The aim of the Statistics unit is to get a deep knowledge on constructing statistical models and making statistical analysis, and to improve the skills of using statistical computer packages.

1. Organising statistical analysis: collecting and data, their analysis and graphical description.

2. Linear and non-linear statistical models – estimation theory and statistical hypotheses testing.

3. Applications of linear and non-linear statistical models in econometrics and financial mathematics.

4. Parametric tests of significance involving two or more samples.

5. Conformity tests.

6. Non-parametric tests of significance involving two or more samples.

7. Applications of statistical computer software to estimation and statistical testing.

The main aim of the module Problem Workshops is to acquaint students with chosen branches of mathematics with applications to knowledge domains such as: economics, biology, physics, chemistry, and computer science.

Additional aims are: training analytical skills (for example, constructing mathematical models of chosen problems from applied sciences), training methodological skills (for example, use of available technology to prepare a project or analysis), training cognitive skills (for example, an analysis of data or source content given in a form of articles or manuals, also in a foreign language) and training skills of team-work (for example, work in small groups during and outside the workshop).

The aim of Computational mathematics course is to teach students how to use computational (both numerical and symbolic) methods in applications coming from various branches of mathematics.

1. Polynomial algorithms: square-free factorization, polynomial factorization over finite fields, factorization of rational polynomials, monomial orders and Groebner bases;

2. Elimination theory: elimination with Groebner bases, classical elimination with resultants;

3. Inifinite summation and Gosper's algorithm;

4. Numerical integration: Monte-Carlo algorithm.

In this module the students, divided into teams consisting of several people, implement projects associated with the given problem.

The project consists of several phases:

1. Planning for the project. The allocation of roles and responsibilities in the team.

2. Review of available literature on the given matter.

3. Analysis of the problem, seeking methods of its solution.

4. Implementation of the solution. This phase, depending on the project, should include elements such as the analysis of empirical data, calibration, simulation and testing of the solution.

5. Preparation of the final report and presentation of results. Both the final effect and the individual phases of the project are assessed. Laboratory classes serve to current reporting and discussing work progress, and give the opportunity of obtaining assistance in the project implementation.

The course establishes the fundamental concepts of the graph theory and shows several applications in various topics.

In particular, the famous problems of the graph theory will be discussed: Minimum Connector Problem, Hall's Marriage Theorem, the Assignment Problem, the Network Flow Problem, the Committee Scheduling Problem, the Four Color Problem, the Traveling Salesman Problem.

1. Bollobas B., Modern Graph Theory, Springer-Verlag, 2001

2. Diestel G. T., Graph Theory, Springer-Verlag, 1997, 2000

3. Foulds L. R., Graph Theory Applications, Springer-Verlag, 1992

4. Hartland G., Zhang P., A First Course in Graph Theory (Dover Books on Mathematics), 2012

5. Matousek J., Nesetril J., An invitation to discrete mathematics, Oxford, 2008

The aim of the Statistics unit is to get a deep knowledge on constructing statistical models and making statistical analysis, and to improve the skills of using statistical computer packages.

1. Organising statistical analysis: collecting and data, their analysis and graphical description.

2. Linear and non-linear statistical models – estimation theory and statistical hypotheses testing.

3. Applications of linear and non-linear statistical models in econometrics and financial mathematics.

4. Parametric tests of significance involving two or more samples.

5. Conformity tests.

6. Non-parametric tests of significance involving two or more samples.

7. Applications of statistical computer software to estimation and statistical testing.

The main aim of the module Problem Workshops is to acquaint students with chosen branches of mathematics with applications to knowledge domains such as: economics, biology, physics, chemistry, and computer science.

Additional aims are: training analytical skills (for example, constructing mathematical models of chosen problems from applied sciences), training methodological skills (for example, use of available technology to prepare a project or analysis), training cognitive skills (for example, an analysis of data or source content given in a form of articles or manuals, also in a foreign language) and training skills of team-work (for example, work in small groups during and outside the workshop).

The aim of Computational mathematics course is to teach students how to use computational (both numerical and symbolic) methods in applications coming from various branches of mathematics.

1. Polynomial algorithms: square-free factorization, polynomial factorization over finite fields, factorization of rational polynomials, monomial orders and Groebner bases;

2. Elimination theory: elimination with Groebner bases, classical elimination with resultants;

3. Inifinite summation and Gosper's algorithm;

4. Numerical integration: Monte-Carlo algorithm.

In this module the students, divided into teams consisting of several people, implement projects associated with the given problem.

The project consists of several phases:

1. Planning for the project. The allocation of roles and responsibilities in the team.

2. Review of available literature on the given matter.

3. Analysis of the problem, seeking methods of its solution.

4. Implementation of the solution. This phase, depending on the project, should include elements such as the analysis of empirical data, calibration, simulation and testing of the solution.

5. Preparation of the final report and presentation of results. Both the final effect and the individual phases of the project are assessed. Laboratory classes serve to current reporting and discussing work progress, and give the opportunity of obtaining assistance in the project implementation.

In our lecture we present an introduction to the mathematics of finance, and in particular the models with discrete time. We are going to discuss, among others, the following questions: mathematical finance in one period, the fundamental theorem of asset pricing, the multi-period market model, arbitrage opportunities and martingale measures, binomial trees and the CRR model, introduction to optimal stopping and American options, risk measures, indifference valuation and optimal derivative design, optimal risk transfer in principal agent games, bonds and contracts for bonds, contracts swap and swaptions, contracts cap and loor, models with infinite set of simple events.

The lecture will be based on a book by Stanley R. Pliska Introduction to Mathematical Finance: Discrete Time Models, Blackwell Publishing Ltd, Oxford 2004.