## 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.
Title: Maximal Operators and Fourier Restriction on the Moment Curve
Abstract: We prove L^p to L^q bounds for certain maximal Fourier restriction operators.

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.
Title:Harmonic analysis for Laplace eigenfunctions
Abstract: In this talk, we would like to discuss some classical results in the study of Laplace eigenfunctions on general compact Riemannian manifolds. The tools are based on harmonic analysis, especially the Pseudo-differential operators. Then we would discuss the similar problems on compact locally symmetric spaces, where alternative methods in Lie theory are available.
12. Week Twelve(11.25.2020): Thanksgiving break.
13. Week Thirteen(12.2.2020): Jacob Denson Title:Salem Sets Avoiding Nonlinear Patterns
Abstract: If a set has large Hausdorff dimension, and also supports a measure with Fourier decay, does it necessarily contain a family of points forming a particular pattern, such as three points forming a right angle, or four points forming a parallelogram? This talk will discuss new construction techniques to find sets with large dimension of the form above avoiding a particular family of patterns. The proof of these techniques involves some combinatorial estimates, and an understanding of concentration of measure phenomena.
14. Week Fourteen(12.9.2020): Changkeun Oh
Title: Decoupling inequalities for quadratic forms
Abstract: In this talk, I'll introduce decoupling inequalities and their applications. I'll also introduce a recent work with Shaoming Guo, Ruixiang Zhang, and Pavel Zorin-Kranich, where we prove sharp decoupling inequalities for every tuples of quadratic forms.
15. Week Fifteen(12.16.2020): Exam week.