# 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 | Two dimensional deep capillary solitary water waves with constant vorticity | |

4/8 | Taylor Tan | Signal Recovery, Uncertainty Principles, and Restriction | |

4/15 | Kaiyi Huang | A proof of ergodic theorem using Ramanujan’s circle method | |

4/22 | Sam Craig | Restriction operators for quadratic manifolds are not weak-type bounded at the endpoint | |

4/29 | Allison Byars | Dispersive PDEs and long time behavior of DNLS |

#### 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: Two dimensional deep capillary solitary water waves with constant vorticity**

* Abstract.* The existence or non-existence of solitary waves for free boundary Euler equation has long been an important question in mathematical fluid dynamics. In this talk I will talk about the two dimensional capillary water waves with nonzero constant vorticity in infinite depth. The existence of solitary waves is equivalent to the existence of nontrivial solutions of the Babenko equation, which is a quasilinear second order elliptic equation. I will show that when the velocity is closed to the critical velocity, the water waves system has a small frequency-localized solitary wave solution.

**Taylor Tan: Signal Recovery, Uncertainty Principles, and Restriction**

* Abstract.* This talk will try to reconstruct the two talks on this topic given by Alex Iosevich during the PLANT conference in March 2024. Given a signal $f: Z_N \to \mathbb{C}$ we can uniquely decompose the signal into its frequencies via the Fourier transform. If certain frequencies are unknown for some reason (due to noise or interference, etc.), is it still possible to recover your original signal? The goal of this talk is to link this question to uncertainty principles and discrete restriction theory.

**Kaiyi Huang: A proof of ergodic theorem using Ramanujan’s circle method**

* Abstract.* The ingenious circle method, originated by Ramanujan, has been applied to a broad span of areas including ergodic theories. In this talk, I aim to illustrate the circle method by proving a pointwise ergodic theorem on linear polynomial averages (Bourgain, 1988). This is a warmup for an upcoming talk in the spring school on harmonic analysis in Madison, where Franky and I will present the latest result in bilinear cases proven in the same spirit.

**Sam Craig: Restriction operators for quadratic manifolds are not weak-type bounded at the endpoint**

* Abstract.* The Fourier extension operator for the paraboloid in $\mathbb{R}^d$ cannot be bounded $L^p \rightarrow L^p$ for $p = 2d/(d-1)$, since the indicator function of a small ball decays like $r^{-(d-1)/2}$ in at least one direction. This example on its own does not preclude a weak-type bound $L^p \rightarrow L^{p, \infty}$, but in 1988 Beckner, Carbery, Semmes, and Soria proved that a weak-type bound cannot hold either, using a variant on the Perron tree construction for Kakeya sets to construct a counterexample. I will present a generalization of this to prove that any $n$-dimensional quadratic manifold in $\mathbb{R}^d$ cannot be bounded $L^{2d/n} \rightarrow L^{2d/n, \infty}$, using a different Kakeya-type construction for counterexamples.

**Allison Byars: Dispersive PDEs and long time behavior of DNLS**

* Abstract.* In this talk, I’ll discuss the setup of dispersive PDEs and some usual questions that are asked in this field. We’ll discuss local and global well-posedness, i.e. existence and uniqueness of solutions in some spaces, as well as the long time behavior of solutions, which tells us what the solution looks like as time goes to infinity. I’ll then talk about my work on the Derivative Nonlinear Schr\”dinger (DNLS) equation through the method of testing by wave packets to gain insight on the long time behavior of solutions.