Let G be an affine algebraic group over an algebraically closed field k of characteristic zero. In this paper, we consider finite G-equivariant morphisms F : X → Y of irreducible affine varieties. First we determine under which conditions on Y the induced map FG : X//G → Y//G of quotient varieties is also finite. This result is reformulated in terms of kernels of derivations on k-algebras A ⊂ B...

The study of m-primary irreducible ideals in a commutative graded connected Noetherean algebra over a field is in principal equivalent to the study of the corresponding quotient algebras. Such algebras are Poincaré duality algebras. The prototype of such an algebra (apart from the cosmetic difference of being graded commutative rather than commutative graded) is the cohomology with field coeffi...

Abstract. In these notes we discuss some equidistribution problems with the aim to give reasonable error rates, i.e. we are interested in effective statements. We motivate some arguments by studying a concrete problem on a two-torus, and then describe recent results on the equidistribution of semisimple orbits obtained in joint work with G. Margulis and A. Venkatesh. We end by studying the rela...

The critical group of an abelian network is a finite abelian group that governs the behavior of the network on large inputs. It generalizes the sandpile group of a graph. We show that the critical group of an irreducible abelian network acts freely and transitively on recurrent states of the network. We exhibit the critical group as a quotient of a free abelian group by a subgroup containing th...

The braid group Bn is the mapping class group of an n-times punctured disk. The Iwahori-Hecke algebra Hn is a quotient of the braid group algebra of Bn by a quadratic relation in the standard generators. We discuss how to use Hn to define the Jones polynomial of a knot or link. We also summarize the classification of the irreducible representations of Hn. We conclude with some directions for fu...

A Shimura surface is the quotient of either the product H×H of two hyperbolic planes or the unit ball HC in C by an irreducible arithmetic lattice. Examples include the normal quasiprojective varieties associated with the Hilbert and Picard modular groups, along with the solutions to many moduli problems for principally polarized abelian varieties. Special amongst the immersed projective algebr...

Let P be a parabolic subgroup of a semisimple simply connected linear algebraic group G over C and ρ an irreducible homomorphism from P to a complex reductive group H . We show that the associated principal H –bundle over G/P , associated for ρ to the principal P –bundle defined by the quotient map G −→ G/P , is stable. We describe the Harder–Narasimhan reduction of the G–bundle over G/P obtain...

A Shimura surface is the quotient of either the product H×H of two hyperbolic planes or the unit ball HC in C by an irreducible arithmetic lattice. Examples include the normal quasiprojective varieties associated with the Hilbert and Picard modular groups, along with the solutions to many moduli problems for principally polarized abelian varieties. Special amongst the immersed projective algebr...

Abstract. One of the main purposes of this paper is to prove that on a complete Kähler manifold of dimension m, if the holomorphic bisectional curvature is bounded from below by -1 and the minimum spectrum λ1(M) ≥ m, then it must either be connected at infinity or diffeomorphic to R × N , where N is a compact quotient of the Heisenberg group. Similar type results are also proven for irreducible...

We study geodesically complete and locally compact Hadamard spaces X whose Tits boundary is a connected irreducible spherical building. We show that X is symmetric iff complete geodesics in X do not branch and a Euclidean building otherwise. Furthermore, every boundary equivalence (cone topology homeomorphism preserving the Tits metric) between two such spaces is induced by a homothety. As an a...