We study Whittaker functions on nonlinear coverings of simple algebraic groups over a nonarchimedean local field. We produce a recipe for expressing such a Whittaker function as a weighted sum over a crystal graph and show that in type A, these expressions agree with known formulae for the prime-power-supported coefficients of multiple Dirichlet series.

We prove that the stably free modules over a smooth affine threefold over an algebraically closed field of characteristic different from $2$ and $3$ are free.

We prove that the Poisson deformation functor of an affine (singular) symplectic variety is unobstructed. As a corollary, we prove the following result. For an affine symplectic variety $X$ with a good $C^{*}$-action (where its natural Poisson structure is positively weighted), the following are equivalent.

(1) $X$ has a crepant projective resolution.

(2) $X$ has a smoothing by a Poisson deformation.

A typical example is (the normalization) of a nilpotent orbit closure in a complex simple Lie algebra. By the theorem, one can see which orbit closure has a smoothing by a Poisson deformation.

We consider the Kähler-Ricci flow on complete finite-volume metrics that live on the complement of a divisor in a compact Kähler manifold $\stackrel{̲}{X}$. Assuming certain spatial asymptotics on the initial metric, we compute the singularity time in terms of cohomological data on $\stackrel{̲}{X}$. We also give a sufficient condition for the singularity, if there is one, to be type II.

The twistor space of the sphere $S^{2n}$ is an isotropic Grassmannian that fibers over $S^{2n}$. An orthogonal complex structure (OCS) on a subdomain of $S^{2n}$ (a complex structure compatible with the round metric) determines a section of this fibration with holomorphic image. In this article, we use this correspondence to prove that any finite energy OCS on ${\mathbb{R}}^6\subset S^6$ must be of a special warped product form, and we also prove that any OCS on $R^{2n}$ that is asymptotically constant must itself be constant. We give examples defined on $R^{2n}$ which have infinite energy and examples of nonstandard OCSs on flat tori in complex dimension $3$ and greater.