The research group in Algebraic Geometry
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Seminars spring 2007
The following researchers and students are associated with the research group in algebra and algebraic geometry:
Permanent faculty:- Gunnar Fløystad (gunnar@mi.uib.no)
- Audun Holme (holme@mi.uib.no)
- Trygve Johnsen (johnsen@mi.uib.no)
- Stein Arild Strømme (stromme@mi.uib.no)
Algebraic geometry is a classical discipline, with problems that often originate from questions that were asked centuries ago. The field has also proved to be remarkably efficient in explaining and solving problems that appear in other branches of science.
Among the classical disciplines that have been central in the research of the group is the study of algebraic manifolds and their embeddings in spaces of low dimension. One has also been involved in developing algebraic methods and tools to study embedded projective manifolds. One has been particularly interested in initial ideals. Here one has generalized these ideals to higher initial ideals that in a natural way constitute a complete set of invariants for embedded projective manifolds. In particular one has applied these ideals to study and classify space curves, a research topic with great international activity. Also more classical cohomological methods have been used to study space curves and their Hilbert schemes.
Another topic of interest is varieties of low codimension in projective spaces. This is an area well suited for development of efficient methods of calculation. Hartshornes conjecture that such varieties are complete intersections is a guideline for this research. An important topic within the study of projective algebraic varieties is secant structures and varieties, associated varieties as dual variieties and related constructions. Closely associated to this is the study of vector bundles on algebraic manifolds, including projective spaces, and moduli spaces for these bundles.
Enumerative geometry and intersection therory attract various kinds of interest. On one hand one develops further a classical theory, with lines back to the nineteenth century, through investigating new methods and concepts. The study of various moduli spaces, parameter spaces, and cohomology rings is central. Examples are Hilbert schemes, moduli spaces for vector bundles, and other geometric quotients. In association with this activity one uses computers for symbolic and numerical calculations, and one develops software for calculations of Chern classes and intersection numbers.
On the other hand one establishes connections to other disiplines, such as string theory within theoretical physics. Key words for the research in this area is mirror symmetry and rational curves on Calabi-Yau threefolds. The number of curves of specified topological types on certain Calabi-Yau threefolds has been the object of special attention. Just to establish that these numbers are finite can be interesting. Also K3 surfaces are studied actively, partly with an eye on question about curves and threefolds. Often physics is not linked to algebraic geometry alone, but to algebraic geometry and fields like topology, differential geometry, analysis simultaneously.
Among other applications of algebraic geometry that the group works with, is algebraic-geometric (Goppa) codes. We will also mention representation theory for algebraic groups and quantum groups. These are fields with great international activity, where one wishes to participate.
