# Erasmus+

Coordinator of Erasmus+ dr hab. Krzysztof Topolski (e-mail: Krzysztof.Topolski(at)math.uni.wroc.pl, contact hours).

Students consultations Student Local Advisor - every thuersday 16:30 - 17:30 in D Pavillon in Faculty of Law, Administration and Economics, room 1.09 D (Uniwersytecka Street 22/26).

Partner Universities - required language, studies: available places

Lectures offered

• DIFFERENTIAL TOPOLOGY - Prof. dr hab. Jacek Świątkowski
• Differetial topology provides tools and techniques which enable to study
smooth manifolds and various structures on them, as well as smooth maps
between them. It is crucial in such areas of mathematics as: singularity
theory and the theory of stable mappings, dynamical systems, theory of
catastrophes, classification of smooth manifolds. index of a vector field,
etc.

The course will present basic notions and tools of differential topology:
manifolds of jets, Whitney topology, general position and transversality,
degree of a map, Morse function, and others. Several applications of
differential topology will be presented, including classification of
smooth surfaces, handle decomposition of smooth manifolds, embedding
theorems.

COURSE CONTENT:
1. Sard's theorem.
2. Manifolds of jets.
3. Whitney topology on spaces of smooth maps between manifolds and Baire's
property.
4. Transversality and Thom's theorem.
5. Embeddings of manifolds in euclidean spaces.
6. Morse functions and handle decompositions of manifolds.
7. Classification of compact smooth surfaces.
8. Degree of a map and index of a vector field, and their applications.
• INTRODUCTION TO NONCOMMUTATIVE PROBABILITY - dr hab. Janusz Wysoczański
The lecture presents basic notions of classical probability and their
analogues in noncommutative probability: noncommutative probability space
and noncommutative expectations, noncommutative random variables and their
distributions, noncommutative universal notions of independence: freeness,
monotone and Boolean independence; related central limit theorems and
Poisson type limit theorems with associated combinatorics of partitions,
creation and annihilation operators on (free, monotone, Boolean) Fock
space, generalized Gaussian operators and models of these independences,
noncommutative stochastic processes with real time (Brownian Motions and
generalized Levy processes) analytic tools (Cauchy transform, Stieltjes
inversion theorem, Voiculescu&#8217;s R-transform and cumulant functions
in free, monotone and Boolean probability, related combinatorics of
partitions, relations with continued fractions and orthogonal
polynomials); (non-universal) mixtures of these notions related to random
variables indexed by partially ordered sets: bm-independence,
bf-independence and cf-independence; related central limit theorems and
Poisson type theorems with associated combinatorics of labelled
partitions, construction of Brownian motions.

• HEALTH INSURANCE MATHEMATICS - dr Marek Arendarczyk
• Multiple state models and multiple decrement models provide a powerful tool for application in many areas of actuarial science, particularly in the actuarial assessment of sickness and disability income benefits. The aim of this course is to introduce the theory of multiple state models emphasizing their essential ideas and concepts. This powerful mathematical framework, based on Markov and semi-Markov stochastic processes, is used to describe and analyze disability and related insurance benefits. During the course we focus on disability and related benefits rather than conventional life insurance. Students interested in the latter subject area should take Life Insurance Mathematics course. During Health Insurance Mathematics course the time-continuous as well as time-discrete approach to multiple state models for life and other contingencies is presented. We start with description of basic notions and tools for time-continuous and time-discrete Markov processes. Then, based on Markov processes theory, methods of construction increment-decrement life tables, calculation of actuarial values of benefits, premiums and reserves are presented. In particular, three general models and their modifications are studied- health insurance model, disability insurance model and model for critical illness cover. Beside solving theoretical problems, at the end of the semester students solve practical case study using tools learned during the course.
• DISCRETE POTENTIAL THEORY - Prof. dr hab. Alexander Bendikov
• CLASSICAL POTENTIAL THEORY - Prof. dr hab. Alexander Bendikov