Research Projects

The Mathematics Division of the Science Institute counts among its members all the faculty members in pure mathematics as well as the more mathematically inclined of the theoretical physicists. In addition there are four research scientists in the mathematics division. The research projects in the division are all in pure mathematics or mathematical physics. The members also work occasionally on various problems in applied mathematics. In addition the division contributes to the development of the mathematical infrastructure in Iceland through writing of textbooks, participating in the review of school curricula and organizing the Icelandic Mathematics Olympiad.

The following research projects were active in the year 1996:

Eggert Briem

• Functions operating on Banach spaces
  Quite a lot is known about functions which operate by composition on Banach spaces of continuous functions on compact spaces. Less is known about Banach spaces of continuous functions vanishing at infinity on locally compact spaces. Two problems were investigated. In the first problem the spaces are sup-norm closed. The work on this problem is still going on. In the second problem the spaces are complete in a norm which dominates the sup-norm. This work is mostly finished and the results have appeared in a preprint from the Institute.

• Harmonic maps
  The goal of this research project is to construct a variational integral which has a harmonic map as a stationary point in each homotopy class of mappings from one compact manifold to another. The idea is to find a stationary point by following a steepest descent path for the variational integral and prove uniform convergence by estimating the change in the local energy of the flow. Up to now an estimate of uniform continuity has been missing in the case when the local energy functional is bounded. During the year a theorem was proven about this which is stronger than originally expected. The result and its proof which is quite long and involved appeared in a research report from the Science Institute in the fall of 1996.

• Classical thermodynamics
  The goal is to give an axiomatic treatment of classical equilibrium thermodynamics and deduce rigourously from simple postulates the existence and essential uniqueness of entropy.
  This project was pursued in collaboration with E.H. Lieb, Princeton.
• Bounded Bose fields
  Quantum fields obeying canonical commutation relations lead necessarily to unbounded field operators on Hilbert space and for a long time the existence of Bose fields with bounded field operators was questioned by field theorists. Recently, however, D. Buchholz and H. Rehren have given examples of such fields in space-time of dimension 2. In the project the possibility of such fields in higher space-time dimensions is investigated.
  Collaboration: H. Steier, Vienna.
• Magnetic Thomas Fermi theory at nonzero temperatures
  The Thomas Fermi theory for atoms and molecules in a nonzero magnetic field at finite temperature was developed in the MS thesis of B. Hauksson, Univ. of Iceland 1996. A part of this work is a quantum mechanical limit theorem that links the grand canonical pressure of quantum statistical mechanics with the pressure computed in Thomas Fermi theory. The aim is to extend this limit theorem to other thermodynamical quantities.
• The Thomas Fermi equation of state for matter in strong magnetic fields
  Thomas Fermi theory of atoms in the spherical approximation offers the possibility to compute approximately the equation of state for compressed matter with manageable numerical effort. In this project this method has been applied to matter in the surface layers of magnetized neutron stars.
  Collaboration: E.H. Gudmundsson, Ö.E. Rögnvaldsson, A. Thorolfsson.
• The ground state of an interacting Bose gas
  This project aims at an understanding of the rigorous link between the exact quantum mechanical ground state of an interacting Bose gas and the Gross-Pitajevskii energy functional that has been widely used to interpret recent experiments with Bose-Einstein condensation.
• Modular group of wedge domains in KMS states
  One of the most useful results of algebraic quantum field theory is the Bisognano Wichmann theorem, that links Lorentz transformations and PCT symmetry with the modular objects in the sense of Tomita-Takesaki theory of von Neumann algebras. More precisely, Lorentz boosts correspond to the modular group of the the vacuum state and the von Neumann algera generated by observables localized in a space-like wedge, and the modular conjugation is the PCT operator combined with a rotation. This project aims at a computation of the modular objects when the vacuum state is replaced by a finite temperature state. Collaboration: H.J. Borchers, Göttingen.

• Construction of meromorphic functions on cycle spaces by means of integration
• Algebraic description of some geometric operations on complex analytic cones

• Plurisubharmonic functions
  Functionals operating on analytic discs on manifolds are used to construct plurisubharmonic functions.

Reynir Axelsson

• Resolutions of analytic sheaves
  The project concerns the question when coherent analytic sheaves on complex manifolds have resolutions by locally free sheaves. Since a coherent analytic sheaf always has a resolution consisting of locally free connected systems of sheaves one more generally investigates the connection between sheaves and connected systems of sheaves.
• Hyperbolic geometry without continuity axioms
  The project concerns the development of hyperbolic geometry without continuity axioms.

• Meromorphic operator-valued functions on Riemann surfaces
  The idea of this research is to view the theory of a meromorphic operator-valued function on a Riemann surface M, with values in the algebra of continuous operators on a Banach space E, as a generalization of the classical spectral theory of a single operator. Thus it is possible to define spectrum, eigenspaces and multiplicity. The spectrum is a subset of M; eigenvectors and a multiplicity are associated with each compact, isolated subset of the spectrum. Eigenvectors belong to the Banach space E whilst the multiplicity is a certain Banach space, unlike the case of ordinary spectral theory where the multiplicity of an eigenvalue is a natural number.

  During the year I completed one article which has been accepted for publication. It is entitled Spectrum and eigenspaces of meromorphic operator-valued functions. I proved a generalization of Runge's Theorem to vector-valued functions on a Riemann surface. This appeared as a Science Institute report but will be published in an appendix to the above mentioned paper.

  Another paper is in preparation entitled Meromorphic operator-valued functions: multiplicity and equivalence but was still unfinished at the end of the year.

• Topological groups, permutation groups and graphs
  This project is about the connections between a certain topology that can be defined on permutation groups and the properties of the group. In 1995 I managed to find a new proof of a theorem about group actions on graphs due to a russian mathematician. A paper containing this proof has now been polished and will appear in the journal Discrete Mathematics. I have also continued with these investigations.
• Infinite permutation groups
  In the summer 1996 I and three other mathematicians gave a lecture course on infinite permutation groups at the Indian Institue of Technology at Guwahati. Based on these lectures the four of us have now prepared a book on this topic. The book will be published by the Hindustan Book Agency in India. The manuscript is now almost finished and the book is scheduled to appear in the summer of 1997. The other three authors of this book are Meenaxi Bhattacharjee at the Indian Institue of Technology at Guwahati, H. Dugald Macpherson at Leeds University og Peter M. Neumann at Oxford University.

• Random surfaces and random manifolds
  Random surfaces and random higher dimensional manifolds are generalizations of random walk which are used to model many different physical phenomena, in particular quantized strings, phase boundaries in statistical mechanics, membranes and quantum gravity. The problems currently under study include the influence of matter fields on the underlying fluctuating geometrical structure, the spectral dimension of random manifolds and the problem of defining causal structures on random surfaces.
• Mathematical models of earthquake faults and avalanches
  A two-dimensional block-spring model with deterministic chaotic behaviour has been studied numerically and the structure of the slip zone analyzed in detail. A random walk model for avalanches was solved analytically and the critical exponents determined, including the Gutenberg--Richter slope.