The notion of a partial product space is a relatively recent unification of various combinatorial constructions in topology. This construction is variously known as the generalized moment-angle complex, or (more euphoniously) as the polyhedral product functor. Some instances of it are closely related to Davis and Januszkiewicz's quasitoric manifolds: these include the moment-angle complexes (Buchstaber and Panov) and homotopy orbit spaces for quasitoric manifolds. By making suitable choices, one also obtains classifying spaces for right-angled Artin groups and Coxeter groups, as well as certain real and complex subspace arrangements.

One advantage to this generality is that some topological information about such spaces can sometimes be expressed directly in combinatorial terms: presentations of cohomology rings; a homotopy-theoretic decomposition of the suspension of a partial product space; descriptions of rational homotopy Lie algebras and the Pontryagin algebra. I will give an introductory overview of some remarkable results along these lines.

Norm inequalities determine whether or not an operator acts as a bounded map between two Banach spaces. For a large range of indices an explicit parameterization gives, with best constant, all possible Lebesgue norm inequalities for positive integral operators. This result is outlined and extended to a class of nonlinear integral operators.

According to the Langlands Philosophy, every algebraic variety defined over the rationals or a number field should be modular (automorphic).

In this talk, I will concentrate on a special class of algebraic varieties, called Calabi-Yau varieties (of dimension at most three), defined over the rationals, and report on the current status of their modularity (automorphy).

A smooth proper variety with a zero cycle of degree one (that is, closed points of relatively prime degrees) need not have a rational point. In this talk we aim to explain how this phenomenon relates to the difference between unstable and stable $A^1$-homtopy theory.