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Abstract: I will give a brief historical introduction and then will report on joint work with M. Tvalavadze, where we prove, in particular, that the first integral homology group of an irreducible component of the variety of graded n-dimensional complex Lie algebras, under certain assumptions, is trivial.

Abstract: I will show how to associate a convex body to a finite dimensional subspace L of rational functions on an n-dimensional variety X. This rather simple construction generalizes the well-known construction of a Newton polytope (in toric geometry). We will see how volume of this convex body is responsible for the number of solutions of a generic system of equations f_1= ...= f_n = 0 from L, which can be regarded as a far generalization of Kushnirenko theorem. This then enables one to apply inequalities in convex geometry to algebraic geometry. As an example, using classical isoperimetric inequality, we get simple and elementary proofs of Hodge index theorem as well as Alexander-Fenchel inequality (for mixed volume of convex bodies). If time permits I'll discuss the relation with the concept of integral closure of a subspace of rational functions.

Abstract: