Application Areas

## Life Sciences Top

"Life sciences" describe a wide research area with enormous technological and social impact. However, there exist specific areas where mathematics has just begun to take on an active role.In medicine, the already traditional role of mathematics in medical imaging (e.g., computer tomography) has been successfully extended. Mathematical progress has proved to directly influence medical progress towards the design of patient-specific therapies - e.g., in the cancer therapy hyperthermia. As another example, computer-assisted surgery planning allows the comparison of various operation options before the actual operation on the basis of a simulation of more and more realistic models describing soft tissue, bone, or typical human gaits such as stair climbing. Further mathematization of the field is expected to open entirely new perspectives for the optimal design of joint prostheses adapted to individual anatomy.

In biotechnology, the present situation is clearly dominated by the generation of huge datasets about biomolecular, genetic, metabolic or other bio-processes. Algorithms from discrete mathematics or computer science (e.g., in multiple alignment) already play a publicly visible role in the decoding of the human and other genomes. In contrast to that, the mathematical treatment of the dynamics of bio-processes is still rather limited - even though this aspect seems to be crucial for the detailed understanding of virus diseases or the design of narrow band drugs. Therefore, beyond the well-established core areas of bioinformatics, numerical biocomputing has recently become more and more accepted as one of the keys to data-based reliable prediction, control and design of real-life bio-processes: As it turns out, a significant increase in our ability in a reliable quantitative simulation of the dynamics of large biomolecules is essential for a detailed understanding of, e.g., the enzymatic mechanisms of prion diseases (like the mad cow disease or its human counterpart, the Creutzfeldt-Jacob syndrome).

Topics:

- computer-assisted surgery
- patient-specific therapy planning
- protein data base analysis
- protein conformation dynamics
- systems biology
- pharmacokinetics

### Scientists in Charge

## Networks Top

Networks, such as telephone networks, the internet, airline, railway, and bus networks are omnipresent and play a fundamental role for communication and mobility in our society. We almost take their permanent availability, reliability, and quality at low cost for granted. However, traffic jams, ill-designed train schedules, canceled flights, break-downs of telephone and computing networks, and slow internet access are reminders that networks are not automatically good networks.In fact, designing and operating communication and traffic networks are extremely complex tasks that lead directly to mathematical problems. A good example is the design of telecommunication networks. They were implemented with simple low-cost tree topologies until 15 years ago. Then, in 1988, a telco hub broke down in Chicago. This brought O Hare airport to a stand-still and caused an estimated business loss of billions of US dollars. Disasters of this kind made it clear that more sophisticated designs were needed. Nowadays, telecommunication companies use mathematically designed networks with built-in failure safety and rerouting capacities. Similar developments are expected in road traffic. We are now facing the installation of the first generation of load measuring, signalling, pricing, and route finding devices. These will soon integrate into a network-wide telematic system based on mathematical methods of traffic prediction, simulation, and control.

Network design and operation tasks of this type are traditionally handled under the responsibility of various engineering disciplines (electrical engineering, traffic management and logistics, industrial engineering). While these disciplines can contribute to the improvement of the engineering components of such networks, todays demand on global optimization of the entire system poses problems where qualitative progress has to come from a better theoretical understanding of the structural aspects of the networks.

This is where mathematics must come into play. The appearance of the word "network" in all the systems described above is not accidental, but hints at a common feature that has deep mathematical roots: networks are fundamental structures of graph theory and combinatorial optimization. Their study has become a prosperous subject in recent years, with impressive successes in many applications. The groups in Berlin are among the driving forces in this development.

Nowadays, mathematical optimization techniques are used to locate switches and hubs in a phone system, to schedule buses and bus drivers in metropolitan transportation systems, etc. These tasks are individual steps in a hierarchical and sequential network planning process. In public transport, for example, this sequential process encompasses line planning, finding a periodic time table, assigning buses to lines, and creating individual bus driver schedules.

Topics:

- planning of optical, multilayer, and UMTS telecommunication networks
- line planning, periodic timetabling, and revenue management in public transport networks
- optimization in logistics, scheduling and material flows
- optimization under uncertainty
- symmetries in integer programming
- game theoretic methods in network design

### Scientists in Charge

## Energy and Materials Top

Production is one of the most important parts of the economy and at the very heart of the creation of value. Due to the central importance of production, big efforts have been made to improve production processes ever since the beginning of the industrial revolution. Nowadays, many production processes are highly automated. Computer programs based on numerical algorithms monitor the processes, improve efficiency and robustness, and guarantee high quality products. Consequently, mathematics is playing a steadily increasing role in this field. The possibilities of applying mathematical methods in production are wide-ranging. The Application Area cannot cover their full scale. For that reason, the projects concentrate on the development of new mathematical methods for special topics in manufacturing and production planning, two central aspects of production, in which the participating groups have longstanding expertise in mathematical modeling, simulation and optimization.In the field of manufacturing, we focus on innovative technologies having a big impact on technological progress: growth and processing of semiconductor bulk single crystals, phase transitions in modern steels and solder alloys, modeling of active and passive behavior of functional materials like shape-memory materials, growth of thin films. In the projects devoted to production planning, the main aim is the effective control of the whole production flow. Among the subjects to be studied, there is also electricity portfolio management.

Topics:

- phase transitions in steels and solder alloys
- production of semiconductor crystals
- modeling of active and passive behavior of functional materials
- online production planning
- growth of thin films

### Scientists in Charge

## Electronic and Photonic Devices Top

The technical progress of the last decades has been enormously stimulated by two technological revolutions: the invention of the transistor in 1947 (Nobel prize 1956) and the invention of the laser in 1958 (Nobel prize 1964). The impact of both inventions on modern life is an evident fact.Already in 1950, a system of partial differential equations was published that models adequately the essential charge transport processes in semiconductor devices. On the basis of this drift-diffusion model the first bipolar transistor was successfully simulated in 1964. Just in that time the first integrated circuits containing a few transistors became commercially available. Since then, the electronics industry has achieved a phenomenal growth, mainly due to the rapid advances in integration technologies, large-scale systems design and numerical simulation. The number of applications of integrated circuits in high-performance computing, telecommunications, and consumer electronics has been rising steadily, and at a very fast pace. As microelectronic research moves into the nanometer scale device regime with GHz or higher operating speeds, the physics of electron flow through devices becomes more complicated, and physical effects, which previously could be safely ignored, become significant. Consequently, models of a higher abstraction level are needed. Conversely, faster simulation is typically required, which places a constraint on the model refinement if conventional simulation techniques are applied.

Like the invention of the transistor triggered research in circuit simulation, the invention of the laser had a major impact on optical technologies. Classical optics turned into photonics. In todays telecommunication technologies, photons have already become the main carrier of information, regardless of the fact that even today most of the applied optical devices are based on conventional optical fibers and low index-contrast waveguides. Recently, a number of pioneering developments - all based on nanotechnologies - opened up the door to completely new working principles, hence to new classes of optoelectronic devices. Among them are nanostructured periodic materials (photonic crystals) and optically active nanostructures like quantum layers and quantum dots. A proper modelling of such structures has to describe simultaneously electrical charge transport, light generation, light propagation and scattering. Moreover, optical active nanostructures have to be described by quantum mechanics.

In spite of the achievements of electronic/optoelectronic device and circuit simulation obtained so far, new nanotechnologies create new challenging tasks for mathematical modeling and numerical simulation in this field.

Topics:

- shape memory alloys in airfoils
- production of semiconductor crystals
- methanole fuel cell optimization
- online production planning metamaterials

### Scientists in Charge

## Geometric Design and Visualization Top

Visualization has the task to create insight from given data. Image analysis is to extract information from data and to make it explicit in the form of a geometric model. The two areas have tight relations on both the methodical and the application level. Prominent examples are image-based rendering and visual analysis of 3D image data, e.g. in tomography or confocal microscopy.In recent years we have seen a rapid development of fundamentally new techniques for the visualization of complex physical phenomena as well as for imaging applications. At the very heart of these new technologies we encounter fundamentally new data structures and algorithms, all with a quest for a new level of abstraction. Here is where mathematics enters the scene.

Especially in visualization, it is still a challenge to give the underlying objects a solid mathematical description. Here the research field of mathematical visualization faces the challenge to develop precise abstractions which eventually enables the development of new algorithms and visualization tools. Efforts in this direction can build on broad mathematical foundations, laid among others in the fields of discrete geometry, computational geometry, discrete differential geometry, and combinatorial topology. Results of this development are not only needed in research, where scientifically correct visualization is essential, but also meant to provide a solid basis for applications, for example, in computer graphics as well as in mathematics education projects.

Even more so, the fields of visualization and image processing are key technologies for very current fields of research, among them many of the natural sciences (physics, chemistry, climate research), the life sciences (medicine, biochemistry, biotechnology, pharmacy), but also for various problems of engineering and production. Due to the multiple applications, but also due to technological reasons such as the availability of new imaging devices and display technology, imaging and visualization have been - and will be - areas of impressive growth.

Topics:

- discrete differential geometry
- geometry processing
- image processing
- virtual reality PORTAL

### Scientists in Charge

## Education, Outreach, Administration Top

The recent TIMSS studies have displayed and highlighted considerable deficits in the mathematical education in Germany, in particular on the gymnasium level. According to the study, in general, German pupils seem to be able to master calculations in a satisfactory way, but their abilities to solve application oriented problems are below average. Moreover, the mathematics taught at schools is not experienced as something interesting and attractive, so pupils are not well-motivated. This in turn leads to the fact that the mathematical knowledge acquired by the pupils is not sufficient for their orientation in the real-world. In particular, we observe corresponding problems in our mathematical education of students at the university. Such deficits were also stressed in the influential lecture Drawbridge Up by the noted German poet and essayist H. M. Enzensberger.There are a number of reasons for the unfavourable situation. Among others, we mention first that the sensitivity for mathematics in the German public is rather low, despite the fact that mathematics is more and more present in everyday life: most of the public many acknowledge that mathematics is difficult and impressive, but they do not view it as something interesting, or as a genuine part of culture. Secondly, the mathematics taught at schools often misses a certain amount of attractivity: very little of what the pupils see or learn is new, and there are typically very few references to any current mathematical developments; it does not become clear that there are new mathematical discoveries made every day, that there is recent and current progress on many different questions. A third reason is a deficit in the practice orientation of the mathematics taught at high schools: pupils do not see that mathematics is relevant and important in the real-world, that there are lots of interesting applications and developments.

To improve the situation the following measures seem promising. The very experts have to give more emphasis to the popularization of current mathematics. Moreover, the teaching of mathematics, including the corresponding mathematics curricula, at schools and universities has to be made more attractive and problem-solving-oriented. Last but not least, the teacher students education has to obtain a more practice-oriented component.

The basis to attack and solve the problems that we have described lies in greater educational activity of university mathematicians, and in a much closer cooperation between schools and universities than the present one.

Topics:

- modern mathematics at school
- school teachers at universities
- network of math-science oriented schools
- public awareness of mathematics
- media presence