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Francisco Villarroya, Temple University
In this talk I will introduce some relatively new results that make a T1 Theory for compactness. These results completely characterize those Calderon-Zygmund operators that extend compactly on the appropriate Lebesgue spaces and at the standard endpoint spaces. The presentation will start with a brief introduction to the classical T1 Theory.
Nick Salter, Columbia University
Families of algebraic varieties exhibit a wide range of fascinating topological phenomena. Even families of zero-dimensional varieties (configurations of points on the Riemann sphere) and one-dimensional varieties (Riemann surfaces) have a rich theory closely related to the theory of braid groups and mapping class groups. In this talk, I will survey some recent work aimed at understanding one aspect of the topology of such families: the problem of (non)existence of continuous sections of “universal” families. Informally, these results give answers to the following sorts of questions: is it possible to choose a distinguished point on every Riemann surface of genus g in a continuous way? What if some extra data (e.g. a level structure) is specified? Can one instead specify a collection of n distinct points for some larger n? Or, in a different direction, if one is given a collection of n distinct points on CP^1, is there a rule to continuously assign an additional m distinct points? In this last case there is a remarkable relationship between n and m. For instance, we will see that there is a rule which produces 6 new points given 4 distinct points on CP^1, but there is no rule that produces 5 or 7, and when n is at least 6, m must be divisible by n(n-1)(n-2). These results are joint work with Lei Chen.
Durgesh Sinha, Temple University
Nick Miller, Indiana University
Both Reid and McMullen have independently asked whether a non-arithmetic hyperbolic 3-manifold necessarily contains only finitely many immersed geodesic surfaces. In this talk, I will discuss recent results where we show that a large class of non-arithmetic hyperbolic n-manifolds has only finitely many geodesic hypersurfaces, provided n is at least 3. Such manifolds are called hyperbolic hybrids and include the manifolds constructed by Gromov and Piatetski-Shapiro. These constitute the first examples of hyperbolic n-manifolds where the set of geodesic hypersurfaces is known to be finite and non-empty. Time allowing, I will also discuss the extension of these results to higher codimension. This is joint work with David Fisher, Jean-Francois Lafont, and Matthew Stover.
Zach Cline, Temple University
There are no conferences next week.