This course is offered in cooperation between the University of L'Aquila, Italy (1^st semester) and the TU Wien (2^nd semester); The 3^rd and 4^th semester can be completed at one of the two universities, depending on the chosen focus. The program is offered in English.

Mathematical modelling refers to the use of modern analytical and numerical techniques to describe or simplify real physical, industrial or socio-scientific problems, so that a "good" solution can be identified within a reasonable time - mostly through numerical simulations. This requires a sense of application as well as sound knowledge of mathematics and computer science. This master's programme is therefore interdisciplinary between mathematics, computer science and engineering.

The design of aircraft wings today is not carried out in the wind tunnel, but by numerical simulations on the computer. First, the air flow (with or without friction) is modeled by a partial differential equation from fluid mechanics. The finite element method provides the numerical-mathematical tool for their exact solution. Due to the complex geometry, real problems can only be implemented on parallel high-performance computers.

While in physics and mechanics many equations of motion have been known in their basic features for some centuries, the collective behavior of large clusters of people or flocks of animals is a new field of research. At first one tries to fathom from experiments the typical movement behavior of individuals. Averaging over a large number of participants often leads to a partial differential equation for group dynamics. Corresponding numerical simulations are e.g. essential for the planning of emergency exits in public buildings or major events.

With the aid of mathematical models, medically relevant information such as stroke volume, elasticity and pulse waveform in the aorta can be calculated from the easily measured pulse and pressure curves.

In order to provide better help in the future to patients with nerve damage, models using partial and general differential equations are being developed and analysed. The simulation results thus obtained will provide the basis of medical improvements (e.g. design of hearing prostheses).

Sounds waves spread through the sea across large distances, almost without attenuation. By comparing the simulated and experimentally measured wave field, it is possible to determine the density and speed of sound in water and on the sea bed, in order to pinpoint the location of oil reserves or shoals of fish.

Computer components are becoming ever smaller and doing more and more work. The current flow heats the tiny components to such an extent that they can get as hot as a light bulb. Numeric simulations can discover the source of the heat in order to channel it away.

For some years it has been possible to calculate the properties of materials using computational methods alone. The basis for this is the density functional theory, for which Walter Kohn from Vienna was awarded the Nobel Prize in Chemistry in 1998. Calculations in material science make it possible to develop ideal new technical materials or medicines.

Modern computer algebra systems contain techniques, embedded into the software, for solving mathematical problems at a precise symbolic level. However, natural limitations are quickly reached here, particularly in the field of applied analysis. Numerical simulation is based on structural implementation of mathematical models, the solution for which cannot be determined precisely using finite complexity. Complexity of calculation can thus be balanced against precision.

The modern information society places ever higher demands on the transfer, security and reliability of data. In information theory, the terms entropy (uncertainty), information and redundancy in information systems are analysed and questions regarding the relationship between transmission speed and reliability as well as the optimal compression of data are dealt with. Coding theory is concerned with the issue of error recognition and correction. Neither CDs nor satellite transmissions would be possible without it. Cryptography today is far removed from any espionage clichés and represents an indispensable foundation of electronic payments and all forms of e-commerce and e-government.

Special graphic models are used, for example, to model the growth of the internet, the spread of infections or social networking structures.
The mathematical analysis of the structure of such graphs, but also of other objects (e.g. data structures), is important to the performance analysis of various algorithms and for designing more efficient algorithms, amongst other things.

Econometrics is defined as the field of economic sciences which is concerned with the application of mathematical statistics and the tools of statistical inference to problems of empirical measurement with regard to relationships postulated by economic theory. Since the foundation of the Econometric Society by Ragnar Frisch in 1933, the unification of the three disciplines of statistics, economic theory and mathematics has been called econometrics.

The research group ECON is concerned with that area of exploration for which the other research groups are developing methods: economics. Teaching includes overview modules as well as some more in-depth subjects on which we are also conducting research. Our main focus subjects are macroeconomics, evolutionary economics, economic policy simulation, monetary economics, political economics and European integration.

OR is concerned with the interdisciplinary solution (support for decision-making) of planning problems in economics, technology, informatics, medicine etc. by exploiting modern developments in mathematics, statistics and informatics.

Did you know that mathematicians are much sought after in Wall Street and other financial markets? In the last 20 years, mathematics has become a key technology in the financial sector. Sophisticated mathematical models are used in the management of financial risk.

The classic model for a stock market price is based on a model from molecular physics. It describes the movement of a particle as a result of random collisions with other particles.
Share price development is influenced in the same way by the constant flow of buy and sell orders. Each one of these orders nudges the share price up or down slightly. In 1973, F Black and M Scholes used this model to derive a formula for valuing stock options. In 1997, this formula was awarded the Nobel prize for economics. Modern research is working intensively to develop this model further.

Insurance and banking live on risk. They have to assess the probability of losses, which must be actively budgeted for. Today, highly complex mathematical models are used for the management of financial risks. Mathematics is at the core of probability theory. It allows one to bring order to chaos.

For a long time, insurance companies have been using probability theory to determine premiums and to calculate the financial reserves that are required to meet their insurance obligations.
In recent years, the management of investment risk has also become increasingly important. Mathematicians who are qualified in these fields receive attractive and lucrative job offers in the insurance industry.