Olav Arnfinn Laudal
Universitetet i Oslo
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ResearchI have, for most of my life, been interested in deformation theory. This fascinating theory, first tuched upon by Bernhard Riemann, holds that a mathematical object contains all the information needed to construct the space of all iso-classes of "nearby" objects, and, in particular, all the moduli, i.e. the parameters necessary to controle all infinitesimal changes of the object. This theory is the basis for the theory of moduli, in algebraic and analytic geometry. However, treating moduli problems in general one quickly runs into problems in classical algebraic geometry. It is f.ex. difficult, or hopeless, to asssign to two orbits of a Lie-group, one sitting in the closure of the other, two different points of a genuine "space". Here is where non-commutative algebraic geometry comes to the rescue. In non-commutative theory, different points, infinitelyI close, exist, and will be linked by a tangent. This is what non-commutative deformation theory tells us, and using this construction, I propose a non-commutative algebraic geometry, where points exist, and where moduli problems in classical algebraic geometry, left open for a long time, have a solution. The construction of non-commutative moduli spaces, like phase spaces in physics, turns also out to give new insights in the relationship between relativity theory and quantum theory. This idea, and its "realization", has been my main interest the last 10 years. I am, today, convinced that the main tool of physics should be the mathematical notion of moduli, in the non-commutative version hinted to above, and that the crucial point is to define time as a "metric" on the relevant moduli space. See the popular texts ...., and the papers ... (til toppen av siden)
Areas of ResearchI have, in very early years, been interested in homological algebra, in particular in the notions of projective and inductive limits, and in spectral sequences. This led me to work on valuations, and on cohomology theory for algebras and groups. Later on I got involved in deformation theory, for commutative algebras, and globally for schemes, see Springer Lecture Notes 754. An easy application of deformation theory turned out to give a result on how curves cuts hyperplanes ("Laudal's lemma"), and this led to some papers on algebraic geometry. In particular I worked, with Knud Lønsted, on the moduli space of hyperelliptic curves. Later on, and as a result of this study, I turned to singularities. The main paper here is the Springer Lecture Notes 1083, with Pfister. As a natural extension of this study, i worked for some time on deformations of Lie algebras, and produced together with Harald Bjar a paper on the relationship between the moduli space of isolated curve singularities and the corresponding moduli space of Lie algebras of the automorphism groups. Realizing that the theory of moduli for mathematical objects, like singularities and Lie algebras, could not be adequately developed within classical algebraic geometry, I turned to non-commutative geometry. First I constructed a non-commutative deformation theory, and later extended this to a non-commutative algebraic geometry. The last years I have been working on mathematical physics, within this non-commutative algebraic geometry, see latest papers, and the point above.
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Some papers on homological algebra, algebraic geometry, deformation theory and singularities.
Some papers on general subjects