Please join us for pre-seminar Tea

I will discuss some recent work concerning variable coefficient extensions of Lp local smoothing estimates for the Schrodinger propagator. This can be thought of as a counterpart to a classical oscillatory integral operator bound of Bourgain (1991). Whilst Bourgain’s result relies on studying Kakeya sets of curves, our Lp local smoothing result relies on studying Nikodym sets of curves. An important observation of Wisewell (2005) is that the Nikodym theory is surprisingly different from the Kakeya theory. Our work aims to further investigate and exploit these differences. 

https://yale.zoom.us/j/91759982176?from=addon [5]

I will describe a joint project with Tobias Ekholm, Pietro Longhi, and Vivek Shende constructing a map from the HOMFLYPT skein module of a 3-manifold M to that of its branched cover arising from the projection of a Lagrangian 3-manifold L in the cotangent bundle of M. The map is defined by counting holomorphic curves and is a vast generalization of the quantum UV-IR map of Neitzke and Yan, which is a close cousin of the quantum trace map of Bonahon and Wong. The existence of this map has some interesting consequences in the theory of skein-valued curve counts, and I will discuss some of them if time permits.