The following stochastic flow
\[
d\mathbf{r}=\mathbf{v}dt,\quad d\mathbf{v}=-(\mathbf{v/}\tau \mathbf{)}%
dt+d\mathbf w(t,\mathbf{r)},\quad \mathbf{r,v\in }R^{2}
\]
is considered which \ is used to describe tracer particles in turbulent
flow, drifters in the upper ocean, cloud formation, ultrasonic aggregation
of aerosols, mammal migration, iterating functions, and other phenomena. An
exact expression for the top Lyapunov exponent of the flow is given for
isotropic Brownian forcing $\mathbf w(t,\mathbf{r)}$ in terms of Airy functions.

Alfors $d$-regular set is a class of fractal sets which
contains geometrically self-similar sets.
In this paper, we investigate symmetric jump-type processes
on $d$-sets with jumping intensities comparable
to radially symmetric functions on $d$-regular sets.
A typical example is the symmetric jump process with jumping intensity
$$
\int_{\alpha_1}^{\alpha_2} \frac{c(\alpha, x, y)}{|x-y|^{d+\alpha}} \,
\nu (d\alpha),
$$
where $\nu$ is a probability measure on $[\alpha_1, \alpha_2]\subset (0, 2)$, and $c(\alpha, x, y)$ is a jointly measurable function that is symmetric in $(x, y)$ and is bounded between two positive constants.
We establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such jump-type processes.

This is a joint work with Takashi Kumagai

It is a classic result of Kollar, Miyaoka and Mori, that the family of all smooth Fano varieties is bounded. Batyrev conjectured that the same holds if one drops the hypothesis on smoothness and adds the hypothesis that some fixed multiple of the canonical divisor is Cartier and that the singularities are log terminal. We prove Batyrev's conjecture.
It suffices to find a bound on the degree of the anticanonical. The
classic proof proceeds in two steps. The first is to find an element of the linear system of some high multiple of the anticanonical and exhibit an element of this linear system which is very singular at any given point. This is the method of Fano. The second, the hardest step, is to exhibit a rational curve, through two general points, of low degree. Unfortunately it seems hard to generalise this idea to the singular case, since it is hard to compute intersection multiplicities on singular varieties. Instead we produce covering families of low degree subvarieties, which automatically have large intersection multiplicities with elements of the pluri anti canonical system.