In the upper half of the unit ball $B\sp + =\{ |x|<1,x\sb 1>0\}$, let $u$ and $\Omega$ (a domain in $\mathbf {R}\sp n\sb + =\{x\in \mathbf {R}\sp n : x\sb 1>0\}$) solve the following overdetermined problem:

\Delta u =\chi_\Omega\quad \text{in}\ B^+, \qquad u=|\nabla u |=0 \quad \text{in}\ B^+\setminus \Omega, \qquad u=0 \quad \text{on}\ \Pi\cap B,

where $B$ is the unit ball with center at the origin, $\chi\sb \Omega$ denotes the characteristic function of $\Delta,\Pi=\{ x\sb 1=0\} ,n\geq 2$, and the equation is satisfied in the sense of distributions. We show (among other things) that if the origin is a contact point of the free boundary, that is, if $u(0)=|\nabla u(0)|=0$, then $\partial\Delta\cap B\sb {r\sb 0}$ is the graph of a $C\sp 1$-function over $\Pi\cap B\sb (r\sb 0)$. The $C\sp 1$-norm depends on the dimension and $\sup\sb {B\sp +}|u|$. The result is extended to general subdomains of the unit ball with $C\sp 3$-boundary.

We give a link–through the affine Schur algebra–between the representations of the $p$-affine Schur algebra of ${\rm GL}(n)$ over $R$ and the smooth $R$-representations of the $p$-adic group ${\rm GL}(n,\mathbf {Q}\sb p)$ over any algebraically closed field $R$ of characteristic not equal to $p$.

This paper addresses J. Cheeger and D. Gromoll's question about which vector bundles admit a complete metric of nonnegative curvature, and it relates their question to the issue of which sphere bundles admit a metric of positive curvature. We show that any vector bundle that admits a metric of nonnegative curvature must admit a connection, a tensor, and a metric on the base space, which together satisfy a certain differential inequality. On the other hand, a slight sharpening of this condition is sufficient for the associated sphere bundle to admit a metric of positive curvature. Our results sharpen and generalize M. Strake and G. Walschap's conditions under which a vector bundle admits a connection metric of nonnegative curvature.

Let $\lambda$ be the space of symmetric functions, and let $V\ sb k$ be the subspace spanned by the modified Schur functions $\{S\sb \lambda[X/(1-t)]\}\sb {\lambda\sb 1\leq k}$. We introduce a new family of symmetric polynomials, $\{A\sp {(k)}\sb \lambda[X;t]\}\sp {\lambda\sb 1\leq k}$, constructed from sums of tableaux using the charge statistic. We conjecture that the polynomials $A\sp {(k)}\sb \lambda[X;t]$ form a basis for $V\sb k$ and that the Macdonald polynomials indexed by partitions whose first part is not larger than $k$ expand positively in terms of our polynomials. A proof of this conjecture would not only imply the Macdonald positivity conjecture, but also substantially refine it. Our construction of the $A\sp {(k)}\sb \lambda[X;t]$ relies on the use of tableau combinatorics and yields various properties and conjectures on the nature of these polynomials. Another important development following from our investigation is that the $A\sp {(k)}\sb \lambda[X;t]$ seem to play the same role for $V\sb k$ as the Schur functions do for $\lambda$. In particular, this has led us to the discovery of many generalizations of properties held by the Schur functions, such as Pieri-type and Littlewood-Richardson-type coefficients.

We show that a certain class of varieties with origin in physics generates (additively) the Denef-Loeser ring of motives. In particular, this disproves a conjecture of M. Kontsevich on the number of points of these varieties over finite fields.