New antiparasitic drugs are urgently needed to treat and control diseases such as malaria, leishmaniasis, sleeping sickness and filariasis, which affect millions of people each year. However, because the majority of those infected live in countries in which the prospects of any financial return on investment are too low to support market-driven drug discovery and development, alternative approaches are needed. In this article, challenges and opportunities for antiparasitic drug discovery are considered, highlighting some of the progress that has been made in recent years, partly through scientific advances, but also by more effective partnership between the public and private sectors.

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Summary.We propose a conjecture combining the Mordell-Lang conjecture with an important special case of the André-Oort conjecture, and explain how existing results imply evidence for it.

The general problem underlying this article is to give a qualitative classification Ž . of all compact subgroups ⌫ ; GL F , where F is a local field and n is arbitrary. Ž .It is natural to ask whether ⌫ is an open compact subgroup of H E , where H is a linear algebraic group over a closed subfield E ; F. We show that ⌫ indeed has this form, up to finite index and a finite number of abelian subquotients. When ⌫ is Zariski dense in a connected semisimple group, we give a precise openness result Ž . for the closure of the commutator group of ⌫. In the case char F s 0 the answers have long been known by results of Chevalley and Weyl. The motivation for this work comes from the positive characteristic case, where such results are needed to study Galois representations associated to function fields. We also derive openness results over a finite number of local fields. ᮊ

Let F q be a fixed finite field of cardinality q. An F -zip over a scheme S over F q is a certain object of semi-linear algebra consisting of a locally free sheaf of O Smodules with a descending filtration and an ascending filtration and a Frob q -twisted isomorphism between the respective graded sheaves. In this article we define and systematically investigate what might be called "F -zips with a G-structure", for an arbitrary reductive linear algebraic group G over F q .These objects come in two incarnations. One incarnation is an exact F q -linear tensor functor from the category of finite dimensional representations of G over F q to the category of F -zips over S. Locally any such functor has a type χ, which is a cocharacter of G k for a finite extension k of F q that determines the ranks of the graded pieces of the filtrations. The other incarnation is a certain G-torsor analogue of the notion of F -zips. We prove that both incarnations define stacks that are naturally equivalent to a quotient stack of the form [E G,χ \G k ] that was studied in our earlier paper [18]. By the results of [18] they are therefore smooth algebraic stacks of dimension 0 over k. Using [18] we can also classify the isomorphism classes of such objects over an algebraically closed field, describe their automorphism groups, and determine which isomorphism classes can degenerate into which others.For classical groups we can deduce the corresponding results for twisted or untwisted symplectic, orthogonal, or unitary F -zips, a part of which has been described before in [16]. The results can be applied to the algebraic de Rham cohomology of smooth projective varieties (or generalizations thereof) and to truncated BarsottiTate groups of level 1. In addition, we hope that our systematic group theoretical approach will help to understand the analogue of the Ekedahl-Oort stratification of the special fibers of arbitrary Shimura varieties.

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Generalizing a classical theorem of Jordan to arbitrary characteristic, we prove that every finite subgroup of GL n \operatorname {GL}_n over a field of any characteristic p p possesses a subgroup of bounded index which is composed of finite simple groups of Lie type in characteristic  p p , a commutative group of order prime to  p p , and a p p -group. While this statement can be deduced from the classification of finite simple groups, our proof is self-contained and uses methods only from algebraic geometry and the theory of linear algebraic groups. We believe that our results can serve as a viable substitute for classification in a range of applications in various areas of mathematics.

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