Elective courses for 2nd stage studies in Mathematics (course group defined by Faculty of Mathematics, Informatics, and Mechanics)
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2022Z  Winter semester 2022/23 2022L  Summer semester 2022/23 (there could be semester, trimester or oneyear classes) 
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2022Z  2022L  
1000135AF*  n/a 
Classes
Winter semester 2022/23
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. 


1000135GEA  n/a 
Classes
Summer semester 2022/23
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. 


1000135MGT  n/a 
Classes
Winter semester 2022/23
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. 


1000135TA  n/a 
Classes
Summer semester 2022/23
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 selfmaps 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. 


1000135APZ  n/a 
Classes
Winter semester 2022/23
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. 


1000135STB  n/a 
Classes
Winter semester 2022/23
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. 


1000135ALP  n/a 
Classes
Winter semester 2022/23
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. 


1000135ANZ  n/a 
Classes
Summer semester 2022/23
Groups
Brief description
Main topics: Weierstrass theorem, MittagLeffler theorem, Runge theorem. Manyvalued 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, CauchyRiemann equations, power series expansions, analytical extensions, Cousin problems. 


1000135ROZ  n/a 
Classes
Winter semester 2022/23
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. 


1000135MOF  n/a 
Classes
Summer semester 2022/23
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 BlackScholes 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. 


1000135GK  n/a 
Classes
Winter semester 2022/23
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 threedimensional geometry, elements of computational geometry, geometric modelling, visibility algorithms and illumination models. 


1000135TST  n/a 
Classes
Winter semester 2022/23
Groups
Brief description
It is an introduction to modern mathematical Control Theory. The theory is illustrated by numerous examples from economy, biology, medicine, physics and technology. In particular: controllability for linear and nonlinear systems, bangbang principle, timeoptimal control, Pontriagin Maximum Principle, transversality, dynamic programming. 


1000135GR  n/a 
Classes
Summer semester 2022/23
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. LeviCivita connection on Riemannian manifold. Ricci tensor.Geodesically complete manifolds. Manifolds of constantt curvature (space form problem). Lie groups and algebras. 


1000135UD  n/a 
Classes
Winter semester 2022/23
Groups
Brief description
The theory of dynamical systems studies the longterm 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). 


1000135EKN  n/a 
Classes
Summer semester 2022/23
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 nonlinear models, large sample theory, time series: stationarity, ARIMA,heteroskedasticity and forecasting. 


1000135IFI  n/a 
Classes
Winter semester 2022/23
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. 


1000135ASW  n/a 
Classes
Winter semester 2022/23
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 semisimple 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 


1000135AF 
Classes
Winter semester 2022/23
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. 


1000135TOG  n/a 
Classes
Summer semester 2022/23
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. 


1000135WAS  n/a 
Classes
Winter semester 2022/23
Groups
Brief description
Main topics: General theory of stochastic processes, Wiener process; Introduction to the theory of continuostime martingales; Definition and basic properties of stochastic integral; Ito's formula; Stochastic differential equations and their connections with partial differential equations. 


1000135AGL  n/a 
Classes
Summer semester 2022/23
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. 


1000135LOM  n/a 
Classes
Winter semester 2022/23
Groups
Brief description
An introduction to classical topics of mathematical logic with elements of model theory. 


1000135MMN  n/a 
Classes
Winter semester 2022/23
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. 


1000135MBM  n/a 
Classes
Summer semester 2022/23
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. 


1000135IP1  n/a 
Classes
Summer semester 2022/23
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. 


1000135IP2  n/a 
Classes
Winter semester 2022/23
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. 


1000135MUZ  n/a 
Classes
Winter semester 2022/23
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. 


1000135TM  n/a 
Classes
Winter semester 2022/23
Groups
Brief description
This lecture is selfcontained 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. 


1000135MAG  n/a 
Classes
Winter semester 2022/23
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. 


1000135MI1  n/a 
Classes
Summer semester 2022/23
Groups
Brief description
In any type of school Computer Science lessons should both provide students with general knowledge about computer science and information theory and teach programming. The course will show how to structure a lesson and how to teach programming and theoretical computer science. Methods of programming teaching and the construction of auxiliary models in the teaching of computer science will be discussed and possible traps will be unveiled. 

