# Year 2 in Katowice - Mathematics in finance and economics

• ## Computational mathematics (3 credits)

• ECTS credits 3
• Semester 1
• University University of Silesia in Katowice
• Topics

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.

The course covers the following subjects:

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.

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• ## Mathematical methods in physics (6 credits)

• ECTS credits 6
• Semester 1
• University University of Silesia in Katowice
• Prerequisites

Basic linear algebra is enough. A bit of number theory can be useful but not necessary.

• Objectives

My aim is to present mathematical methods for quantum information processing. As in most applications it is enough to work with qubits and systems of qubits, mathematical methods originate from linear algebra, which is usually one of first curses taught. It makes quantum information accessible for very 'fresh' students. I would like to convince students that quantum information processing is useful, interesting, counter-intuitive, sometimes seemingly as mysterious as the Schroedinger cat.

• Topics

Mathematical formalism of quantum mechanics.

Postulates of quantum mechanics.

Quantum information: quantum gates, no-go theorems, measurement.

Quantum entanglement: mathematical basis.

Selected applications: teleportation, dense coding.

Quantum cloning and applications.

Basic protocols for quantum cryptography: BB84, B92.

Quantum nonlocality: Bell and Leggett-Garg inequalities, contextuality.

Dynamics of quantum systems, open quantum systems.

Quantum error correction.

• Books

Quantum Computation and Quantum Information by Michael A. Nielsen & Isaac L. Chuang

Lecture notes by John Preskill http://www.theory.caltech.edu/people/preskill/ph229/

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• ## Decision Making Techniques and Tools (6 credits)

• ECTS credits 6
• Semester 1
• University University of Silesia in Katowice

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• ## Statistics (3 credits)

• ECTS credits 3
• Semester 1
• University University of Silesia in Katowice
• Topics

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.

The contents of this unit are the following:

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

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• ## Wavelet transforms (6 credits)

• ECTS credits 6
• Semester 1
• University University of Silesia in Katowice
• Topics

The main goal of the lecture is to present basic properties of wavelet transforms and some methods of construction of wavelet bases. We will pay special attention to these wavelet transforms which have used to the analysis and the synthesis of sound signals. We also will pay special attention to structures of bases with special properties which have used to the data compression in digital transmissions.

• Books

[1] C.K. Chui, An Introduction to Wavelets, Academic Press, Boston, 1992.

[2] I. Daubechies, The wavelet transform, Time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), 961-1005.

[3] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philidelphia, 1992.

[4] C. Heil, D. Walnut, Continuous and discrete wavelet transforms, SIAM Review 31 (1989), 628-666.

[5] G. Kaiser, A Friendly Guide to Wavelets, Birkhauser, Boston, 1994.

[6] D. Kozlow, Wavelets. A tutorial and a bibliography, Rendiconti dell’Instituto di Matematica dell’Universita di Trieste, 26, supplemento (1994).

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• ## Workshop on Problem solving (2 credits)

• ECTS credits 2
• Semester 1
• University University of Silesia in Katowice
• Topics 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).

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• ## Polish language and culture for foreigners (level A1) (3 credits)

• ECTS credits 3
• Semester 1
• University University of Silesia in Katowice
• Topics The aim of the module is to develope all language skills (listening, reading, speaking and writing) and to prepare students for quite easy communication in Polish, necessary while studying in Poland. Students acquire not only linguistic and communicative competence, but also sociocultural: they get to know selected aspects of Polish culture, basic habits and holidays celebrated in Poland, taking into account the pragmatic and sociolinguistic efficiency. Programme includes basic communication situations: greetings and farewells, shopping, ordering food, traveling, etc.

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• ## Applications of the theory of functional equations (6 credits)

• ECTS credits 6
• Semester 2
• University University of Silesia in Katowice
• Topics

Applications in Geometry:

1. Joint characterization of Euclidean, hyperbolic and elliptic geometries.

2. Characterizations of the cross ratio.

3. A description of certain subsemigroups of some Lie groups.

Applications in Functional Analysis:

1. Analytic form of linear-multiplicative functionals in the Banach algebra of integrable functions on the real line.

2. A characterization of strictly convex spaces.

3. Some new characterizations of inner product spaces.

4. Birkhoff-James orthogonality.

5. Addition theorems in Banach algebras; operator semigroups.

• Books

1. J. Aczel & J. Dhombres, Functional equations in several variables, Cambridge University Press, Cambridge, 1989. 2. J. Aczel & S. Gołąb, Funktionalgleichungen der Theorie der Geometrischen Objekte, PWN, Warszawa, 1960. 3. J. Dhombres, Some aspects of functional equations, Chulalongkorn Univ., Bangkok, 1979. 4. D. Ilse, I. Lehman and W. Schulz, Gruppoide und Funktionalgleichungen, VEB Deutscher Verlag der Wissenschaften, Berlin, 1984. 5. M. Kuczma, An introduction to the theory of functional equations and inequalities, Polish Scientific Publishers & Silesian University, Warszawa-Kraków-Katowice, 1985.

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• ## Mathematics of finance, discrete models (6 credits)

• ECTS credits 6
• Semester 2
• University University of Silesia in Katowice
• Topics 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 floor, models with infinite set of simple events.
• Books The lecture will be based on a book by Stanley R. Pliska Introduction to Mathematical Finance: Disrete Time Models Blackwell Publishing Ltd, Oxford 2004.

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• ## Collective project (4 credits)

• ECTS credits 4
• Semester 2
• University University of Silesia in Katowice
• Objectives

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

• Topics

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.

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• ## Master’s Thesis (US) (18 credits)

• ECTS credits 18
• Semester 2
• University University of Silesia in Katowice
• Objectives
The topic of the thesis can be proposed to the student by the local InterMaths coordinator or by the student him/herself. In any case, the InterMaths executive committee is the responsible to approve the thesis project before its formal start. The taste and expectations of the students are respected whenever possible. The local InterMaths coordinator in the hosting institution is the responsible to provide an academic advisor to the student, although proposals from the students will always be heard in this respect.

In some cases, after the agreement with the local InterMaths coordinator, the thesis topic can be related to a problem proposed by a private company. In this case, a tutor will be designated by the company as responsible person of the work of the student, especially if he/she is eventually working in the facilities of the company; however, the academic advisor is, in any case, the responsible to ensure the progress, adequacy and scientific quality of the thesis. The necessary agreements between the university and the company will be signed in due time, according to the local rules, in order that academic credits could be legally obtained during an internship, and the students be covered by the insurance against accidents outside the university.

NOTE: Although the thesis is scheduled for the 4th semester, some preliminary work may be anticipated due to the local rules - such as preliminary local courses in the 3rd semester, ensuring that the student can follow the main courses of the 3rd semester without problems. In this point, the personalised attention to the students has to be intensified, and decisions taken case by case.

In addition to previously mentioned, inludes the Master's Thesis at University of Silesia in Katowice also following two local courses: Seminar 1 (1st semester, 4 credits) and Seminar 2 (2nd semester, 14 credits).

Seminar 1 (1st semester, 4 credits): The module is aimed for skills, both spoken and written, precise mathematical language, to formulate and justify mathematical content of the topic related to the Master’s theses. Due to the nature of the module is expected that the curriculum will be closely related to the topics of the Master’s theses.

Seminar 2 (2nd semester, 14 credits): The module is aimed for skills, both spoken and written, precise mathematical language, including understanding the role of proof in mathematics. Due to the nature of the module is expected that the curriculum will be closely related to the module content Seminar 1.

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#### Programme Structure

»Year1 L'Aquila: IntMat | SciCom
»Year2 L'Aquila: IntMat | SciCom
»Year2 BrnoMath. Engineering
»Year2 Katowice: MatMod | MatFin
»Year2 LvivApplied Maths

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