Research M. Bergvelt

I am interested in completely integrable systems and their relations with representation theory and algebraic geometry.

At the moment I am working on the geometry of Flag manifolds related to vertex algebras and quantization of integrable systems. Another project deals with Toda lattices and multiplicative vertex algebras. If you don't know what this means and are interested you are welcome to step by my office and chat with me about this.

This paper classifies all integrable systems coming from Heisenberg algebras in loopgroups.

Partitions, Vertex Operator Constructions and Multi-component KP equations, M.J. Bergvelt, A.P.E. ten Kroode, Pacific Jnl. of Mathematics, 1995, vol 171, pp23-88.

For every partition of a positive integer $n$ in $k$ parts and every point of an infinite Grassmannian we obtain a solution of the $k$ component differential-difference KP hierarchy and a corresponding Baker function. A partition of $n$ also determines a vertex operator construction of the fundamental representations of the infinite matrix algebra $gl_\infty$ and hence a $\tau$ function. We use these fundamental representations to study the Gauss decomposition in the infinite matrix group $Gl_\infty$ and to express the Baker function in terms of $\tau$-functions. The reduction to loop algebras is discussed.

Paper: Source or Postscript

The relation with the algebraic geometry of finite ramified coverings of an algebraic curve is :

The Krichever Map, Vector Bundles over Algebraic Curves and Heisenberg Algebras, M.R. Adams, M.J. Bergvelt, Commun. Math. Phys. 1993, vol 154, pp. 265-305.

A large part of classsical algebraic geometry deals with polynomial algebras, i.e., with the symmetric algebra of a vector space. There is a meaningful algebraic geometry based on the exterior algebra of a vector space, or more generally on the polynomial algebra in even and odd variables.

Super Curves, their Jacobians, and super KP equations

Author(s): M.J. Bergvelt , J.M. Rabin, Duke Jnl. of Math., Vol. 98, 1999, pp 1-58.

We study the geometry and cohomology of algebraic super curves, using a new contour integral for holomorphic differentials. For a class of super curves (``generic SKP curves'') we define a period matrix. We show that the odd part of the period matrix controls the cohomology of the dual curve. The Jacobian of a generic SKP curve is a smooth supermanifold; it is principally polarized, hence projective, if the even part of the period matrix is symmetric. In general symmetry is not guaranteed by the Riemann bilinear equations for our contour integration, so it remains open whether Jacobians are always projective or carry theta functions. These results on generic SKP curves are applied to the study of algebro-geometric solutions of the super KP hierarchy. The tau function is shown to be, essentially, a meromorphic section of a line bundle with trivial Chern class on the Jacobian, rationally expressible in terms of super theta functions when these exist. Also we relate the tau function and the Baker function for this hierarchy, using a generalization of Cramer's rule to the supercase.

Paper: Source or Postscript

Maarten Bergvelt  
Department of mathematics
University of Illinois at
Urbana-Champaign
Urbana-Champaign, IL 61801
Office:Illini Hall 333
tel: 217-333-6326, 	
email:bergv@uiuc.edu

last modified August 18th, 1999

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