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Based on large N Chern-Simons/topological string duality, in a series of papers [36, 21, 19], Labastida, Marino, Ooguri, and Vafa conjectured certain remarkable new algebraic structure of link invariants and the existence of infinite series of new integer invariants. In this paper, we provide a proof of this conjecture. Moreover, we also show these new integer invariants vanish at large genera.

We apply a study of orders in quaternion algebras, to the differential geometry of Riemann surfaces. The least length of a closed geodesic on a hyperbolic surface is called its systole,and denoted sysπ1. P. Buser and P. Sarnak constructed Riemann surfaces X whose systole behaves logarithmically in the genus g(X). The Fuchsian groups in their examples are principal congruence subgroups of a fixed arithmetic group with rational trace field. We generalize their construction to principal congruence subgroups of arbitrary arithmetic surfaces. The key tool is a new trace estimate valid for an arbitrary ideal in a quaternion algebra. We obtain a particularly sharp bound for a principal congruence tower of Hurwitz surfaces (PCH), namely the 4/3bound sysπ1(XPCH) ≥ 43 log(g(XPCH)). Similar results are obtained for the systole of hyperbolic 3-manifolds, relative to their simplicial volume.

Being inspired by a work of Curtis T. McMullen about a very impressive automorphism of a K3 surface of Picard number zero, we shall clarify the structure of the bimeromorphic automorphism group of a non-projective hyperk¨ahler manifold, up to finite group factor.

For all open Riemann surface N and real number  2 (0, /2), we construct a conformal minimal immersion X = (X1,X2,X3) : N ! R3 such that X3+tan()|X1| : N ! R is positive and proper. Furthermore, X can be chosen with an arbitrarily prescribed flux map. Moreover, we produce properly immersed hyperbolic minimal surfaces with non-empty boundary in R3 lying above a negative sublinear graph.