Elective courses for Mathematics (course group defined by Faculty of Mathematics, Informatics, and Mechanics)
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2023Z - Winter semester 2023/24 2023L - Summer semester 2023/24 2024Z - Winter semester 2024/25 2024L - Summer semester 2024/25 (there could be semester, trimester or one-year classes) |
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| 1000-135AMD | n/a | n/a | n/a |
 
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Classes
Summer semester 2024/25
Groups
- (from 2024-10-01) Elective courses for 1st degree studies in mathematics
Brief description
(in Polish) Wykład jest wprowadzeniem do algorytmiki opartym na prezentacji wybranych problemów obliczeniowych oraz algorytmów związanych z klasycznymi strukturami matematyki dyskretnej takimi jak grafy, drzewa, sieci przepływowe oraz języki regularne. |
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| 1000-135AF* |
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n/a | n/a |
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Classes
Winter semester 2023/24
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. |
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| 1000-135EAR | n/a |
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n/a |
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Classes
Summer semester 2023/24
Groups
Brief description
(in Polish) Splot funkcji i jego zastosowania do aproksymacji. Szeregi Fouriera i badanie ich zbieżności. Przestrzeń Schwartza i transformata Fouriera. Funkcja maksymalna Hardy’ego-Littlewooda. Funkcje monotoniczne, o wahaniu ograniczonym i absolutnie ciągłe. Funkcje lipszycowskie: ich rozszerzenia i własności aproksymacyjne. Przykłady powiązań pomiędzy teorią równań cząstkowych, teorią aproksymacji, analizą harmoniczną i zespoloną oraz teorią interpolacji. |
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| 1000-135OPL | n/a |
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n/a |
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Classes
Summer semester 2023/24
Groups
Brief description
We will focus on solving linear programming problems mainly by applying various simplex methods . Special attention will be given to duality in linear programming and to geometric interpretations. |
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| 1000-135WPS | n/a |
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n/a |
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Classes
Summer semester 2023/24
Groups
Brief description
Introduction of basic concepts of the theory of stochastic processes. Definition and properties of the Poisson process and the Wiener process. Preliminary information about Markov processes and continuous time martingales. |
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| 1000-135WUD | n/a |
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n/a |
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Classes
Summer semester 2023/24
Groups
Brief description
The course is an introduction to some aspects of the dynamical systems theory based on the analysis of some model examples. This includes a description of the dynamics of transformations of the interval, circle, torus and complex plane. |
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| 1000-135WTL |
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n/a |
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n/a |
Classes
Winter semester 2023/24
Groups
Brief description
(in Polish) Podstawowym celem wykładu jest przedstawienie wstepu do teorii liczb, jako jednego z najwazniejszych działów matematyki. W dalszej jego czesci przedstawione sa przykłady zastosowania tej teorii do kryptografii oraz teorii kodowania. |
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| 1000-134AG2 | n/a |
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n/a |
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Classes
Summer semester 2023/24
Groups
Brief description
Elements of group theory, field theory, theory of noncommutative rings and modules. Group theory: free groups, solvable groups, semidirect products of groups. Fields: Galois theory and applications. Modules: finitely generated modules over principal ideal domains. Noncommutative rings: matrix algebras, division algebras, Frobenius theorem, algebras of skew polynomials and Weyl algebras. |
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| 1000-135GEA | n/a |
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n/a |
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Classes
Summer semester 2023/24
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. |
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| 1000-135MGT |
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n/a |
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n/a |
Classes
Winter semester 2023/24
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. |
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| 1000-135TA | n/a |
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n/a |
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Classes
Summer semester 2023/24
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. |
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| 1000-135APZ |
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n/a |
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n/a |
Classes
Winter semester 2023/24
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. |
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| 1000-135STB |
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n/a |
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n/a |
Classes
Winter semester 2023/24
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. |
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| 1000-135ALP |
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n/a |
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n/a |
Classes
Winter semester 2023/24
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. |
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| 1000-135ANZ | n/a |
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n/a |
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Classes
Summer semester 2023/24
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. |
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| 1000-135ROZ |
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n/a |
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n/a |
Classes
Winter semester 2023/24
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. |
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| 1000-135MOF | n/a |
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n/a |
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Classes
Summer semester 2023/24
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. |
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| 1000-135GK |
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n/a |
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n/a |
Classes
Winter semester 2023/24
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. |
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| 1000-135TST | n/a | n/a |
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n/a |
Classes
Winter semester 2024/25
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, bang-bang principle, time-optimal control, Pontriagin Maximum Principle, transversality, dynamic programming. |
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| 1000-134BAD |
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n/a |
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n/a |
Classes
Winter semester 2023/24
Groups
Brief description
Database concepts and role. Data models. Relational databases. Relational query languages. SQL. Database dasign theory, normal forms, modeling using entity-relationship models. Transaction processing. Physical models, query optimization. Databases based on non-standard data models: object-oriented, deductive (Datalog) and distributed. |
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| 1000-135SYD |
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n/a | n/a |
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Classes
Winter semester 2023/24
Groups
Brief description
An overview of classification methods and intelligent decision support systems. Methods deriving from different approaches, such as machine learning, statistics, fuzzy set theory and rough set theory, will be presented. During practical laboratory classes decision support software will be used. There will be some takehome projects for individual investigation. |
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| 1000-135GR | n/a |
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n/a |
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Classes
Summer semester 2023/24
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. |
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| 1000-134MAD | n/a |
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n/a |
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Classes
Summer semester 2023/24
Groups
Brief description
Selected topics in combinatorics and graph theory, in particular: counting principles, combinatorial proofs, recurrence relations, Burnside's lemma, Eulerian circuits and Hamiltonian cycles, trees, planarity, colorings, matchings. |
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| 1000-135UD |
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n/a |
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n/a |
Classes
Winter semester 2023/24
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). |
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| 1000-135EKN | n/a |
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n/a |
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Classes
Summer semester 2023/24
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. |
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| 1000-135IFI |
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n/a |
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n/a |
Classes
Winter semester 2023/24
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. |
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| 1000-135ASW |
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n/a |
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n/a |
Classes
Winter semester 2023/24
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 |
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| 1000-135AF |
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Classes
Winter semester 2023/24
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. |
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| 1000-135TOG | n/a |
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n/a |
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Classes
Summer semester 2023/24
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. |
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| 1000-135GM1 |
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n/a |
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n/a |
Classes
Winter semester 2023/24
Groups
Brief description
The lecture presents selected properties of Euclidean figures and transformations. |
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