Abstract: In this talk, I will give some basic definitions and facts about topological and Lie groupoids. I will describe some examples involving group actions and configuration spaces of geometric structures. I will outline some work analogs of standard constructions in differential geometry, such as differential forms, in this context. I will also describe how these same examples can be described in terms of other kinds of structures: stacks (specifically, topological and differentiable stacks), and $C*$-algebras associated to the groupoids. This talk will be a fairly introductory presentation of the subjects of this seminar.

Abstract: The subject of the talk is resolution of singularities in algebraic or analytic geometry. Resolution of singularities leads to a space with only normal crossings singularities (i.e., transverse self-intersections). It therefore makes sense to consider normal crossings singularities acceptable from the start, and try to resolve singularities except for normal crossings. We will discuss the following question (a variant of a problem of Janos Kollar). Can we find the smallest class of singularities S with the following properties: (1) S includes all normal-crossings singularities; (2) every variety or space X admits a proper mapping f: X' --> X such that X' has only singularities in S, and f is an isomorphism over the locus of points of X having only singularities in S? No technical background will be assumed.