and their geometric properties. The next topic of the course concerns linear functionals and operators in these spaces and their properties. The course gives also basic informations on spectra and spectral properties of linear operators. In particular, spectra of compact operators in Hilbert spaces are discussed.

Course page
1000-135GEA n/a
Classes
Summer semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

This is an introductory course in algebraic geometry. The aim is to introduce students to algebraic varieties and their basic geometric properties. At the end of the course examples of applications of algebraic geometry will be shown.

Course page
1000-135MGT n/a
Classes
Winter semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

Fundamental notions of the category theory, additive and abelian categories. Tensor product in the category of modules. Projective and injective modules, resolvents. Graded groups, chain complexes and their homologies.

Derived functors of Hom and of the tensor product. Presheaves, sheaves and their cohomologies.

Simplicial cohomologies and Cech cohomologies. Coverings and principal bundles; cohomological interpretation.

Course page
1000-135TA n/a
Classes
Summer semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

Homotopy groups. Fibrations and cofibrations. Long exact sequence of homotopy grooups of fibration. Axioms for generalized (co-)homology. Singular (co-)homology. Degree of self-maps of spheres. Cellular (co-)homology. De Rham cohomology. Multiplicative structure in singular (co-)homology. Orientation of topological manifolds and duality theorems. Intersection number and linking number.

Course page
1000-135APZ n/a
Classes
Winter semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

Introduction to two key concepts in numerical analysis: approximation and complexity. Classical polynomial approximation of smooth functions. Approximation based on partial information. Construction of optimal algorithms in prescribed model of computation.

Course page
1000-135STB n/a
Classes
Winter semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

Systematic introduction to Bayesian statistics. The subject of this course is now becoming more popular, has many important

applications, but is treated marginally or entirely omitted in standard courses of statistics. The course is dedicated to students of mathematics and also students of informatics who are interested in statistics.

Course page
1000-135ALP n/a
Classes
Winter semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

This lecture class provides an introduction to commutative algebra; it is required for algebraic geometry lecture. The topics concern commutative rings and modules over such rings. An important class of rings considered are noetherian rings.

Course page
1000-135ANZ n/a
Classes
Summer semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

Main topics: Weierstrass theorem, Mittag-Leffler theorem, Runge theorem. Many-valued functions, analytical extensions, monodromy.

Analytical functions on Riemann surfaces; Problems in Riemann surface theory: basic information and examples.

Fundamental notions of the theory of analytic functions in many complex variables, Cauchy-Riemann equations, power series expansions, analytical extensions, Cousin problems.

Course page
1000-135ROZ n/a
Classes
Winter semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

The course covers the following topics: local complex geometry, complex differential forms, Kaehler manifolds, Dolbeault cohomologies, Hodge theory, vector bundles, Chern classes.

Course page
1000-135MOF n/a
Classes
Summer semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

The course will present methods for pricing financial assets. The following topics will be discussed: tree algorithms, Monte Carlo methods, solutions to the Black-Scholes PDE. The course will present convergence problems for SDE of Ito type, parabolic PDE and properties of their solutions, and convergence problems for numerical solutions to parabolic PDE. Mathematical content will be enlarged by examples of numerical valuation of selected instruments.

Course page
1000-135GK n/a
Classes
Winter semester 2023/24
  • Classes - 15 hours
  • Lab - 15 hours
  • Lecture - 30 hours
Groups

Brief description

Our aim is to introduce the students to basic algorithms and data structures used in computer graphics. These include basic raster graphics algorithms, two- and three-dimensional geometry, elements of computational geometry, geometric modelling, visibility algorithms and illumination models.

Course page
1000-135GR n/a
Classes
Summer semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

Abstract smooth manifolds, smooth maps. Tangent vectors and derivative of a smooth map. Vector fields as differentatials and flows. Lie bracket. Tangent bundle. Ventor bundles and operations on them. Tensor fields. Foliations and Frobenius theorem. Differential forms, exterior derivative and the Stokes theorem. Covariant derivative and affine connection, parallel transport and geodesics. Curvature tensor. Levi-Civita connection on Riemannian manifold. Ricci tensor.Geodesically complete manifolds. Manifolds of constantt curvature (space form problem). Lie groups and algebras.

Course page
1000-135UD n/a
Classes
Winter semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

The theory of dynamical systems studies the long-term evolution of systems, which takes place under constant and deterministic rules. The evolution may thus be given (in the discrete time case) by the iterates of a certain map, or (in the continuous time case) by the solutions of an ODE, etc. The aim of the theory is to describe regular and chaotic properties of some classes of systems, study their stability and determine their invariants (such as entropy).

Course page
1000-135EKN n/a
Classes
Summer semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

The aim of the course is to present the theoretical background, main concepts and basic methodology of modern econometrics. We will discuss classification and examples of econometric models, in particular one equation linear model, estimation with least squares, applications in non-linear models, large sample theory, time series: stationarity, ARIMA,heteroskedasticity and forecasting.

Course page
1000-135IFI n/a
Classes
Winter semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

The course presents fundamental financial instruments: forward rate agreements, interest rate swaps, forward and futures contracts, options - plain vanilla options, selected simple exotic options and interest rate options. For each of these instruments the following is shown: the structure of the instrument, its market role, pricing method and sensitivity analysis - all that with market practice aspects.

Course page
1000-135ASW n/a
Classes
Winter semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

The lecture aims to present the classical results concerning the structure and linear representation theory of

finite dimensional algebras over fields. The following will be discussed: correspondence between theory of

modules and representation theory, simple modules, radical algebras and classification of semi-simple associative algebras.

Applications will be given to the representation theory of finite groups, through results concerning

group algebras and the theory of group characters. Examples of applications will be discussed. Basic information on finite dimensional Lie algebras and their representations will be given. As a tool in this theory,

universal enveloping algebras and their properties will be discussed

Course page
1000-135AF
Classes
Winter semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Summer semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

This is a fundamental course in functional analysis. The course starts with basic notions on Banach and Hilbert spaces and their properties. The next topic of the course concerns linear functionals and operators in these spaces and their properties. The course gives also basic information on spectra and spectral oroperties of linear operators. Spectra of compact operators on Hilbert spaces are discussed.

Course page
1000-135TOG n/a
Classes
Summer semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

The aim of this course is to present a series of main concepts and theorems of general topology which are both important and elegant from the point of view of this field, as well as essential for applications in topology and mathematics as a whole. The notion of compactness and its variants is of central importance to the course.

Course page
1000-135WAS n/a
Classes
Winter semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

Main topics: General theory of stochastic processes, Wiener process; Introduction to the theory of continuos-time martingales; Definition and basic properties of stochastic integral; Ito's formula; Stochastic differential equations and their connections with partial differential equations.

Course page
1000-135AGL n/a
Classes
Summer semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

Classical linear groups, abstract Lie groups, compact groups.

Classical Lie's theory: correspondence between Lie groups and Lie algebras. The exponential map.

Abstract approach to Lie algebras. Classification of simple Lie algebras.

Representations of classical Lie groups and Lie algebras by the highest weight. Homogeneous spaces.

Course page
1000-135LOM n/a
Classes
Winter semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

An introduction to classical topics of mathematical logic with elements of model theory.

Course page
1000-135MMN n/a
Classes
Winter semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

The aim of the lecture is to present some basic methods of dynamical systems and partial differential equations that are essential in the modern description of natural and social processes.

Course page
1000-135MBM n/a
Classes
Summer semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

The lecture is devoted to the widely understood mathematical modelling in biology and medicine. We mainly focus on ecological models which are built using differential and difference equations. We also consider models of immune reactions and those of classical genetics (Mendel theory) based on Markov chains.

Course page
1000-135MMK n/a
Classes
Summer semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description
No brief description found, go to course home page to get more information.
Course page
1000-135IP1 n/a
Classes
Summer semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

During this course we describe and solve problems related to the modelling of financial markets and to the pricing and hedging of financial derivatives.

Course page
1000-135IP2 n/a
Classes
Winter semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

The course will desribe: Interest rate securities. Models of short rate. HJM model. Interest rate derivatives (FRA, caps, floors, swaptions etc.). Market models. Callibration of models to market data.

Course page
1000-135MUZ n/a
Classes
Winter semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

This is a basic course providing theoretical principles for computation of premiums and reserves in a life insurance company. The required mathematical background comprises calculus and the first course in probability. The computational models for individual and multiple - life policies (including multiple - decrement model) are developped in a systematic way. The carefully selected problems and exercises reinforce working knowledge of theoretical issues. The course can serve as a good preparation for future actuaries.

Course page
1000-135TM n/a
Classes
Winter semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

This lecture is self-contained and is based only on the obligatory courses from the first two years of studies. We will recall some elements of measure theory. The aim of this course is to improve the knowledge of tools from Measure Theory in Functional Analysis, Partial Differential Equations, Probability Theory, and many other fields of mathematics.

Course page
1000-135MAG n/a
Classes
Winter semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

The course intends to systematize the fundamental notions of number theory, algebra and analysis in the scope of the national curriculum in mathematics.

It will also present methods of teaching of school algebra and the good practices, at the level of an elementary school and of a high school. The course will also include topics that can inclrase the interest of students in mathematics.

Course page
1000-135MGE n/a
Classes
Winter semester 2023/24
  • Classes - 30 hours
  • Lecture - 30 hours
Groups

Brief description

The course presents a wide variety of methods for teaching geometry at primary and secondary school level.

Course page
Krakowskie Przedmieście 26/28
00-927 Warszawa
tel: +48 22 55 20 000 https://uw.edu.pl/
contact accessibility statement USOSweb 7.0.1.0-2 (2024-02-19)