Georgia Institute of Technology College of Sciences

The Fourier restriction conjecture and the Bochner-Riesz conjecture ask for Lebesgue space mapping properties of certain oscillatory integral operators. They both are central in harmonic analysis, are open in dimensions $\geq 3$, and notably have the same conjectured exponents. In the 1970s, H\"{o}rmander asked if a more general class of operators (known as H\"{o}rmander type operators) all satisfy the same $L^p$-boundedness as in the above two conjectures. A positive answer to H\"{o}rmander's question would resolve the above two conjectures and have more applications such as in the manifold setting. Unfortunately H\"{o}rmander's question is known to fail in all dimensions $\geq 3$ by the work of Bourgain and many others. It continues to fail in all dimensions $\geq 3$ even if one adds a ``positive curvature'' assumption which one does have in restriction and Bochner-Riesz settings. Bourgain showed that in dimension $3$ one always has the failure unless a derivative condition is satisfied everywhere. Joint with Shaoming Guo and Hong Wang, we generalize this condition to arbitrary dimension and call it ``Bourgain's condition''. We unify Fourier restriction and Bochner-Riesz by conjecturing that any H\"{o}rmander type operator satisfying Bourgain's condition should have the same $L^p$-boundedness as in those two conjectures. As evidence, we prove that the failure of Bourgain's condition immediately implies the failure of such an $L^p$-boundedness in every dimension. We also prove that current techniques on the two conjectures apply equally well in our conjecture and make some progress on our conjecture that consequently improves the two conjectures in higher dimensions. I will talk about some history and some interesting components in our proof.

In this talk, we will explore and make comparisons between various models that exist for spherical tensor categories associated to the category of representations of the quantum group U_q(SL_n). In particular, we will discuss the combinatorial model of Murakami-Ohtsuki-Yamada (MOY), the n-valent ribbon model of Sikora and the trivalent spider category of Cautis-Kamnitzer-Morrison (CKM). We conclude by showing that the full subcategory of the spider category from CKM, whose objects are monoidally generated by the standard representation and its dual, is equivalent as a spherical braided category to Sikora's quotient category. This proves a conjecture of Le and Sikora and also answers a question from Morrison's Ph.D. thesis.

A fundamental conjecture of tight closure theory is every weakly F-regular ring is strongly F -regular. There has been incremental progress on this conjecture since the inception of tight closure. Most notably, the conjecture has been resolved for rings graded over a field by Lyubeznik and Smith. Otherwise, known progress around the conjecture have required assumptions on the ring that are akin to being Gorenstein. We extend known cases by proving the equivalence of F -regularity classes for rings whose anti-canonical algebra is Noetherian on the punctured spectrum. The anti-canonical algebra being Noetherian for a strongly F -regular ring is conjectured to be a vacuous assumption. This talk is based on joint work with Ian Aberbach and Craig Huneke.