Nonnegative weak solutions of quasi-linear degenerate parabolic equations of $p$-Laplacian type are shown to be locally bounded below by Barenblatt-type subpotentials. As a consequence, nonnegative solutions expand their positivity set. That is, a quantitative lower bound on a ball $B_{\rho }$ at time $\stackrel{̲}{t}$ yields a quantitative lower bound on a ball $B_{2\rho }$ at some further time $t$. These lower bounds also permit one to recast the Harnack inequality of [4] in a family of alternative, equivalent forms

We introduce Weil representations for loop symplectic groups and prove the convergence and modularity of the related theta functions

We prove that the center of a regular block of parabolic category $O$ for the general linear Lie algebra is isomorphic to the cohomology algebra of a corresponding Springer fiber. This was conjectured by Khovanov [K]. We also find presentations for the centers of singular blocks, which are cohomology algebras of Spaltenstein varieties

Let $h_R$ denote an $L^{\infty }$-normalized Haar function adapted to a dyadic rectangle $R\subset [0,1{]}^3$. We show that there is a positive $\eta <1/2$ so that for all integers $n$ and coefficients $\alpha (R)$, we have ${\displaystyle 2^{-n}\sum _{\left|R\right|=2^{-n}}|\alpha (R)|\lesssim n^{1-\eta }\Vert \sum _{\left|R\right|=2^{-n}}\alpha (R)h_R{\Vert }_{\infty }.}$ This is an improvement over the trivial estimate by an amount of $n^{-\eta }$, while the small ball conjecture says that the inequality should hold with $\eta =1/2$. There is a corresponding lower bound on the $L^{\infty }$-norm of the discrepancy function of an arbitrary distribution of a finite number of points in the unit cube in three dimensions. The prior result, in dimension three, is that of József Beck [1, Theorem 1.2], in which the improvement over the trivial estimate was logarithmic in $n$. We find several simplifications and extensions of Beck's argument to prove the result above

We study the Wakimoto modules over the affine Kac-Moody algebras at the critical level from the point of view of the equivalences of categories proposed in our previous works, relating categories of representations and certain categories of sheaves. In particular, we explicitly describe geometric realizations of Wakimoto modules as Hecke eigen-D-modules on the affine Grassmannian and as quasi-coherent sheaves on the flag variety of the Langlands dual group