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

Winter semester 2023/24

• Classes - 30 hours
• Monographic lecture - 30 hours

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.

Summer semester 2023/24

• Classes - 30 hours
• Lecture - 30 hours

Summer semester 2024/25

• Classes - 30 hours
• Lecture - 30 hours

(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.

Summer semester 2023/24

• Classes - 30 hours
• Lecture - 30 hours

No brief description found, go to course home page to get more information.

Summer semester 2023/24

• Lab - 30 hours
• Lecture - 30 hours

Summer semester 2024/25

• Lab - 30 hours
• Lecture - 30 hours

(in Polish) Przyjrzymy się specyfice wybranych zadań obliczeniowych spotykanych w zagadnieniach analizy danych oraz uczenia maszynowego oraz własnościom algorytmów używanych do ich rozwiązywania.

Summer semester 2023/24

• Classes - 30 hours
• Monographic lecture - 30 hours

Convex geometry deals mainly with convex sets in Euclidean spaces. During the lectures we shall focus on certain important inequalities in this field, including isoperimetric and concentration inequalities, Brunn-Minkowski type inequalities and Khinchine inequalities.

Winter semester 2023/24

• Classes - 30 hours
• Lecture - 30 hours

Winter semester 2024/25

• Classes - 30 hours
• Lecture - 30 hours

The lecture is an introduction to the supervised learning, in other words statistical prediction, focused on modern linear methods and largely (about 2/3) based on "The Elements of Statistical Learning" by Hastie, Tibshirani and Friedman.

Winter semester 2023/24

• Classes - 30 hours
• Monographic lecture - 30 hours

No brief description found, go to course home page to get more information.

Summer semester 2023/24

• Classes - 30 hours
• Lecture - 30 hours

Winter semester 2024/25

• Classes - 30 hours
• Lecture - 30 hours

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.

Winter semester 2024/25

• Classes - 30 hours
• Lecture - 30 hours

No brief description found, go to course home page to get more information.

Winter semester 2023/24

• Classes - 30 hours
• Lecture - 30 hours

Winter semester 2024/25

• Classes - 30 hours
• Lecture - 30 hours

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.

Summer semester 2023/24

• Lab - 30 hours
• Lecture - 30 hours

Summer semester 2024/25

• Lab - 30 hours
• Lecture - 30 hours

The aim of the course is to introduce the techniques of data analysis and visualization to the students.

Winter semester 2024/25

• Classes - 30 hours
• Lecture - 30 hours

No brief description found, go to course home page to get more information.

Summer semester 2024/25

• Classes - 30 hours
• Lecture - 30 hours

(in Polish) Wykład poświęcony będzie prezentacji kilku najważniejszych pojęć wymiaru, które odgrywają szczególnie istotną rolę, jako narzędzia, w teorii pierścieni łącznych (na ogół nieprzemiennych). A mianowicie:

• klasycznego wymiaru Krulla,

• wymiaru Gelfanda-Kirillova,

• wymiaru Goldiego,

• wymiaru Gabriela-Rentschlera,

• wymiarów natury homologicznej (projektywny, injektywny, globalny, itd.).

Winter semester 2023/24

• Classes - 30 hours
• Monographic lecture - 30 hours

"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.

Winter semester 2023/24

• Classes - 30 hours
• Lecture - 30 hours

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.