We relate the negative $K$-theory of a normal surface to a resolution of singularities. The only nonzero $K$-groups are $K\sb {-2}$, which counts loops in the exceptional fiber, and $K\sb {-1}$, which is related to the divisor class groups of the complete local rings at the singularities. We also verify two conjectures of Srinivas about $K\sb 0$-regularity and $K\sb {-1}$ of a surface.

This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety $X$. The motivation for this is M. Kontsevich's homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when dim $X\geq 2$, our braid group actions are always faithful.

We describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKay correspondence and with exceptional sheaves on Fano manifolds are given. Moreover, the case of an elliptic curve is worked out in some detail.

We prove several finiteness theorems for the normal bundles to souls in nonnegatively curved manifolds. More generally, we obtain finiteness results for open Riemannian manifolds whose topology is concentrated on compact domains of "bounded geometry."

Given a dynamical twist for a finite-dimensional Hopf algebra, we construct two weak Hopf algebras, using methods of P. Xu and of P. Etingof and A. Varchenko, and we show that they are dual to each other. We generalize the theory of dynamical quantum groups to the case when the quantum parameter q is a root of unity. These objects turn out to be self-dual—which is a fundamentally new property, not satisfied by the usual Drinfeld-Jimbo quantum groups.

Let x, y, and z be coprime positive integers with x+y=z. In this paper we give upper bounds for z in terms of the greatest square-free factor of xyz.

Using vertex operators, we construct explicitly Lusztig's ℤ[q,q^−4]-lattice for the level one irreducible representations of quantum affine algebras of ADE type. We then realize the level one irreducible modules at roots of unity and show that the character is given by the Weyl-Kac character formula.