ar X iv : m at h / 04 07 28 6 v 2 [ m at h . PR ] 6 M ar 2 00 5 1 ASYMPTOTIC BEHAVIOR OF RANDOM HEAPS

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 / 05 04 08 2 v 4 [ m at h . D G ] 1 5 A pr 2 00 6 COMPLETE PROJECTIVE CONNECTIONS

The first examples of complete projective connections are uncovered: on surfaces, normal projective connections whose geodesics are all closed and embedded are complete. On manifolds of any dimension, normal projective connections induced from complete affine connections with slowly decaying positive Ricci curvature are complete.

ar X iv : m at h / 02 04 28 5 v 1 [ m at h . G T ] 2 3 A pr 2 00 2 FIBER SUMS OF GENUS 2 LEFSCHETZ FIBRATIONS

Using the recent results of Siebert and Tian about the holomorphicity of genus 2 Lefschetz fibrations with irreducible singular fibers, we show that any genus 2 Lefschetz fibration becomes holomorphic after fiber sum with a holomorphic fibration.

ar X iv : m at h / 06 04 09 1 v 1 [ m at h . Q A ] 5 A pr 2 00 6 QUANTUM INVARIANTS , MODULAR FORMS , AND LATTICE POINTS II

We study the SU(2) Witten–Reshetikhin–Turaev invariant for the Seifert fibered homology spheres with M-exceptional fibers. We show that the WRT invariant can be written in terms of (differential of) the Eichler integrals of modular forms with weight 1/2 and 3/2. By use of nearly modular property of the Eichler integrals we shall obtain asymptotic expansions of the WRT invariant in the large-N l...

ar X iv : m at h / 05 04 37 3 v 1 [ m at h . Q A ] 1 9 A pr 2 00 5 Lax Operator for the Quantised Orthosymplectic