# Specialization "Applied Mathematics and Scientific Computing"

Here we present the area of specialization "Applied Mathematics and Scientific Computing", which covers the major part of the offered courses on applied mathematics. We include hints on possible topics for bachelor and master's theses for all students of mathematics. Nearly all of these topics are directly linked to an application in the real wold and usually cover several areas of mathematics. Often computer programmes are used for solving these problems and are created by the students in the course of their theses. This makes graduates of this area of specialization particularly attractive for many industrial companies. The information is sorted by study programme.

## Teacher training programme

"Applied mathematics" is one of the compulsory modules in the bachelor curriculum. It teaches how, originating from real-world application problems that have frequently only been formulated verbally, mathematical descriptions (mathematical models) can be built that serve as a starting point for the computation of (approximate) solutions and qualitative analyses. The results and insights gained are then translated back from the mathematical model into the language of the user.

Thus applied mathematics offers material that can also serve as a motivation for topics for teaching at secondary schools. This important part of school mathematics is taught in the compulsory module "Application orientation in teaching mathematics".

There are numerous topics for bachelor and master's theses in the area of applied mathematics for students of the teacher training programme.

## Bachelor programme

In the bachelor programme applied mathematics is first contained in the compulsory module "Numerical mathematics", where the basics of mathematical modelling are acquired as well. Subsequently it can mainly be found in the elective modules "Career-oriented mathematics: XXX", within which, among other things, introductions to financial mathematics, statistics, graph theory and discrete optimization, linear and smooth optimization, biomathematics and game theory, computational geometry (basics of CAD), cryptography, mathematical modelling, and numerical methods for differential equations are offered. Applied mathematics is a very extensive area. It uses the major part of the other courses as a direct basis and often combines the knowledge of several branches of mathematics - depending on the application.

- In the compulsory module "Numerical mathematics" the basic techniques for the computer-aided solution of mathematical problems from the fields of linear algebra and analysis are taught. The correct handling of rounding errors is an important point since, if they are not taken into account, they can make the results of numerical computations useless. Further covered topics are the solution of systems of linear equations, of least squares problems, the singular value decomposition, interpolation, the solution of nonlinear equations and systems of equations, unconstrained optimization, the approximate computation of definite integrals, numerical differentiation and the basic methods for the approximate solution of ordinary differential equations.
- The elective module "Career-oriented mathematics" gives a first insight into different applications how mathematicians encounter them in their future careers:
- Financial mathematics: In this module a basic understanding for the application of mathematical and particularly stochastic methods to economic issues is provided. Topics of financial mathematics like the binomial options pricing model, European and American options, random walk, interest rate models, foreign currency models and the Black-Scholes formula are taught.
- Stochastics: As a sequel to probability theory important applications like random walks, extreme value theory, large deviations, queueing theory and random graphs are dealt with.
- Applied statistics: In this module the possibilities and limitations of statistical methods are shown and the practical application of statistical standard software is conveyed.
- Graph theory and discrete optimization: In this module methods of discrete mathematics in different application areas are taught. In the course of this graph-theoretic concepts like matchings, connections, flows, colouring, planarity, random graphs etc. are treated and optimization in graphs and other discrete structures is covered.
- Optimization: This module offers an introduction to and a basic understanding of the role of smooth opimization problems in various application areas of mathematics. Methods for solving such problems are taught. The students get to know the classification of optimization problems, modelling languages in optimization (e.g. AMPL, CVX, GAMS, NEOS etc.), optimization software and their usage. Moreover, the simplex method for linear programming, heuristic methods for the solution of MINLPs and black box optimization are introduced.
- Biomathematics and game theory: This elective module focuses on the treatment of selected concepts and models of biomathematics (e.g. evolution theory, genetics, ecology, epidemiology, systems biology) and/or of game theory (e.g. prisoner's dilemma, zero-sum games and minimax, Nash equilibrium, evolutionary game theory).
- Computational geometry: This module deals with selected problems and approaches from the field of computational geometry as they can be encountered in modern CAD software.
- Cryptography: In this module the students develop a basic understanding for the role of algebraic methods in application problems from the fields of encryption and coding theory. Computational complexity theory, various primality tests and integer factorization algorithms, Diffie-Hellman, RSA and ElGamal method are presented. Cryptanalysis is another important topic.
- Mathematical modelling: In the course of this module the students get to know mathematics in its role as a modelling language for selected applications from natural science, economics, or social sciences.This is done on the basis of specific examples from these areas, technology or industry. The various impacts on the process of modelling (understanding the problem, knowledge of the facts and data, mathematical formulation of the problem, mathematical and numerical analysis, interpretation of the results) are examined. The ability to transform specific problems from science and technology into mathematical models is promoted and deepened.
- Numerical methods for differential equations: Here students acquire important methods for the numerical solution of differential equations and possible applications are shown.

Due to the wide range of the topic there is an abundance of possible topics for bachelor and master's theses from various applications. In addition, they are usually a good basis for later job applications in industrial companies and financial institutions.

## Master programme

In the master programme "Applied mathematics and scientific computing" is one of seven areas of specialization. You have to choose one of these seven areas and the chosen main area of specialization results from the completion of the compulsory module group "Basic courses in the area of specialization ...". The further modules of the master programme can be divided into courses from the chosen area of specialization and courses from other areas of specialization.

The basic courses in the area of specialization "Applied mathematics and scientific computing" consist of four compulsory modules:

- In the module "Numerical mathematics" the methods from the bachelor programme are expanded and supplemented. Here the focus is on the solution of large systems of linear equations, eigenvalue problems, multiple integrals, Monte Carlo methods, and the foundations of the numerical solution of partial differential equatons. Special attention is paid to the practical implementation of algorithms on computer systems.
- The compulsory module "Applied analysis" provides an introduction to one or two important branches of analysis (like differential equations, Fourier analysis, asymptotic analysis etc.) with special reference to their application to problems from the natural sciences.
- In the module "Optimization and variational calculus" the theoretical and practical foundations of the solution of finite- and infinite-dimensional optimization problems are developed. One aspect is the application of the methods to the solution of problems from economy and natural sciences.
- In the module "Seminars: Applied mathematics and scientific computing" you have to complete one seminar and one project seminar from the branches applied mathematics, image and signal processing, mathematical modelling, numerical mathematics or optimization. Usually the seminar is theoretically oriented, but the project seminar includes the development of computer programmes.

The offer of advanced courses for the master programme is closely linked to the research interests of the faculty members working in this area. It comprises courses on the branches ordinary and partial differential equations, dynamical systems, mathematical modelling, harmonic analysis, optimization, variational calculus, image and signal processing, numerical mathematics, scientific computing etc.

## Doctoral programme

As usual at the faculty of mathematics, there is no real difference between advanced courses for the master programme and courses for the doctoral programme in the specialization "Applied mathematics and scientific computing". An abundance of advanced lecture courses and seminars from this area is offered. The recognition of courses for the doctoral programme will be specified individually in an agreement ("Dissertationsvereinbarung"). In particular it is irrelevant for the recognition whether a course is announced with a course number for mathematics (25XXXX) or for the doctoral programme (44XXXX). You can find general information on the doctoral programme on the web pages of the SSC mathematics and the Center of Doctoral Studies of the University of Vienna.

The research interests of the individual faculty members play a much larger role in the choice of a topic and supervisor for a doctoral thesis than for a master's thesis. The topics are usually related to the (more or less) immediate research area of the supervisor. Therefore it does not make sense to give global information on these questions. It is worth mentioning that many research groups are devoted to applied mathematics and scientific computing at our faculty and various research grants for doctoral students are offered.

It is extremely important that you contact a potential supervisor before starting the doctoral programme and talk about a possible supervision. It does not make sense to enroll for the doctoral programme first and then look for a supervisor.