# Mathematics

**The programme:**

At the Department of Mathematics there are several directions of research in Mathematics, both pure and applied; these encompass both theoretical aspects within mathematics itself as well as investigations of interest in other fields of science, such as for example physics, coding theory, computer science, and mathematical finance. Also, there are cooperations with other sciences and practical work in connection with medical and engineering problems, including wind power.

The main research areas in Pure Mathematics are: Algebra and Discrete Mathematics: Algebraic geometry and commutative algebra, algebraic groups and quantum groups, computer algebra, coding theory and cryptography, representation theory for Lie algebras, computational algebraic geometry.

Analysis: Mathematical physics, spectral theory, spectral geometry, representation theory, analytic number theory, harmonic analysis, Lie groups, K-theory, C*-algebras, scattering theory, mathematical physics of fields and particles, magnetic fields and superconductivity.

Geometry and Topology: Algebraic K-theory, homotopy theory, differential geometry, Kaehler geometry, gauge theory, quantum field theory, geometric quantization of symplectic varieties, quantization of moduli spaces, topology of smooth mapping germs, singularity theory.

At the Department of Mathematics, there are also several directions of research in Applied Mathematics; the most important ones are:

Theoretical Statistics: Likelihood theory, statistical models, spatial statistics, stereology, asymptotic methods, geometry and statistics, inference for stochastic processes, time series analysis, simulation. Some important applications of statistics, particularly within biology, geology and economics are also studied: spatial biological structures, medical image processing, evolutionary development of DNA, turbulence and sand transport, insurance mathematics, financial markets and volatility.

Probability Theory: Stochastic analysis, infinite dimensional probability theory, stationary processes, multiparameter processes, Levy theory, martingale theory, free probability and the connection to operator algebras.

Mathematics - Economics and Operations research: Financial theory, discrete optimization, production economics.

Energy production and distribution: The study of networks for reliable energy distribution, statistical methods, problems of wind mill parks.

The research in theoretical statistics and probability theory includes in addition ambit processes, turbulence modelling, insurance mathematics and mathematical finance, simulation, asymptotic likelihood theory, inference for stochastic processes, time series analysis, Edgeworth and saddlepoint approximations, statistical models for DNA, geometrical probability and stereology, spatial statistics, connections between stochastics and quantum physics, planning of experiments in particular with a view to industry, and analysis of statistical models.

The research in operations research is focused on three main fields: Optimization, production and financial theory. stochastic programming, multicriteria optimization, combinatoric and discrete optimization and the application of Markov decision processes on inventory control and pricing.

The central topics within financial theory are pricing and hedging of contingent claims, theory of term structure of interest rates and the interface between financial theory and the theory of insurance.

For the problems of renewable energy resources, the department cooperates with some of the large companies in the area as well as with engineering schools.

**Affiliated centres/departments:**

Much of the research in pure mathematics at the department is supported by external grants, in particular for the international cooperations, that are very active; affiliated with the department is the center QGM, Quantum Geometry of Moduli Spaces, that cooperates with several institutions outside the department in the field of moduli spaces and the connections to modern physics.

The Thiele Center is also affiliated with the department. The Thiele Center cooperates with several institutions outside the department within statistics and probability theory; also affiliated is "Center for Stokastisk Geometri og Avanceret Bioimaging" (CSGB), a VKR Center of Excellence, working on developing new computer based methods within stochastic geometry and spacial statistics for analyzing advanced bioimaging data. Both of these centers are in part supported by external grants. In general there is a very active international cooperation with universities abroad.

**Examples of employment after graduation:**

The employment of PhDs takes place in a wide range of positions, from statistical and medical companies to research institutions. At the moment there are about 10 active PhD students at the department in applied mathematics.

**Approx. number of PhD students:**

At the moment there are a total of 29 PhD students in pure mathematics at the department, and usually employment of PhDs takes place in a wide range of positions, for example in industry, such as the wind mill industry, in financial institutions, and in research or teaching.