The Gelfand Mathematical Seminars, 1990-1992, 1993

This Seminar began in Moscow in November 1943 and has continued without interruption up to the present. We are happy that with this vol­ ume, Birkhiiuser has begun to publish papers of talks from the Seminar. It was, unfortunately, difficult to organize their publication before 1990. Since 1990, most of the talks have taken place at Rutgers University in New Brunswick, New Jersey. Parallel seminars were also held in Moscow, and during July, 1992, at IRES in Bures-sur-Yvette, France. Speakers were invited to submit papers in their own style, and to elaborate on what they discussed in the Seminar. We hope that readers will find the diversity of styles appealing, and recognize that to some extent this reflects the diversity of styles in a mathematical society. The principal aim was to have interesting talks, even if the topic was not especially popular at the time. The papers listed in the Table of Contents reflect some of the rich variety of ideas presented in the Seminar. Not all the speakers submit­ ted papers. Among the interesting talks that influenced the seminar in an important way, let us mention, for example, that of R. Langlands on per­ colation theory and those of J. Conway and J. McKay on sporadic groups. In addition, there were many extemporaneous talks as well as short discus­ sions.

The Degeneracy of Two Spectral Sequences.- Hopf Algebra Structures for the Heisenberg Algebra: I.- Avalanches, Sandpiles and Tutte Decompositions.- On the Dimension and Degree of the Projective Dual Variety: A q-Analog of the Katz-Kleiman Formula.- Crofton Densities and Nonlocal Differentials.- On the Local Geometry of a Bihamiltonian Structure.- The Classical Polylogarithms, Algebraic K-Theory and ?F(N).- A Brief History of the Sporadic Simple Groups.- Vertex Operator Algebras and Operads.- Representations of the Quantized Function Algebras, 2-Categories and Zamolodchikov Tetrahedra Equation.- Formal (Non)-Commutative Symplectic Geometry.- Constructible Functions, Lagrangian Cycles and Computational Geometry.- Quantum Groups and Perverse Sheaves. An Example.- Linearly Recursive Sequences, Witt Algebras and Quantum Groups.- Complexes of Connected Graphs.