Mathematical research still focuses primarily on the areas algebra, analysis and algebraic topology. Almost all mathematics staff are specialized within one of these main areas. Research activities increasingly focus on tasks related to more than one of these disciplines and to other areas, including application of mathematics within physics, computer science, genetics, economics etc. The Department is particularly active within research on: Algebraic K-theory and geometry of manifolds, algebraic geometry, Lie algebras and their representations, quantum algebras, category theory, homotopy theory, topology of smooth germs, dispersal theory, convexity theory, operator algebras, Bose algebras, combinatorics, algebraic number theory, representation of Lie groups, coding theory, cryptology, invariants of 3-dimensional manifolds and Gauge theory.

Research also includes singularity theory, topological quantum field theory, moduli space, mapping-class groups, computational biology, partial differential equations relating to mathematical physics, including scattering theory, super conductivity, operator theory and quantum mechanics, and analysis of manifolds. Research also revolves around harmonic analysis, analytical number theory, Diophantine analysis, K-theory for operator algebras and dynamical systems.

Research within theoretical statistics concentrates on probability theory and basic concepts of statistical inference, particularly stochastic analysis, Markov processes and associated potential theory, probability theory in infinite-dimensional space, actuarial mathematics and mathematical financing, simulation, asymptotic likelihood theory, inference for small random samples and stochastic processes, Edgeworth – and saddle point approximation, advanced statistical DNA models, time series analysis, geometric probability and stereology, spatial statistics, relationship between stochastic processes and quantum physics, analysis of concrete statistical models.

Research also includes stochastic geometry and bio imaging, space-time modeling and statistical methods within bioinformatics.

Operations research concentrates on optimization, optimal control theory, production economics and financing. The optimization theory concentrates on stochastic programming, multicriteria optimization and combinatorial optimization problems. Within production economics the application of Markov decision processes for stock control and pricing are analysed. Central topics within financing theory encompass pricing of contingent claims, theory of interest rates structure and the interaction between finance theory and actuarial science. Models to be applied within the energy sector are also analysed.

- Centre for Science Studies (CSS)
- Centre for Quantum Geometry of Moduli Spaces (QGM)
- Centre for Stochastic Geometry and Advanced Bioimaging (CSGB)
- T.N. Thiele Centre for Applied Mathematics in Natural Science (Thiele)
- Center for Research in Econometric Analysis of Time Series (CREATES)
- Sapere Aude - Semi Classical Quantum Mechanics
- Sapere Aude - Intuitions in Science and Philosophy
- Sapere Aude – Time-wise Behavior of Fractal Processes

The aim of the Centre for Science Studies is to do:

- Research in the fields of history and philosophy of science and technology, science and society, and science communication
- Undergraduate and post-graduate teaching across research areas
- Research-based development of education and communication activities

Homepage: css.au.dk

Centre for Quantum Geometry of Moduli Spaces (QGM) was established in 2009 as a ’Center of Excellence’ funded by the Danish National Research Foundation (DNRF) with a grant of DKK 50 million. In 2013 the DNRF awarded a five-year extension to QGM until 2019, so that the total QGM funding from DNRF amounts to DKK 90 million over the 10 year funding period.

QGM’s research objective is to address fundamental mathematical problems at the interface between geometry and theoretical physics. Directed by Professor Jørgen Ellegaard Andersen, QGM hosts a strong team of high-profile, internationally acclaimed researchers, and with the continuous generation of groundbreaking results, the Centre together with its international collaborators are recognized throughout the mathematics community worldwide as one of the leading research institutions within its research field.

Homepage: qgm.au.dk

Centre for Stochastic Geometry and Advanced Bioimaging (CSGB) is a VKR Centre of Excellence established for the purpose of developing new computer-based methods within stochastic geometry and spatial statistics for the analysis of microscopy and other advanced bioimaging data. The center has received a grant of DKK 25mill from the Villum Foundation.

Four research groups cooperate in CSGB: The Spatial Statistics Group, Institute of Mathematical Sciences, Aalborg University; The Image Group, Department of Computer Science, Copenhagen University; Biomedical Group, Stereology and EM Research Laboratory, Aarhus University; Stochastic Geometry Group, Institute of Mathematical Sciences, Aarhus University.

Homepage: csgb.dk

This centre is hosted by the School of Economics and Management, Faculty of Science and Technology. Researchers of the Institute of Mathematical Sciences specializing in financing participate in time series and financial econometrics research performed in CREATES.

The centre was established in 2007 by the Danish National Research Foundation (Centers of Excellence) with a grant of DKK 40 million.

Homepage: creates.au.dk

The project Semiclassical Quantum Mechanics is funded by a Sapere Aude: DFF-Topresearcher Grant from the Danish Council for Independent Research with Søren Fournais as Principal Investigator. In this project research is carried out in the area of mathematical quantum mechanics, in particular, spectral problems in the presence of magnetic fields. When the magnetic field is very strong and/or when many interacting particles are considered, one can often show that the quantum mechanical systems can be described by simpler - sometimes even classical - models. This can be seen as an aspect of the Correspondence Principle by Niels Bohr.

Homepage: Semiclassical Quantum Mechanics

The Sapere Aude project Intuitions in Science and Philosophy, funded by the Danish Council for Independent Research and led by Samuel Schindler, investigates the role and nature of intuitive judgements in science and philosophy. Whereas intuitive judgements in philosophy have been much debated in recent years, little attention has been paid to intuitive judgments in science. This is where the project steps in. In particular, it investigates intuitive judgements in thought experiments in physics and in the form of acceptability judgements in linguistics. The results of these investigations will be related to debates about the evidential function of intuitive judgements in philosophy.

Homepage: projects.au.dk/intuitions-in-science-and-philosophy/

The project TBFP is funded by a Sapere Aude: DFF Starting Grant awarded by the Danish Council for Independent Research with Andreas Basse-O'Connor as principal investigator. The main aim of the project is to provide new mathematical descriptions of a class of models called *fractional processes*, which can be perceived as random fractals. Such models are often used to describe phenomena that have a very irregular behavior over time, as for example, seen in stock prices. It is of particular interest to characterize how these processes behaved in very small time intervals, which say something about how much uncertainty there are in the future predictions made from these models.

Homepage: TBFP