Basic theoretical science research is a prerequisite for all scientific, technical and modern medical research. It is thus also a prerequisite for concentrated future efforts within all high-technology research fields that theoretical science is both very strong and of a very high quality.

Theoretical science at the University of Aarhus is so strong that the university is regarded as being among the elite. The Faculty of Science in Aarhus is listed as the best Nordic faculty and among the top 16 in Europe in a list of rankings of science faculties compiled by the Times Higher Education Supplement.

The research priority area will ensure this strong position for the benefit of increased research efforts throughout Denmark.

The Centre for Theory in Natural Science (CTN) acts as host for several theoretical research centres, and manages and coordinates interdisciplinary activities within the Faculty of Science.

Below we present the PhD students, visitors and projects partly or fully financed by CTN (list not complete):

**Dr. Robert C. Albers, Los Alamos National Laboratory, New Mexico (USA).**Robert C. Albers, Los Alamos Natl. Lab., stayed in May 2007 at the Department of Physics and Astronomy, University of Aarhus, as a guest professor financed by the Centre for Theory in Natural Science, CTN. This was the first one of two visits planned in connection with a collaboration agreement between Dr. Albers and Professor Niels Egede Christensen. During his stay in Aarhus he gave two seminars, both on electronic and other physical properties of actinides, mainly plutonium, and other materials with strongly correlated electrons. During the visit new common research activities were initiated in theoretical research in electronic properties of solids, mainly dealing with strongly correlated electron systems. The new projects involve in USA (R.C. Albers and A. Chantis, Los Alamos, M. van Schilfgaarde, Arizona State University), Japan (T. Kotani, Osaka University) and Denmark (A. Svane and N.E. Christensen, University of Aarhus). Van Schilfgaarde and Kotani have developed a so-called self-consistent GW method for advanced quantum theoretical calculations of electronic states in solids, and we intend to apply these in studies of the actinides and actinide compounds. Axel Svane shall visit Arizona State University in august 2007 in order to become familiar with the new computing codes and start the first calculations together with our collaborators. The major programs are now installed on the computer systems at the

*Centre for**Scientific Computing in Aarhus*(CSCAA), where many of the computations will be performed in the future.The visit by Dr. Albers was very valuable, and we do not doubt that the new research projects will lead to interesting results. Further, availability of the new methods will be an essential supplement to our presently used approaches, and they will enable studies of materials and processes beyond the specific cases referred to above. The project will have a high priority, and we look forward to the next time when Dr. Albers visits Aarhus as a part of the agreement with CTN.

**Niels Buhl, PhD Student, Department of Physics and Astronomy and Aarhus University Hospital**Diffusion weighted nuclear magnetic resonance (DW-NMR) provides a number of exciting opportunities for probing the microscopic structure of disordered media via the use of diffusing molecules. These techniques which are already in wide use in geology, biology and medicine allows quantities such as e.g. pore size and surface to volume fraction in porous media and local diffusion constants and flow rates in living organisms to be determined noninvasively. In medicine DW-NMR measurements of apparent diffusion constants provide the most important modality for diagnosing acute stroke. The latter use hinges on a still not fully understood change in water diffusivity associated with this disease. However DW-NMR holds promise of a much more detailed insight into the microstructure and state of living tissue by relating the measured DW-NMR signal via mathematical models to the microstructural and physiological parameters characteristic for the given type of tissue in the given physiological state.

The aim of the project will be to develop and validate such new mathematical models for brain tissue based on prior knowledge of tissue morphology.

**Jeremy Faupin, Postdoc, Department of Mathematical Sciences, University of Aarhus.**Much remains to be done in the construction of a rigorous mathematical theory of Quantum Electrodynamics. Recently, mathematical models of non-relativistic matter coupled to quantized radiation fields have been widely studied. The so-called standard model of non-relativistic quantum electrodynamics describes an atom, an ion or a molecule, minimally coupled to the transverse photon field. Due to the fact that the photon mass vanishes, serious difficulties appear in the spectral analysis of the associated Pauli-Fierz Hamiltonian. Questions such as the existence of a ground state, perturbation of embedded eigenvalues and the existence of resonances, the study of the life-time of metastable states, are of great interest, both from a physical and a mathematical point of view.

**Jens Fjelstad, Postdoc, Department of Mathematucal Sciences, University of Aarhus**Among all quantum field theories there are two subclasses that presently allow a rigorous understanding, topological field theories in two and three dimensions and rational conformal field theories in two dimensions. These classes of theories are related, and in recent developments topological field theory is used to construct conformal theories. The current projects are focused on understanding the general structure of conformal field theory and reside mathematically in the intersection of algebra, topology and category theory. Avenues for extending the understanding beyond rational models are investigated, in particular by reformulating rational conformal field theory using certain bicategories of bimodules in braided tensor categories.

**Maurice Jansen, Postdoc, Department of Computer Science, University of Aarhus.**When it comes to proving complexity lower bounds for explicit problems, the current challenge to humanity is

*whether we can do this at all*in any general model of computation to extents beyond the barely non-trivial. Efforts to resolve the famous**P**vs.**NP**problem, in the non-uniform setting at least, should be retargeted to first showing the mathematically strictly weaker statement that**VP**does not equal**VNP**, over for example the field of complex numbers. The latter classes are Valiant's analogues of**P**and**NP**for the algebraic model of computation. In the algebraic model potentially more mathematical lower bound tools are available.During the first four months of his year long stay, Maurice Jansen has been giving a lecture series on algebraic complexity theory. In his research he has been working on determinantal complexity, algebraic branching programs and various other restricted models of algebraic computation.

**Professor Jerzy Karczmarczuk, University of Caen, France.**Jerzy Karczmarczuk stayed five months at the Department of Computer Science, University of Aarhus, as a guest associate professor co-financed by the Centre for Theory in Natural Science, CTN and by the Faculty of Science. His stay in Aarhus was in two parts: three months from September, October, and November 2007, and two more months in January and February 2008. During the first part, he gave a quarter course and one seminar at the Department of Computer Science. The quarter course was about a functional-programming approach to quantum structures. The seminar was about using lazy functional-programming techniques for generating the sounds of musical instruments. Both topics are parts of Prof. Karczmarczuk's forthcoming Habilitation in France (the French equivalent of a DSc here in Denmark).

**Michael Knudsen, PhD student, Bioinformatics Research Center, University of Aarhus.**Within biology networks are widely used. They provide a convenient way to graphically describe complex, biological systems. An extensive mathematical theory behind networks makes it possible to develop mathematically-founded, biological theories about networks and their evolution. The protein interaction networks (PINs) form a special class of networks. The proteins (in practice, only a subset of them) are represented by nodes, and two nodes (proteins) are connected if they interact. Large datasets for several small organisms are already present and the description of human PINs are on the way.

The work in the present project aims at developing mathematical models that can describe PINs, in particular PINs from different organisms in order to describe their individual evolution. This is carried out by comparing local and global network structures. A central part of the work is to develop methods for selecting the best models.

**Bjarke Hammersholt Roune, PhD student, Department of Computer Science, University of Aarhus.**Bjarke Hammersholt Roune has stayed in the United States at the University of Minnesota for the past year. He and his co-advisor, Associate Professor Niels Lauritzen, has been attending the inter-disciplinary mathematics and computer science event "Thematic Year on Applications of Algebraic Geometry" at the Institute for Mathematics and its Applications. This year Bjarke's research has primarily been on the algorithmic aspects of the Frobenius Problem, maximal lattice free bodies and monomial ideals.

Due to a paper of his on the computation of Frobenius numbers, Bjarke was recently invited to give a talk at an AMS Special Session on The Linear Diophantine Problem of Frobenius at the Joint Mathematics Meeting in San Diego. The paper is available at www.broune.com and has recently been accepted for publication in the Journal of Symbolic Computation.

**Professor Viacheslav Belavkin, University of Nottingham (England)**In the spring of 2008, Professor Viacheslav Belavkin from the University of Nottingham, UK, was employed for three months at the Department of Physics and Astronomy as guest professor. Professor Belavkin was recipient of the Gold Medal for Outstanding Achievements in Science and Technology, 1996 (formerly, the Lenin prize, the most prestigious research award in the Soviet Union). He invented in the 1980’es the quantum filtering equations, which form the basis for quantum measurement and feedback theory. During his stay in Aarhus, Professor Belavkin gave a series of lectures, and he conducted research for a joint publication with researchers at the Aarhus University.