ar X iv : m at h / 03 07 23 0 v 1 [ m at h . A G ] 1 6 Ju l 2 00 3 EXPLICIT EQUATIONS OF SOME ELLIPTIC MODULAR SURFACES

ar X iv : m at h / 04 03 11 6 v 1 [ m at h . N T ] 6 M ar 2 00 4 Elliptic Curves x 3 + y 3 = k of High Rank

We use rational parametrizations of certain cubic surfaces and an explicit formula for descent via 3-isogeny to construct the first examples of elliptic curves Ek : x 3 + y = k of ranks 8, 9, 10, and 11 over Q. As a corollary we produce examples of elliptic curves over Q with a rational 3-torsion point and rank as high as 11. We also discuss the problem of finding the minimal curve Ek of a give...

ar X iv : m at h / 03 06 23 5 v 1 [ m at h . D G ] 1 6 Ju n 20 03 GEOMETRIC CONSTRUCTION OF MODULAR FUNCTORS FROM CONFORMAL FIELD THEORY JØRGEN

We give a geometric construct of a modular functor for any simple Lie-algebra and any level by twisting the constructions in [48] and [51] by a certain fractional power of the abelian theory first considered in [32] and further studied in [2].

ar X iv : 0 70 7 . 33 03 v 1 [ m at h . O A ] 2 3 Ju l 2 00 7 OPERATOR - VALUED FRAMES

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

ar X iv : 0 90 7 . 41 78 v 1 [ m at h . PR ] 2 3 Ju l 2 00 9 An Introduction to Stochastic PDEs

2 Some Motivating Examples 2 2.1 A model for a random string (polymer) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 The stochastic Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 The stochastic heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4 What have we learned? . . . . . . . . . . . . . . . . . . . . . ...