2020 Fall: Graduate student seminar (921)

Co-organised with Betsy Stovall

Zoom ID: 8129551821

1. Week One(9.9.2020): Ben Bruce
Title: The Fourier restriction problem for hyperboloids
Abstract: In this talk, I will discuss recent work on the Fourier restriction problem for hyperboloids, with a focus on the one-sheeted hyperboloid in R^3. I will show how polynomial partitioning can be used to prove local restriction estimates for this surface, and I will also present global estimates obtained in joint work with Betsy Stovall and Diogo Oliveira e Silva.

2. Week Two(9.16.2020): Rajula Srivastava
Title: Orthogonal Systems of Spline Wavelets as Unconditional Bases in Sobolev Spaces
Abstract: We exhibit the necessary range for which functions in the Sobolev spaces $L^s_p$ can be represented as an unconditional sum of orthonormal spline wavelet systems, such as the Battle-Lemari\'e wavelets. We also consider the natural extensions to Triebel-Lizorkin spaces. This builds upon, and is a generalization of, previous work of Seeger and Ullrich, where analogous results were established for the Haar wavelet system.

3. Week Three(9.23.2020):
4. Week Four(9.30.2020): Polona Durcik. Joint with the analysis seminar.
5. Week Five(10.7.2020): Geoff Bentsen
Title: Averages over curves on the Heisenberg group
Abstract: In this talk I will investigate the L^p regularity of generalized Radon transforms through several model examples, culminating in a sharp result for averaging operators over certain families of curves invariant under translation in the Heisenberg group. This noncommutative setting involves degeneracies not seen in the Euclidean case. We prove L^p regularity in the "worst" case of these degeneracies, a fold blowdown singularity, using almost orthogonality arguments and iterated decoupling on the cone.

6. Week Six(10.14.2020):
7. Week Seven(10.21.2020): Niclas Technau. Joint with the analysis seminar.
Title: Number theoretic applications of oscillatory integrals
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.

8. Week Eight(10.28.2020): Michael Spoerl.
9. Week Nine(11.4.2020): Liding Yao.
Title: Frobenius Theorem for Log-Lipschitz Subbundles
Abstract: In differential geometry, Frobenius theorem says that if a (smooth) real tangential subbundle is involutive, i.e. that X,Y are sections implies that [X,Y] is also a section, then this subbundle is spanned by some coordinate vector fields. Recently we prove that the theorem in the log-Lipschitz setting. In the talk I will go over the formulation of the theorem and show how harmonic analysis involves in the proof.
10. Week Ten(11.11.2020):
11. Week Eleven(11.18.2020): Xiaocheng Li.
12. Week Twelve(11.25.2020): Thanksgiving break.
13. Week Thirteen(12.2.2020): Jacob Denson
14. Week Fourteen(12.9.2020): Changkeun Oh
15. Week Fifteen(12.16.2020): Exam week.