GAPS

The Graduate Analysis and PDEs Seminar (GAPS) is intended to build community for graduate students in the different subfields of analysis and PDEs. The goal is to give accessible talks about your current research projects, papers you found interesting on the arXiv, or even just a theorem/result that you use and think is really cool!

We currently meet Mondays, 1:20pm-2:10pm, in Van Vleck 901. Oreos and apple juice (from concentrate) are provided. If you have any questions, please email the organizers: Summer Al Hamdani (alhamdani (at) wisc.edu) and Allison Byars (abyars (at) wisc.edu).

To join the mailing list, send an email to: gaps+subscribe@g-groups.wisc.edu.

Spring 2024

Date	Speaker	Title	Comments
2/26	Organizational Meeting
3/4	skip-bc of PLANT
3/11	Amelia Stokolosa	Inverses of product kernels and flag kernels on graded Lie groups	1:20-1:50
3/11	Allison Byars	Wave Packets for DNLS	1:55-2:10
3/18	Mingfeng Chen	Nikodym set vs Local smoothing for wave equation
4/1	Lizhe Wan	TBD
4/8	Taylor Tan	TBD
4/15	Kaiyi Huang	TBD
4/22	Sam Craig	TBD
4/29	Allison Byars	TBD

Spring 2024 Abstracts

Amelia Stokolosa: Inverses of product kernels and flag kernels on graded Lie groups

Abstract. Consider the following problem solved in the late 80s by Christ and Geller: Let $Tf = f*K$ where $K$ is a homogeneous distribution on a graded Lie group. Suppose $T$ is $L^2$ invertible. Is $T^{-1}$ also a translation-invariant operator given by convolution with a homogeneous kernel? Christ and Geller proved that the answer is yes. Extending the above problem to the multi-parameter setting, consider the operator $Tf = f*K$, where $K$ is a product or a flag kernel on a graded Lie group $G$. Suppose $T$ is $L^2$ invertible. Is $T^{-1}$ also given by group convolution with a product or flag kernel accordingly? We prove that the answer is again yes. In the non-commutative setting, one cannot make use of the Fourier transform to answer this question. Instead, the key construction is an a priori estimate.

Allison Byars: Wave Packets for DNLS

Abstract. Well-posedness for the derivative nonlinear Schrödinger equation (DNLS) was recently proved by Harrop-Griffiths, Killip, Ntekoume, and Vișan. The next natural question to ask is, "what does the solution look like?", i.e. does it disperse in time at a rate similar to the linear solution? In 2014, Ifrim and Tataru introduced the method of wave packets in order to prove a dispersive decay estimate for NLS. The idea of wave packets is to find an approximate solution to the equation which is localized in both space and frequency, and use this to prove an estimate on the nonlinear solution. In this talk, we will explore how this method can be applied to the DNLS equation.

Mingfeng Chen: Nikodym set vs Local smoothing for wave equation

Abstract. This talk is about classifying maximal average over planar curves. It is well-known that if we consider the maximal operator defined by averaging over planar line, then the maximal operator is not bounded on $L^p(\mathbb{R}^2)$ for any $p<\infty$ because of the existence of Nikodym set. On the other hand, if we replace line by parabola or circle, the celebrated Bourgain's circular maximal theorem shows that such operator is bounded for every $p>2$. We classify all the maximal operator, that is: we find all the curves such that Nikodym sets exist, thus the corresponding maximal operator is not bounded on $L^p$ for any $p<\infty$; for other curves, we prove sharp $L^p$ bound for the maximal operator.

Lizhe Wan: TBD

Abstract.

Taylor Tan: TBD

Abstract.

Kaiyi Huang: TBD

Abstract.

Sam Craig: TBD

Abstract.

Allison Byars: TBD

Abstract.