Let X be a rational nonsingular compact connected real algebraic surface. Denote by Aut(X) the group of real algebraic automorphisms of X. We show that the group Aut(X) acts n-transitively on X, for all natural integers n. As an application we give a new and simpler proof of the fact that two rational nonsingular compact connected real algebraic surfaces are isomorphic if and only if they are h...

We discuss the algebraic structure of the spin chains related to high energy scattering in QCD. We study the sl(2) Yangian symmetry and possible generalizations to nonzero spin and anisotropy parameter.

Discriminating groups were introduced by G.Baumslag, A.Myasnikov and V.Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. However they have taken on a life of their own and have been an object of a considerable amount of study. In this paper we survey the large array results concerning the class of discriminating groups that have been developed over the past decade.

This week we will be studying the tropical analogue of Bezout’s theorem, which in classical geometry gives a precise count of the number of intersections between two projective algebraic curves. It turns out, interestingly, that although all the definitions (except that of intersection) are different in tropical geometry, we can also prove a theorem with the same statement. Next time, we will s...

When a fixed algebraic variety in a multiplicative group variety is intersected with the union of all algebraic subgroups of fixed dimension, a key role is played by what we call the anomalous subvarieties. These arise when the algebraic variety meets translates of subgroups in sets larger than expected. We prove a Structure Theorem for the anomalous subvarieties, and we give some applications,...

The additivity theorem in algebraic K-theory, due to Quillen and Waldhausen, is a basic tool. In this paper we present a new proof, which proceeds by constructing an explicit homotopy combinatorially.

Chow varieties are a parameter space for cycles of a given variety of a given codimension and degree. We construct their analog for differential algebraic varieties with differential algebraic subvarieties, answering a question of [11]. The proof uses the construction of classical algebro-geometric Chow varieties, the theory of characteristic sets of differential varieties, the theory of prolon...

Let Z be a germ of a reduced analytic space of pure dimension. We provide an analytic proof of the uniform BriançonSkoda theorem for the local ring OZ ; a result which was previously proved by Huneke by algebraic methods. For ideals with few generators we also get much sharper results.

Let F be an algebraically closed field and let G be a semisimple F-algebraic group for which the characteristic of F is very good. If X ∈ Lie(G) = Lie(G)(F) is a nilpotent element in the Lie algebra of G, and if C is the centralizer in G of X, we show that (i) the root datum of a Levi factor of C, and (ii) the component group C/Co both depend only on the Bala-Carter label of X; i.e. both are in...

My purpose here is to convey to you some idea of the scope and the depth of the work for which we are today honoring Pierre Deligne with the Fields Medal. Deligne's work centers around the remarkable relations, first envisioned by Weil, which exist between the cohomological structure of algebraic varieties over the complex numbers, and the diophantine structure of algebraic varieties over finit...