Sunrise and the continuity of maximal operators
Abstract: We will explore how some insightful constructions related to the classical sunrise lemma in harmonic analysis can be used in connection to the endpoint continuity of certain maximal operators in Sobolev spaces. The statement of the main theorem of the lecture will look rather innocent and inviting, which will turn out to be somewhat deceiving. The proof will reveal the beautiful and subtle maze to be explored. This is based in a joint work with Jose Madrid and Cristian Gonzalez-Riquelme.
This talk will be available through the following link: https://meet.google.com/mvu-djoo-oes
Separating periods of quartic surfaces
Abstract: Kontsevich--Zagier periods form a natural number system that extends the algebraic numbers by adding constants coming from geometry and physics. Because there are countably many periods, one would expect it to be possible to compute effectively in this number system. This would require an effective height function and the ability to separate periods of bounded height, neither of which are currently possible. In this talk, we introduce an effective height function for periods of quartic surfaces defined over algebraic numbers. We also determine the minimal distance between periods of bounded height on a single surface. We use these results to prove heuristic computations of Picard groups that rely on approximations of periods. Moreover, we give explicit Liouville type numbers that can not be the ratio of two periods of a quartic surface. This is ongoing work with Pierre Lairez (Inria, France).