Courses in registration (in Polish) Rejestracja na przedmioty z semestru letniego 2023/24 1000-2023L

(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.

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.

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.

The course it is highly recommended for students who plan to attend Master Seminar on Distributed Systems, which is partly devoted to operating systems and in particular distributed operating systems. The course will have a form of lectures and labs.

We plan to view in detail the structure of a specific operating system. The chosen case study is Linux which is modern operating system, popular in the Faculty of Mathematics, Computer Science and Mechanics, University of Warsaw, often used as server platform, but also on desktops, mobiles, as embedded system. Source code of Linux is freely available which gives a unique opportunity to analyze in detail used algorithms, data structures, and also to run experiments and do research in the area of operating systems.

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.

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.

The lecture deals with issues at the intersection of computer science, artificial intelligence and economics. We will discuss key issues from game theory (cooperative and non-cooperative), social choice theory, mechanism design and social network analysis. The lecture will focus on algorithms and solutions that are practically relevant.

Algorithm design and analysis. Survey of fundamental algorithms and data structures. Improving practical programming and algorithm development skills. Applying ready to use libraries of algorithms and data structures.

The course is focused on using intelligent methods for solving problems that are difficult or impractical to solve with other methods. Accordingly, we discuss, among the others, various approaches based on heuristics, approximations, randomized, as well as deductive and inductive schemes of reasoning, often designed by analogy to the human way of problem solving. The main topics include also intelligent search through large spaces of states and solutions, intelligent game strategies, reasoning in logic and logical foundations of planning, foundations of machine learning in relation to artificial intelligence, foundations of modeling of uncertainty, as well as various specialized applications.

We will present techniques and tools for processing Big data sets on clusters of commodity computers. The main covered technologies are Hadoop and Spark. We will start with introducing architecture of those systems and programming models they assume like MapReduce and Resilient Distributed Dataset. Then we will cover most important algorithmic techniques and methods for analysing and comparing algorithms. Finally, we will discuss typical problems like skew and typical bottlenecks like limited reducer memory as well as methods to deal with those problems. This course will combine theory and practice.

Educational content: tissues, organs, organ systems in plants and animals, populations; growth and differentiation of plants and animals; elements of immunology; plant and animal reproduction; plant responses to environmental changes; animal behavior; systemic signaling in plants and animals; regulation of metabolism; population dynamics.

Combinatorics of words is a study of properties of words understood as in formal languages.

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.

Working, desing and configuration principles of computer networks. A survey of the most important network protocols. Network security basics.

The course provides an introduction to modern cryptography, focusing on the CS foundations of the field. We present the most essential cryptographic techniques used today to secure digital communications. Particular emphasis is placed on formal security analysis and understanding its assumptions. The last few lectures are devoted to an introduction to more advanced aspects of cryptography. These topics will be deepened in the subject "Cryptography 2", to which the subject "Cryptography 1" is an introduction.