• PROGRAM (updated 12.07.00, final)

  • ABSTRACTS (updated 12.07.00, complete)

    Vertex (operator) algebras are a special kind of infinite-dimensional algebras which were first introduced by R. Borcherds around 1986. Since then their theory has been developed by a number of people (Frenkel, Lepowsky, Meurman, Zhu, Dong, Li, Mason, Huang, ...) and it provides actually a field of growing interest in both mathematics and theoretical physics. On the one hand, one of the major achievements of 20th century's discrete mathematics was certainly the establishment of the classification of finite simple groups -- going along with the discovery of 21 until then unknown so-called sporadic groups. These groups are extremely interesting objects because unlike the other finite simple groups (${\mathbb Z}_p$, $p$ prime, the alternating groups, and the finite simple groups of Lie type) they do not belong to any infinite series and no unified description of all of them is known. The largest of the sporadic groups -- called the Monster -- had first been constructed by R. Griess as automorphism group of a 196 883-dimensional complex algebra. Later this algebra was recovered as subspace in a certain vertex operator algebra (usually denoted $V^{\natural}$) and it could be shown that the Monster is actually the automorphism group of $V^{\natural}$. The algebra $V^{\natural}$ seems to be the natural object to define the Monster and it is conjectured that other (maybe even all) sporadic simple groups arise as automorphism group of some vertex operator algebra as well. On the other hand, vertex operator algebras are essentially the chiral algebras of conformal quantum field theories and the latter -- providing a concept to describe symmetry in two-dimensional critical systems -- are one of the objects of greatest interest in modern physics. There exists a special class of quantum field theories called minimal models which are believed to constitute in a certain sense the building blocks of all other conformal field theories and the classification of all such minimal models is actually one of the most important problems in theoretical physics and has attracted growing research interest during the last years. The minimal models are naturally related to affine Kac-Moody algebras and the question about their existence turns into a question about the existence of certain modular invariant linear combinations of characters of these algebras. This problem has so far only been solved for certain cases -- the most important result being the so-called ``ADE-classification'' of all conformal field theories related to the algebra $A_1^{(1)}$ by Cappelli, Itzykson, and Zuber. (Partial) results for other algebras have been obtained among others by Coste, Ebholzer, Gannon, Gepner, Kato, Qiu, Roberts, Ruelle, Schellekens, Verstegen, ... and actual research is going on in this direction.\par Amazingly, modular invariance also arises in connection with the sporadic Monster group where it is usually referred to as ``Monstrous moonshine''. The most famous of these ``Moonshine conjectures'' states that the coefficients of the classical number theoretical $j$-function are the graded dimensions of a certain (infinite-dimensional) representation of the Monster. The construction of such a vector space with these properties -- namely $V^{\natural}$ -- had been provided by Frenkel, Lepowsky, and Meurman (1984/1988) and they also proved that the Monster is the full automorphism group of $V^{\natural}$. Later, Borcherds endowed $V^{\natural}$ with the structure of VOA and proved the part of the Conway-Norton conjecture that had not been established by Frenkel, Lepowsky, Meurman. \par There are other connections of the Monster as well as of conformal field theory with various areas of mathematics. E.g., in analogy to the well-known construction of lattices from codes, one can construct (affine) Lie algebras from lattices and vertex operator algebras from Lie algebras or directly from lattices or codes and properties of the lattice or code (even, self-dual, etc.) can be translated into properties of the obtained vertex operator algebra. The Monster vertex operator algebra $V^{\natural}$ is in fact obtained as ${\mathbb Z}_2$-orbifold from the Leech lattice. All the above suggests that the area of vertex operator algebras, finite simple groups, and conformal field theories is currently a very active field of mathematical and physical research and that there exist a lot of open problems whose solutions might have important consequences in various areas of mathematics and physics. Because of the interdependence of so different fields, it is important for researchers on any of the related topics to share their results and to collaborate. We suggest therefore to hold a conference on ``Algebraic Combinatorics, the Monster, and Vertex Operator Algebras'' from July 24th to 28th , 2000, at the University of California, Santa Cruz. We propose to have a series of survey talks covering the areas: vertex operator algebras, Kac-Moody algebras and their representations, constructions from codes and lattices, the sporadic simple Monster group, conformal field theory and modular invariance. Additionally, in each area there will be a number of accompanying 20-minute talks presenting current research. Such a gathering should give a better understanding of the philosophy and research ideas in all areas and hopefully lead to new result and avenues of study.

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