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'''''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. | '''''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. | ||
===== '''[https://sites.google.com/wisc.edu/allisonbyars Allison Byars]: Wave Packets for | ===== '''[https://sites.google.com/wisc.edu/allisonbyars Allison Byars]: Wave Packets for DNLS''' ===== | ||
'''''Abstract.''''' Well-posedness for the | '''''Abstract.''''' Well-posedness for the derivative nonlinear Schr\"odinger equation (DNLS) was recently proved by Harrop-Griffiths, Killip, Ntekoume, and Vi\c{s}an. 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. | ||
===== '''[https://sites.google.com/view/chenmingfeng/home Mingfeng Chen]: TBD''' ===== | ===== '''[https://sites.google.com/view/chenmingfeng/home Mingfeng Chen]: TBD''' ===== | ||
Revision as of 21:01, 29 February 2024
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 | TBD | |
| 4/1 | Lizhe Wan | TBD | |
| 4/8 | Taylor Tan | TBD | |
| 4/15 | Kaiyi Huang | TBD | |
| 4/22 | Sam Craig | TBD | |
| 4/19 | 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 Schr\"odinger equation (DNLS) was recently proved by Harrop-Griffiths, Killip, Ntekoume, and Vi\c{s}an. 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: TBD
Abstract.
Lizhe Wan: TBD
Abstract.
Taylor Tan: TBD
Abstract.
Kaiyi Huang: TBD
Abstract.
Sam Craig: TBD
Abstract.
Allison Byars: TBD
Abstract.