Elective courses for 2nd stage studies in 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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2023Z | 2023L | 2024Z | 2024L | |||||||
1000-1M23TMW | n/a |
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Classes
Summer semester 2023/24
Groups
Brief description
Geometric measure theory is the study of geometric objects using the methods of measure theory. A manifold embedded in Rn can be associated a Hausdorff measure restricted to a given manifold or to a tangent bundle of that manifold. Considering a sequence of such measures and passing to the weak limit we get more general objects, e.g., varifolds or currents. We study functionals defined on such objects and their critical points, i.e. stationary varifolds (generalization of minimal surfaces). The lecture aims to present the current knowledge in this field to the extent that allows independent research. We will start with the classic area and coareavformulas and introduce the concept of rectifiability. Next, we shall discuss the basic theory of varifolds. Finally, we will focus on the pivotal concept of ellipticity of functionals, which has not been sufficiently explored so far. |
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1000-1M23GK |
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Classes
Winter semester 2023/24
Groups
Brief description
The course is dedicated to students interested in functional analysis in a broad sense, mathematical formalism of quantum mechanics or modern, yet elementary, mathematics. Quantum graphs arise in quantum information and are natural counterparts of graphs. We will start with a quick introduction to quantum information theory, necessary to motivate the notion of a quantum graph. Afterwards we will introduce three equivalent definitions of quantum graphs, we will explain how to translate between them and will illustrate how each of them is useful in its own way. We will show how to construct plenty of examples of quantum graphs. In the last part of the course we will study properties of random quantum graphs: we will prove that a typical quantum graph does not admit any nontrivial symmetries. |
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1000-1M23ITM | n/a |
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Classes
Summer semester 2023/24
Groups
Brief description
The lecture will be devoted to set-theoretic properties of ideals of Lebesgue measure zero sets and Baire first category sets, as well as other related classes of small subsets of the real line. |
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1000-1M24TDA | n/a | n/a | n/a |
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Classes
Summer semester 2024/25
Groups
Brief description
(in Polish) In this lecture, we will introduce the necessary basic concepts from topology that are needed for the first step into topological data analysis, e.g. topological spaces, simplicial complexes, homology etc. Parallel to introducing these the- oretical concepts, we also discuss options of how to handle geometric objects with the help of computers and learn how to implement the learned techniques to analyze data. The lecture will be accompanied by exercise sessions, in which theoretical and practical (programming) problems will be solved by the students. Prior knowledge of (algebraic) topology will be helpful but not a strict re- quirement. First experiences with Python is necessary for the practical part. |
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1000-1M22IF2 | n/a |
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Classes
Summer semester 2023/24
Groups
Brief description
(in Polish) Wykład jest kontynuacją wykładu Inżynieria Finansowa. Na wykładzie będą przedstawione wybrane metody wyceny instrumentów opcyjnych na stopę procentową oraz praktyki rynkowe wyceny opcji walutowych. Ćwiczenia będą się koncentrowały na przykładach numerycznych ilustrujących omawiane na wykładzie metody. |
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1000-1M20KNW | n/a |
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Classes
Summer semester 2023/24
Groups
Brief description
The course aims to introduce into the theory of quantized and categorified invariants of knots. After explaining the classical Alexander, Conway, Jones and HOMFLY-PT polynomials, we will quantize them with the use of representations of quantum groups and the Reshetikhin- Turaev functor and categorify to the Khovanov-Lee homology. All technicalities will be explained on the spot when needed. The course is based on selected fragments of the literature listed below. The suggestions for further reading will be provided. |
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1000-1M23LNT | n/a |
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Classes
Summer semester 2023/24
Groups
Brief description
No brief description found, go to course home page to get more information.
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1000-1M24MAS | n/a | n/a |
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Classes
Winter semester 2024/25
Groups
Brief description
(in Polish) Program zajęć koncentruje się na metodach analizy nieefektywności rynków finansowych w oparciu o dane. Wykład wprowadza słuchaczy w zakres podstawowych pojęć i teorii ekonomicznych związanych z wyceną instrumentów finansowych. Omówione zostaną techniki agregacji danych różnych typów (numeryczne, tekstowe) pochodzących ze źródeł wielu rodzajów oraz ich analizy w środowisku chmurowym wykorzystywanym również do budowy narzędzi służących do automatyzacji realizacji strategii zarządzania portfelem aktywów, w tym tych bazujących na uczeniu maszynowym. Efektywność zaproponowanych rozwiązań będzie oceniana z uwzględnieniem rzeczywistych modeli bazowych np. indeksów giełdowych. |
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1000-1M22NUM | n/a |
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Classes
Summer semester 2023/24
Groups
Brief description
(in Polish) Przyjrzymy się specyfice wybranych zadań obliczeniowych spotykanych w zagadnieniach analizy danych oraz uczenia maszynowego oraz własnościom algorytmów używanych do ich rozwiązywania. |
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1000-1M18SUM |
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Classes
Winter semester 2023/24
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Brief description
The lecture is an introduction to the supervised learning, in other words statistical prediction, focused on modern linear methods for tabular data and based partly on "The Elements of Statistical Learning" by Hastie, Tibshirani and Friedman. In addition, I will discuss the use of linear methods and deep neural networks (ConvNets, Visual Transformers) for the prediction of image properties such as classification or segmentation. |
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1000-1M15TR2 | n/a |
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Classes
Summer semester 2023/24
Groups
Brief description
No brief description found, go to course home page to get more information.
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1000-1M21WZU |
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Classes
Winter semester 2023/24
Groups
Brief description
No brief description found, go to course home page to get more information.
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1000-2M23STI | n/a |
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Classes
Summer semester 2023/24
Groups
Brief description
No brief description found, go to course home page to get more information.
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1000-1M18ZRR | n/a |
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Classes
Summer semester 2023/24
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Brief description
This course presents a survey of more advanced methods of PDE theory: smoothness of solutions of elliptic equations, method of difference quotients, Fredholm theory, elements of Schauder theory and semigroup theory, variational methods. |
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1000-1M22GAD | n/a | n/a |
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Classes
Winter semester 2024/25
Groups
Brief description
(in Polish) Podstawowym celem jest zrozumienie różnych technik stosowanych do badania rozmaitości zdefiniowanych nad ciałami o dodatniej charakterystyce i zastosowanie tych metod do badania rozmaitości w charakterystyce zero. Pokazane będą też znaczące różnice występujące między rozmaitościami w charakterystyce zero i w charakterystyce dodatniej. |
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1000-1M24SAB | n/a | n/a |
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Classes
Winter semester 2024/25
Groups
Brief description
The notion of amenability is one of the fundamental notions in group theory and harmonic analysis which is connected to the existence of invariant measures and paradoxical decompositions of groups. The goal of the lecture is to present a possibly broad spectrum of applications of some methods concerning amenability and to study analogies between amenability and cohomology theory for Banach algebras, as well as stability problem in the sense of Ulam. |
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1000-1M24STK | n/a | n/a | n/a |
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Classes
Summer semester 2024/25
Groups
Brief description
Basic notions and theorems of the category theory such as categories, functors, natural transformations, adjoint functors, and universal constructions will be presented. An analogy of various constructions in algebra, geometry topology and mathematical analysis will serve as an application of the category theory. Some examples of the categorification will be presented. |
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1000-1M00WA |
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Classes
Winter semester 2023/24
Groups
Brief description
The lecture is addressed to students of mathematics as well as to students of computer science interested in theoretical or applied aspects of approximate reasoning. Issues and problems discussed in the lecture are considered by many famous mathematicians and computer scientists to be among central problems of the current century, as important as deciphering the genetic code was for the second half of the 20th century. We will discuss problems important for making progress in many projects, in particular, interdisciplinary projects, in which mathematicians and computer scientists work together with specialists from other areas such as neuroscience, bioinformatics, psychology, economy, or complex adaptive systems. Different theoretical and applied aspects of methods of concept approximation from experimental data and domain knowledge will be covered. |
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1000-1M13AB | n/a | n/a |
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Classes
Winter semester 2024/25
Groups
Brief description
(in Polish) Wykład "Algebry Banacha" ma na celu zaznajomienie uczestników z podstawową teorią algebr Banacha ze szczególnym uwzględnieniem przypadku przemiennego. |
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1000-719DAV | n/a |
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Classes
Summer semester 2023/24
Groups
Brief description
The aim of the course is to introduce the techniques of data analysis and visualization to the students. |
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1000-2M23ZWL | n/a |
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Classes
Summer semester 2023/24
Groups
Brief description
The lecture presents the state-of-the art knowledge concerning the way a dictionary of symbols one can construct statements from (i.e. quantifiers, connectives, relations and function symbols) impacts the complexity of the satisfiability and provability problems. |
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1000-1M19DPM | n/a | n/a |
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Classes
Winter semester 2024/25
Groups
Brief description
Moduli spaces parameterize other geometric or algebraic objects (e.g. the projective space parameterizes lines through zero). Deformation theory provides a local description of such spaces while moduli theory tells us how to construct them. The lecture is an introduction to these subjects. It is geared towards M.Sc. and Ph.D. students interested in algebra and algebraic geometry. We will do examples. |
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1000-1M15DM | n/a |
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Classes
Summer semester 2023/24
Groups
Brief description
(in Polish) Celem zajęć jest zapoznanie przyszłych nauczycieli matematyki z uwarunkowaniami zawodu nauczyciela. |
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1000-1M24WTP | n/a | n/a | n/a |
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Classes
Summer semester 2024/25
Groups
Brief description
The lecture will be devoted to the presentation of several most important concepts of dimension, which play a particularly important role, as a tool, in the theory of associative rings (generally non-commutative). Namely: • classic Krull dimension, • Gelfand-Kirillov dimension, • Goldie dimension, • Gabriel-Rentschler dimension, • dimensions of a homologous nature (projective, injective, global, etc.). |
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1000-1M07ET |
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Classes
Winter semester 2023/24
Groups
Brief description
Basic notions and theorems of Category Theory will be discussed. Some specific applications will be covered in the last part of the course. I intend to discuss various aspects of Grothendieck topoi. |
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1000-1M23EK |
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Classes
Winter semester 2023/24
Groups
Brief description
"Equivariant Cohomology in Algebraic Geometry" Borel equivariant cohomology theory is introduced topologically and through differential methods. The theory is applied to classical objects of algebraic geometry, such as flag varieties and Grassmannians. Properties of projective manifolds are presented. Applications are based on equivariant formality and localization theorem for torus action. Equivariant Schubert calculus is discussed. |
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1000-1M24FRC | n/a | n/a | n/a |
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Classes
Summer semester 2024/25
Groups
Brief description
No brief description found, go to course home page to get more information.
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1000-1M10AF |
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Classes
Winter semester 2023/24
Groups
Brief description
I would like to present a part of the Fourier analysis appearing currently in the applied mathematics. The main point will be the Paley-Littlewood decomposition, defining structure of functions. This point of view in a natural way introduces us Besov B^s_{p,q} and Triebel F^s_{p,q} function spaces -- being a fractional generalization of the classical Sobolev spaces. To understand the properties of this approach we recall the theory of Fourier multipliers -- the Marcinkiewicz theorem, which extends the elementary features of the L_2- on L_p-spaces. This part of the theory can be relatively easily extended on the nonlinear problems. We introduce the paraproducts to control the multiplication beyond the classical point of view. We plan to consider applications to concrete problems from PDEs, too. |
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1000-1M24GPA | n/a | n/a | n/a |
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Classes
Summer semester 2024/25
Groups
Brief description
The fundamental group of a topological space is one of the basic homotopy invariants. It allows one to translate questions about the "shape" of the space into group-theoretical considerations. As it turns out, similar invariants can be introduced for algebraic varieties defined over arbitrary fields and other algebra-geometric objects. During the course we shall learn some basic facts about fundamental groups of complex algebraic varieties and define the etale fundamental group. This invariant, introduced by Grothendieck, allows one in particular to interpret the Galois group as an example of a fundamental group, providing a foundation for modern arithmetic geometry. |
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1000-1M08MG | n/a |
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Classes
Summer semester 2023/24
Groups
Brief description
The lecture of geometric modelling is devoted to curves and surfaces in Bezier and B-spline representations (including NURBS), commonly used in computer graphics and in CAD software packages. The course will be given in Polish, if no non-polish speaking students register for it. |
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