ar X iv : m at h / 06 09 09 3 v 3 [ m at h . A G ] 2 1 Fe b 20 07 INVARIANTS OF NEWTON NON - DEGENERATE SURFACE SINGULARITIES

ar X iv : m at h / 06 11 07 6 v 4 [ m at h . G T ] 1 4 Fe b 20 09 Virtual Homotopy

Two welded (respectively virtual) link diagrams are homotopic if one may be transformed into the other by a sequence of extended Reidemeister moves, classical Reidemeister moves, and self crossing changes. In this paper, we extend Milnor’s μ and μ̄ invariants to welded and virtual links. We conclude this paper with several examples, and compute the μ invariants using the Magnus expansion and Pol...

ar X iv : m at h / 06 02 37 1 v 1 [ m at h . A G ] 1 7 Fe b 20 06 Braid Monodromy of Hypersurface Singularities

ar X iv : 0 90 2 . 39 12 v 1 [ m at h . G R ] 2 3 Fe b 20 09 Brent Everitt The Combinatorial Topology

ar X iv : m at h / 06 02 64 7 v 1 [ m at h . A G ] 2 8 Fe b 20 06 HIGHER FANO MANIFOLDS AND RATIONAL SURFACES

Let X be a Fano manifold of pseudo-index ≥ 3 such that c 1 (X) 2 − 2c 2 (X) is nef. Irreducibility of some spaces of rational curves on X (in fact, a weaker hypothesis) implies a general point of X is contained in a rational surface.

ar X iv : m at h / 99 06 09 7 v 2 [ m at h . C O ] 3 1 A ug 1 99 9 BOUNDS ON ARITHMETIC PROJECTIONS , AND APPLICATIONS TO THE KAKEYA CONJECTURE

Let A, B, be finite subsets of an abelian group, and let G ⊂ A × B be such that #A, #B, #{a + b : (a, b) ∈ G} ≤ N. We consider the question of estimating the quantity #{a − b : (a, b) ∈ G}. In [2] Bourgain obtained the bound of N 2− 1 13 , and applied this to the Kakeya conjecture. We improve Bourgain's estimate to N