One basic example is the configuration space of unordered k-tuples of distinct points in a space M. When specialized to the case where M is given by the complex numbers, these spaces can be identified as the space of complex, monic polynomials of degree k which have exactly k distinct roots. Features of these spaces as well as their connections to knots, links, and homotopy groups will be addressed.

A second subspace of a product is the generalized moment-angle complex with notable cases given by subspaces of products of infinite dimensional complex projective space 'indexed by a finite simplicial complex'. These spaces appearing in work of Goresky-MacPherson, Buchstaber-Panov-Ray, Denham-Suciu, Franz as well as many others encode information ranging from the structure of toric varieties in one guise as well as 'motions of certain types of robotic legs' in another guise.

Further elementary features of these spaces are developed within the context of classical homotopy theory in joint work with A. Bahri, M. Bendersky, and S. Gitler. These elementary properties provide easily accessible explantions for certain properties of moment-angle complexes as well as further, delicate features

The main direction of this topic is the structure for the space of homomorphisms from a free abelian group to a Lie group as well as associated quotients spaces obtained from the adjoint representation, the associated space of representations. These spaces admit the additional structure of a simplicial space at the heart of work of Jardine. What is the fundamental group or the first homology group of the associated space ? This deceptively elementary question as well as more global information is the subject of this talk based on joint work with A. Adem, E. Torres, and J. Gomez.

Given a nonzero integer d, we know, by Hermite's Theorem, that there exist only finitely many cubic number fields of discriminant d. A natural question is, how to refine the discriminant in such way that we can tell, when two of these fields are isomorphic. Here we consider the binary quadratic form q_K: Tr_{K/ mathbb{Q}}(x2)|_{O^0_K}, and we show that if d is a positive fundamental discriminant, then the isomorphism class of q_K, as a quadratic form over Z², gives such a refinement.