ar X iv : m at h / 04 07 22 3 v 1 [ m at h . D G ] 1 3 Ju l 2 00 4 C 1 HYPERSURFACES OF THE HEISENBERG GROUP ARE N - RECTIFIABLE

ar X iv : m at h / 04 09 02 9 v 1 [ m at h . A G ] 2 S ep 2 00 4 ACM BUNDLES ON GENERAL HYPERSURFACES IN P 5 OF LOW DEGREE

In this paper we show that on a general hypersurface of degree r = 3, 4, 5, 6 in P 5 a rank 2 vector bundle E splits if and only if h 1 E(n) = h 2 E(n) = 0 for all n ∈ Z. Similar results for r = 1, 2 were obtained in [15], [16] and [1].

ar X iv : m at h / 02 07 25 7 v 1 [ m at h . A G ] 2 7 Ju l 2 00 2 RATIONAL CURVES ON HYPERSURFACES OF LOW DEGREE , II

This is a continuation of [7] in which we proved irreducibility of spaces of rational curves on a general hypersurface X d ⊂ P n of degree d < n+1 2. In this paper, we prove that if d 2 + d + 2 ≤ n and if d ≥ 3, then the spaces of rational curves are themselves rationally connected.

ar X iv : m at h / 04 07 28 6 v 1 [ m at h . PR ] 1 6 Ju l 2 00 4 1 ASYMPTOTIC BEHAVIOR OF RANDOM HEAPS

Let Wn be a random walk on the locally free group with m generators subject to periodic boundary conditions, and let T (Wn) be the set of removable elements. We prove that limn→∞ E(#T (Wn)) m ≤ 0.32893 for m ≥ 4. This result disproves a conjecture (due to Vershik et. al. [3]) that the limit tends to 1 3 as m → ∞.

ar X iv : m at h / 04 09 56 9 v 4 [ m at h . A G ] 1 7 Ju l 2 00 6 INTERMEDIATE MODULI SPACES OF STABLE MAPS

We describe the Chow ring with rational coefficients of M0,1(P , d) as the subring of invariants of a ring B(M0,1(P , d);Q), relative to the action of the group of symmetries Sd. We compute B(M0,1(P , d);Q) by following a sequence of intermediate spaces for M0,1(P , d).

ar X iv : m at h / 03 09 40 8 v 2 [ m at h . D G ] 1 2 Ju l 2 00 4 EINSTEIN METRICS ON SPHERES