History of the department of mathematics at TUM

Subsequently, a rough specification of the scientific profile of TUM Mathematics follows, including selected research priorities related to the M-units. The paragraph in the text dedicated to an M-unit is preceded by the former logo of the M-unit concerned, where available. It is to be noted, that the structure of the faculty has been revised and the division of professors together with their research groups has been redesigned according to mathematical research fields in mid-2017. The development in recent years, especially with regard to the narrower research areas currently represented at the CIT-Department of Mathematics founded in October 2022, can be traced on the CIT website.

Optimization (M1)

Mathematical Optimization is concerned with finding the best possible solutions to a wide variety of problems. Often, the goal is to minimize energy or costs in given scenarios or to maximize the return on an investment or to use existing resources efficiently. This requires the quantitative analysis of the underlying technical and economic processes. The methodological core of corresponding modeling approaches and simulation algorithms is often mathematical. One research focus at TUM is the development of efficient methods for solving large-scale nonlinear optimization problems that arise, for example, in connection with the shape optimization of solids or with the control of fluids.

Numerical Mathematics (M2)

A central component of the profile of TUM Mathematics is Numerical Mathematics. It deals with the development and analysis of algorithms for solving mathematical problems, with the implementation of those algorithms on digital computing machines (computers) as well as with applications to explore scientific and technical questions from many different fields in mathematical terms. Important focal points at TUM are the treatment of control problems from Aerospace and Automotive Engineering or Robotics. Mathematical models are often given by differential equations, which are discretized in a suitable form for the purpose of finding approximate solutions by implementation of corresponding approximation algorithms on computers.

Scientific Computing (M3)

Scientific Computing is a modern offshoot of Numerical Mathematics. Efficient and accurate algorithms are designed, analyzed and implemented here on modern high-performance computers for a wide range of problems. Often the task is to reproduce, predict and control highly complex processes in nature or technology with the help of computer simulations and to estimate process parameters. However, important practical problems are usually so complex that relevant progress can only be achieved in interaction with various subfields of Mathematics and other sciences as well as with the application areas. Corresponding mathematical models are often given by partial differential equations or also by so-called algebro-differential equations. For instance, the latter allow to consider a-priori given relations among the state variables in system and process modeling.

Mathematical Statistics (M4)

Mathematical Statistics is, by its very nature, practical. It is an applied offshoot of Probability Theory (mathematics of uncertainty) and together with it plays a role in almost all areas of life. Statistics includes quantitative methods to extract information from data (data mining). In this process, data is considered to be random. An important subfield represented at TUM is quantitative risk management, in which mathematical methods for risk analysis and risk measurement are developed. A significant special case is risk modeling for time series data and high-dimensional network data. A concrete application is risk quantification in the insurance and financial sector or in aviation or also concerning climate and environmental protection.

Mathematical Physics (M5)

Mathematical Physics deals with mathematical problems that have their motivation or application in (Theoretical) Physics. Of particular importance are, on the one hand, the mathematically rigorous formulation of physical theories and the analysis of underlying mathematical structures, and, on the other hand, the application of mathematical concepts and solution methods to physical problems. This includes Statistical Physics, through which systems of many (stochastically) interacting particles, stochastic growth processes and interfacial dynamics are described. Some relevant subfields of Mathematics are Applied Probability Theory as well as the theories of stochastic processes and random matrices. Quantum Physics describes quantum systems, i.e. interacting particle systems on atomic scales. Their mathematical formulation uses, among other things, spectral theory of unbounded operators on Hilbert spaces, in particular spectral theory for Schrödinger operators.

Mathematical Modeling (M6)

One focus of Mathematical Modeling is the analysis and control of complex processes and intelligent systems in nature and technology based on concepts from the relevant natural sciences and using mathematical methods. Examples are material-oriented processes such as the (thermo-visco-) elastic or plastic deformation of materials, phase transitions, heat and wave propagation in various media, or the spreading and interaction of populations of certain species in living nature. The model equations are often differential equations. Besides their derivation, relevant questions concern the existence and properties of their solutions. Both Applied Analysis, especially calculus of variations, and Numerical Mathematics are relevant subfields of Mathematics here. A special challenge is posed by systems and processes that have a kind of memory, i.e., their current behavior is influenced by their past behavior. In this case, hysteresis operators play an important role for mathematical modeling.

Analysis (M7) 

While numerical methods deliver mathematical results in the form of numbers, concepts and methods of Analysis can be used, for example, to derive formulas or to prove the existence and special properties of solution functions of a model equation without having to solve the latter explicitly. Application-relevant subfields of Analysis are function theory, differential and integral calculus, measure theory, calculus of variations, Fourier analysis, functional analysis, approximation theory and operator theory. Analysis forms a foundation of many mathematical disciplines and constantly interacts with concrete questions from various application areas. Research interests at TUM include the analysis of the electron structure of atoms and molecules, calculus of variations, molecular dynamics, multiscale methods, and nonlinear waves.

Dynamical Systems (M8)

In the middle of the 20th century, the subject of Dynamical Systems started to flourish as a subfield of Mathematics again. One topic that received much public attention in the 1970s and 1980s is nonlinear dynamics, casually also referred to as Chaos Theory. The focus is on the mathematical analysis and prediction of the evolution over time of complex systems and processes (mathematics of time). Typical examples are the motion of a rigid body or of components of a system of rigid bodies, the deformation of special materials or the flow of certain fluids. Concepts and methods of the mathematical theory of dynamical systems allow in particular to investigate phenomena of a qualitative nature such as questions about the stability of evolution patterns over long and even arbitrarily long time intervals or potential changes (bifurcations) of an evolution pattern in case of its instability, inclusively the transition to chaotic behavior.

Applied Geometry and Discrete Mathematics (M9)

Discrete structures often play a prominent role in practical applications. Applied Geometry and Discrete Mathematics are an indispensable part of the current profile of TUM Mathematics due to their close connection to algorithmic problems such as route planning and scheduling in public transport, chip layout, discrete tomography or data analysis. The research covers fundamental aspects of Discrete Mathematics and applications to the analysis and algorithmic treatment of practical problems from the Natural Sciences, Engineering, Economics, and Finance, with particular interest in optimization issues. There is a close connection to the mathematical disciplines of Combinatorics and Graph Theory. Concrete applications investigated at TUM are, for example, the optimal planning of land consolidation or of tram schedules.

Geometry and Visualization (M10) 

Geometry focuses on fundamental aspects of geometric patterns and structures as well as on concrete applications that require a deep geometric understanding to analyze or visualize underlying patterns. With the help of Computer Graphics, real objects and processes such as figures or solids and their movements, respectively, can be represented in a clear manner, and many abstract structures and facts can be visualized geometrically. In particular, this is a means of gaining new mathematical insights (experimental mathematics) or presenting mathematical solutions and results impressively (computer simulation). Here, Mathematics, Computer Science and Electrical Engineering often interact intensively.

Algorithmic Algebra (M11)

Like Analysis and Geometry, Algebra is one of the fundamental branches of Mathematics. It deals with properties of arithmetic operations based on special algebraic structures such as groups, rings or fields. The latter are sets of certain objects (elements), usually abstract in nature, for which arithmetic operations and relations are specified in a manner which is consistent with a set of basic assumptions (axioms). Nowadays, there is even the possibility of calculating with such objects using computers, i.e., of systematically manipulating arrays of symbols according to given rules with the aid of computers (Computer Algebra). Thus, for example, certain equations can be solved exactly or even mathematical proofs can be carried out automatically. Algorithmic Algebra is concerned with the search for corresponding algorithms and the determination of their efficiency. Algebraic Geometry deals with relations between algebraic and geometric structures in order to deepen the understanding of both.

Biomathematics (M12)

The aim of Biomathematics is to develop mathematical models and theories to describe or predict the structure and dynamics of systems and processes from living nature. A particular objective is to discover, through abstraction, principles and mechanisms underlying the development and interaction of living organisms. Due to the considerable complexity of living systems, this approach is becoming increasingly important in Biology and other areas of the Life Sciences, including Medicine. Here, many mathematical disciplines interact. Today's findings in Ecology, Epidemiology, Neurology or Evolutionary Theory are inconceivable without Mathematics. Modern health care and personalized medicine also rely on mathematical-statistical models and methods. Since 1997, there have been faculty members simultaneously holding positions at TUM Biomathematics and at the Institute for Biomathematics and Biometry at the Helmholtz Center Munich.

Financial Mathematics (M13)

Financial Mathematics provides access to the complexities of finance. For example, mathematical instruments are developed to determine fair commodity prices, stock prices, and interest rates, or to model loans and insurance products. This in particular helps to evaluate options, quantify financial risks and figure out optimal investment strategies. Methodologically, Financial Mathematics uses a very broad repertoire of techniques from Stochastics, Statistics, Numerics, Optimization, and Analysis.

Probability Theory (M14)

Probability Theory (mathematics of uncertainty) investigates how likely it is that a random event will actually occur. It is used in fields such as Economics or Computer Science. In addition, it has numerous connections to other areas of Mathematics such as Analysis, Graph Theory and Mathematical Physics. Last but not least, with its offshoot, Mathematical Statistics, it forms the area of Stochastics. Stochastics has a significant impact on our everyday life. It deals with stochastic processes and their long-term behavior. Research foci at TUM include so-called random walks in random networks and environments as models for transport in inhomogeneous media. Other probabilistic models which are studied, help to better understand the dynamics of complex biological systems or to study randomized processes in signal processing and data analysis.

Applied and Numerical Analysis and Data Analysis (M15)

Whether in Logistics, Medicine or Mobile Communications – many applications lead to ever increasing data sets. In order to analyze and efficiently use these data sets, suitable mathematical tools are more and more needed. These tools can be used to make forecasts, save resources and reduce costs in many application areas. A particular challenge is posed by very large data volumes, not least for reasons of data storage. Here, mathematical concepts for data compression are of particular importance.

Numerical Methods in Plasma Physics (M16)

One of the greatest challenges of mankind is the search for environmentally and climate friendly as well as economically acceptable solutions to the energy problem. In this context, nuclear fusion is a promising physical principle, although there are considerable difficulties in implementing this principle successfully from a technological point of view. Among other things, Plasma Physics deals with fundamental problems that arise in fusion research. Of particular importance are

• on the one hand, the mathematically rigorous formulation of corresponding physical theories and the analysis of underlying mathematical structures, and
• on the other hand, the application of mathematical solution methods and strategies to solve resulting mathematical model equations.

Due to the enormous complexity of these equations (kinetic models, in case of magnetic fusion also models from Magnetohydrodynamics), numerical solution methods play a prominent role here. It is important that the approximation algorithms used reproduce essential structural features of the exact solutions, at least in sufficient approximation, including symmetries and the existence of quantities which are conserved along the solutions. This poses a big challenge for the choice or development of suitable solution methods. In particular, concepts of Differential Geometry and the theory of Lie groups are used here. The head of the M16-unit simultaneously holds a director position at the Max Planck Institute for Plasma Physics in Garching. 

Optimal Control (M17)

One research focus is the development of efficient numerical algorithms for optimal control of complex processes in nature and technology. The theory of optimal control is closely related to the theory of optimization and to calculus of variations. An optimal control strategy is mathematically described by an input function that minimizes or maximizes a given objective function under a differential equation constraint and possibly other constraints.

For example, if one wants to heat a room in an energy-efficient manner, one might specify the objective to use as little fuel as possible. How does one change the setting of the heating system from time to time during a given period of time in order to achieve that goal? In a mathematical model, it is relatively easy to vary the input function describing the time course of the setting and thus derive a determining equation for optimal controls, i.e., for input functions that minimize the total fuel supply for the heating system during the time period considered. Here, certain constraints must be met, for example that the temperature should not exceed or fall below certain limits anywhere in the room. Starting from a given spatial temperature distribution at the initial time and imposing suitable boundary conditions, the evolution of the temperature in the room over time could be modeled mathematically by means of a heat conduction equation with forcing depending on the heating. The boundary conditions depend in particular on the heat permeability of the walls, doors and windows, and of course temporary ventilation periods must also be taken into account.