## DESCARTES |

## Nowe wykłady w roku 2021/22 |

####
Apes together strong: the nuances and special traits of galaxies in groups (FALL) (*)

Dr Błażej Nikiel-Wroczyński

Galaxies rarely live on their own. In most of the cases, they spend their lifetime in a group: a low-quantity, low-mass system of not more of a few tens of them. Mutual proximity of several galaxies leads to a plenitude of phenomena that are not seen in field ones, and in those found in clusters. As a result, no matter how weird does it sound, the most typical environment is also the most unique one. Starburst activity, gas outflows, tangled lines of the magnetic field, galaxies cannibalizing each other – this is what is left after the interaction. But how exactly does a galaxy behave during the collision? What are the factors that result in one object being more “interesting” than the other? And how can this project be considered a creative, not a destructive one? These are the problems we are going to explore. Over the course of the lecture, we will dissect a few low-quantity galaxy systems to see which physical processes are hidden beneath them. The theoretical course will be enhanced by a hands-on experience carried out using the modern data analysis tools.

####
Machine learning on graphs (SPRING) (*).

Dr hab. Prof. UJ, Brabara Strug

The lecture will cover the domain of graph machine learning starting with a brief introduction to graph theory and basic concepts understanding.The lectures will then cover topics like machine learning models for graph representation, graph neural networks, their purpose, supervised and unsupervised learning models, machine learning pipeline, algorithms and methods for using the full potential of graph data. Some real-world scenarios like for example extracting data from social networks, text analytics, natural language processing (NLP) using graphs or others will be covere. Performing graph analytics to store, query, and process network information and selected latest trends on graphs.will also be introduced.

####
Quasicrystals and geometry (FALL or SPRING) (*).

Dr hab. Prof. UJ, Dominik Kwietniak

In 1984 Shechtman discovered quasicrystals - that is, `forbidden' symmetries in alloy of aluminium and manganese. Its diffraction images showed icosahedral symmetry, long believed to be impossible for matter in the crystalline state. The discovery won the Nobel Prize for Shechtman and posed fascinating and challenging problems in many fields of mathematics, as well as in the solid state sciences. By demonstrating that `order' need not be synonymous with periodicity, it raised the question of what we mean by `order', and how orderliness in a geometric structure is reflected in measures of order such as diffraction spectra. Increasingly, mathematicians and physicists are becoming intrigued by the quasicrystal phenomenon, and the result has been an exponential growth in the literature on the geometry of diffraction patterns, the behavior of the Fibonacci and other nonperiodic sequences, and the fascinating properties of the Penrose tilings and their many relatives. The story of the latter began in the early 1960s with a philosophical problem studied by philosopher Hao Wang... The course is devoted to the theory of quasicrystals. Well, mostly to the mathematical theory of quasicrystals, which is, above all, mathematics, but one hopes that it has something to say about real crystals too.. Understanding quasicrystals requires an unusual variety of specialties, including ergodic theory, functional analysis, group theory and ring theory from mathematics, and statistical mechanics and wave diffraction from physics. But none of the above is required for the course! We will introduce all the necessary backgroud ab initio and only in the extent needed for the course. The course is highly interdisciplinary, and although we will emphasize mathematics, we will not avoid physics nor cristallography. The course should be interesting to mathematicians at all levels with an interest in quasicrystals, dynamical systems and/or geometry, and will also be of interest to graduate students in physics, crystallography and materials science.

####
Time Series Analysis (SPRING) (*).

Dr hab. Prof. UJ, Paweł Góra

Prediction is very difficult, especially about the future (attributed to Niels Bohr) Time Series Analysis attempts to understand the past and predict the future. It belongs to a broad range of Data Science, and its objective is: given a time series, or an ordered, often temporal, string of data points, predict its future values. Time series often arise when monitoring natural or industrial processes, taking consecutive measurements of a quantity or tracking corporate business metrics. Time Series Analysis accounts for the fact that data points taken over time may have an internal structure, such as autocorrelation, trend, or seasonal variations that should be accounted for, but at the same time data points are contaminated by random noise. Methods developed within Time Series Analysis are frequently used in other areas, like signal or image processing. The course will cover the following subjects: Fast Fourier Transform - the power spectrum - smoothing and denoising - digital linear filters - ""classic"" linear models (AR, MA, ARMA, ARIMA) - fractional models (ARFIMA), long memory processes - multivariate time series - wavelets - phase space reconstruction methods and nonlinear prediction.

Prosimy o pre-jestrację celem zapisania się na wykład, rejestracja w USOS bedzie możliwa wkrótce.

(*) Wykład odbedzie się jeżeli zarejestruje się przynajmniej trójka uczestników projektu.

## Wykłady Kartezjusz/Descartes 2020/2021 |

#### Introduction to theory of chaos: the most ubiquitous phenomenon in nature.

Dr Mariusz Tarnopolski (FAIS)

The aim of the lecture is to familiarize with the notion of deterministic chaos. Fundamental mechanisms leading to its onset, like the period doubling cascade or intermittence, will be discussed. Introduced will be basic concepts and computational techniques, applicable to theoretical description of a given phenomenon (dynamical systems) as well as empirical data (time series). Several examples from various fields, like physics, astronomy, chemistry, meteorology, ecology, electronics, engineering etc. will be discussed. Examples of practical applications will be given. Nontrivial connections between chaos and some general statistical properties will be mentioned. The tools of theory of chaos have found their place in the analysis of systems that do not exhibit chaotic behavior per se, nevertheless allow insights into their nature.

strona USOS#### Introduction to neural networks.

Prof. Jacek Tabor (WMII)

In the lecture, we plan to present the introduction to modern neural networks. In particular, the following topics will be discussed: 1. Motivation of neural networks; 2. Basic tasks of machine learning - regression and classification; 3. Introduction to fully connected neural networks; 4. Processing of images and convolutional networks; 5. Generative models: manifold hypothesis, density estimation and generative adversarial models; 6. Processing of sequential data: recurrent networks. The lecture should be accessible to students which have finished the linear algebra, statistics and mathematical analysis courses.

strona USOS## Wykład Kartezjusz/Descartes 2019/20 (SEMESTR LETNI) |

#### Introduction to representations of finite and compact groups.

dr Jakub Byszewski (WMII)

The aim of this lecture series is to introduce the basic concepts and examples of representations of finite and compact topological groups. We will study such topics as: irreducibility, character theory, induction, unitary representations, Peter-Weyl theorem, representations of compact Lie groups of small dimension, and applications to physics (molecular vibration and/or the hydrogen atom). We intend for the main part of the lecture to be fully rigorous, and supplemented by an informal overview of some more advanced topics at the end of the course.

## Wykład Kartezjusz/Descartes 2019/20 (SEMESTR LETNI) |

#### Scientific High-Performance Computing

dr Piotr Korcyl (WFAIS)

This lecture will introduce the student to the fundamentals of parallel scientific computing. We will overview different types of machines from the point of view of large-scale floating-point-heavy workloads. This will include a study of CPU, GPU, and FPGA architectures, interconnects, and forms of parallel memory. Alongside, we will discuss practical aspects of using these architectures in terms of different programming models (MPI, OpenMP, OpenCL/CUDA). We will exemplify these methods using the implementation of Monte Carlo simulations of the Ising model. Important tools for scientific computing such as debuggers, Makefiles, version control systems will be emphasized.

### Wykład Kartezjusz/Descartes 2019/20 (SEMESTR ZIMOWY)

#### Random matrix theory, free random variables and applications
(strona w USOS)

Prof. Maciej A. Nowak

Nowadays, it is hard to find a branch of science where random matrix theory (hereafter RMT) does not have any applications. These lectures represent an attempt to explain this omnipresence of RMT techniques and the wide scope of recent, interdisciplinary applications. Our guiding rule is to look at RMT as an alternative to the classical probability calculus, where instead of single random variable we have to deal with a huge set of random numbers arranged in a matrix-like structure. We will show, that such construction is not only possible, but it shares amazing similarities to classical probability calculus. Moreover, this correspondence can be formalized at the mathematical level in the case where the size of the random matrix tends to infinity. By no means this is a severe restriction, since in contemporary applications the sizes of random matrices can easily reach the order of thousands (financial engineering), millions (wireless networks) or even billions (genetics). So, almost paradoxically, the bigger the size of the random matrix, the more definite prediction we can make on its properties. The cornerstone of this new calculus of large matrices is the study of the spectra (eigenvalues) of the matrices. Alike in classical probability calculus, powerful central limit theorems exist in random matrix theory calculus. This is the reason, why so many similar, macroscopic spectral properties are shared by diverse and unrelated to each other complex random systems, in analogy to the omnipresence of Gaussian distribution in single-valued random structures. We call this phenomenon macroscopic universality of RMT, to make a distinction from so-called microscopic universality of RMT, second powerful phenomenon responsible for wide scope of applications of RMT. Microscopic universality of RMT is the consequence of interactions between the eigenvalues. Since this interaction is long-ranged, certain critical spectral phenomena can emerge locally at the vicinity of the some points of the spectra of the matrices. The spectral behavior in the vicinity of these points (fluctuations) depends usually only on the symmetries of the system, therefore can be categorized into universality classes, shared by very different complex systems respecting the underlying symmetry of the matrix model. Technically, this block of lectures is composed out of three parts. First part covers the principles of random matrix theory, including so-called microscopic universality. Second part is devoted to the link between large random matrices and Voiculescu theory of free random variables, therefore corresponds to so-called macroscopic regime. Third block emphasizes recent applications in interdisciplinary areas of science.

### Wykład Kartezjusz/Descartes 2019/20 (SEMESTR ZIMOWY)

#### Fractals everywhere
(strona w USOS)

dr hab. Dominik Kwietniak (WMI)

The course covers the fundamental approach to fractal geometry through iterated function systems. After a brief introduction to the subject, we will deal with the concepts and principles of spaces, contraction mappings, fractal construction, and the chaotic dynamics on fractals. We will also discuss the fractal dimension and interpolation, the Julia sets, parameter spaces, and the Mandelbrot sets. In the remaining time, we examine the measures on fractals and the practical application of recurrent iterated function systems. This course will prove useful to both undergraduate and graduate students from many disciplines, including mathematics, biology, chemistry, physics, mechanical, electrical, and aerospace engineering, computer science, and geophysical science.

### Wykład Kartezjusz 2018/19

####
Bayesian Data Analysis

prof. Piotr Białas (WFAIS) 15h

Semestr zimowy

Bayesian methods are gaining in popularity as a tool for analyzing scientific (and other) data since nineties. This is however not reflected in the most science programs curricula. My lecture is intended to be a short practical introduction to this topic. Bayesian approach requires us to formulate a full probability model for all observable and unobservable quantities including the model parameters. Those are called priors as they represent the knowledge or beliefs about the phenomena before collecting any data. While some call this part subjective and even arbitrary, Bayesians argue that at least this approach makes our assumptions explicit. In practice many methods for assigning the prior probabilities have been developed. The model is then conditioned on the observed data leading to a posterior distribution of model parameters. This distribution encodes our knowledge gained from collected data. Based on this distribution we can validate our model, make predictions etc.

### Wykład Kartezjusz 2018/19

####
Formy i liczby - związki matematyki i sztuki na przestrzeni dziejów

dr hab. Robert Wolak, prof. UJ (WMI) 15h

Semestr letni

W trakcie wykładu przestawię w ujęciu historycznym najważniejsze przykłady wzajemnego oddziaływania matematyki i sztuki. Omówię następujące zagadnienia: pojęcie kanonu w sztuce Grecji antycznej opartego na bazie rozwijanej w tym czasie teorii liczb, abstrakcyjne dekoracje budowli islamskich, ich strukturę i metody konstrukcji, a także wpływ na rozwój teorii grup krystalograficznych i parkietaży, pojęcie perspektywy linearnej i dziwacznej (anamorfozy), ich związki z rozwojem geometrii rzutowej. W wyniku rozwoju i doskonalenia przedstawień perspektywicznych powstała geometria wykreślna co ułatwiło w znaczącym stopniu rewolucję przemysłową w Europie; wpływ nowej matematyki przełomu XIX i XX w. na kierunki w sztuce XX w. (kubizm, sztuka abstrakcyjna, etc.), a także sztuka będąc inspiracją dla matematyków (np. grafiki Escher'a).