Providing Access to Safe Water: Lessons Learned from Two Decades of Philanthropic Investment in the Rural Poor

The Conrad N. Hilton Foundation has been a leading U.S. funder for increasing safe water access for over 20 years. The author reflects on that history to describe valuable lessons, especially on partnership (in the context of West Africa Water Initiative), and how to think strategically in the long- and short-term about efficient and sustainable WASH funding

Periodicity in the cohomology of symmetric groups via divided powers

A famous theorem of Nakaoka asserts that the cohomology of the symmetric group stabilizes. The first author generalized this theorem to non-trivial coefficient systems, in the form of $\mathrm{FI}$-modules over a field, though one now obtains periodicity of the cohomology instead of stability. In this paper, we further refine these results. Our main theorem states that if $M$ is a finitely generated $\mathrm{FI}$-module over a noetherian ring $\mathbf{k}$ then $\bigoplus_{n \ge 0} \mathrm{H}^t(S_n, M_n)$ admits the structure of a $\mathbf{D}$-module, where $\mathbf{D}$ is the divided power algebra over $\mathbf{k}$ in a single variable, and moreover, this $\mathbf{D}$-module is "nearly" finitely presented. This immediately recovers the periodicity result when $\mathbf{k}$ is a field, but also shows, for example, how the torsion varies with $n$ when $\mathbf{k}=\mathbf{Z}$. Using the theory of connections on $\mathbf{D}$-modules, we establish sharp bounds on the period in the case where $\mathbf{k}$ is a field. We apply our theory to obtain results on the modular cohomology of Specht modules and the integral cohomology of unordered configuration spaces of manifolds.Comment: Fixed some minor mistakes and expanded the section on configuration space

A comprehensive class of harmonic functions defined by convolution and its connection with integral transforms and hypergeometric functions

For given two harmonic functions $\Phi$ and $\Psi$ with real coefficients in the open unit disk $\mathbb{D}$, we study a class of harmonic functions $f(z)=z-\sum_{n=2}^{\infty}A_nz^{n}+\sum_{n=1}^{\infty}B_n\bar{z}^n$ $(A_n, B_n \geq 0)$ satisfying \RE \frac{(f*\Phi)(z)}{(f*\Psi)(z)}>\alpha \quad (0\leq \alpha <1, z \in \mathbb{D}); * being the harmonic convolution. Coefficient inequalities, growth and covering theorems, as well as closure theorems are determined. The results obtained extend several known results as special cases. In addition, we study the class of harmonic functions $f$ that satisfy \RE f(z)/z>\alpha $(0\leq \alpha <1, z \in \mathbb{D})$. As an application, their connection with certain integral transforms and hypergeometric functions is established.Comment: 14pages, 1 figur