We consider the spreading of a droplet on a planar surface covered
with random obstacles. Assuming the obstacles are stationary ergodic
and taller than the droplet, we show that the homogenized limit is
described by a droplet spreading on a flat surface but with a reduced
front speed and surface tension. This is joint work with Inwon Kim.

The continuity of Lyapunov exponent plays an important role for many problems in quasi-periodic cocycles. One example is Ten Martini problem. It is well known that the Lyapunov exponent is continuous in analytic topology and discontinuous in C^0-topology. In this talk, we will provide quasi-periodic cocycles at which the Lyapunov exponent is not continuous in C^l-topology with 0 ≤ l ≤ +∞. This is joint work with Jiangong You.

The parity conjecture is a weak version of Birch-Swinnerton-Dyer
Conjecture or more generally, Beilinson-Bloch-Kato Conjecture. It is
conjectured that the vanishing order of the L-function at the central
point has the same parity as the dimension of the Bloch-Kato Selmer
group. I will explain an approach to this conjecture for modular
forms by varying the modular forms in a p-adic family. This is a joint
work with Kiran Kedlaya and Jay Pottharst.

The parity conjecture is a weak version of Birch-Swinnerton-Dyer
Conjecture or more generally, Beilinson-Bloch-Kato Conjecture. It is
conjectured that the vanishing order of the L-function at the central
point has the same parity as the dimension of the Bloch-Kato Selmer
group. I will explain an approach to this conjecture for modular
forms by varying the modular forms in a p-adic family. This is a joint
work with Kiran Kedlaya and Jay Pottharst.