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 Wednesdays, 1:20pm-2:10pm, in Van Vleck 901. Cookies 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.

Fall 2024 Schedule

Date	Speaker	Title	Comments
September 4	Summer & Allison	Planning / Social!
September 11	Jake Fiedler	Projection theorems in geometric measure theory	A continuation of this talk, "Universal sets for projections," will happen on September 13th in the Graduate Analysis Seminar (Fridays @ 1:20pm-2:10pm in VV B235).
September 18	Sam Craig	Structural properties of sticky Kakeya sets	A continuation of this talk will happen on September 20th in the Graduate Analysis Seminar.
September 25	Kaiyi Huang	A fast algorithm to solve the discrete integrable NLS
October 2	Kaiwen Jin	$L^2$ Schrödinger maximal function estimate via fractal extension estimate	A continuation of this talk (title TBA) will happen on October 4th in the Graduate Analysis Seminar.
October 9	Chiara Paulsen	Norm convergence of ergodic averages
October 16	CANCELLED	good luck with grading midterms!
October 23	Amelia Stokolosa & Allison Byars	Practicing our talks for the AMS sectional (see titles below)
October 30	Multiple	Elevator Pitches
November 6	Amelia Stokolosa	Theory and Applications of the Nash-Moser inverse function theorem by Hamilton
November 13	Summer Al Hamdani	On the ball multiplier theorem
November 20	Adrian Calderon	On the comparison principle and doubling variables method in the viscosity solution theory for Hamilton-Jacobi Equations
November 27	CANCELLED	CANCELLED	Day before Thanksgiving!
December 4	Gustavo Flores	The Haar measure
December 11	Dimas de Albuquerque	TBA

September 11. Jake Fiedler, Projection theorems in geometric measure theory.

Abstract: Geometric measure theory (GMT) investigates how certain geometric properties of sets or operations on sets affect their size. Orthogonal projections are one such operation, and have been closely studied in this context for many years. Marstrand's projection theorem is the most prominent result of this type and states that for any (reasonable) set, the projections of that set in almost every direction have maximal Hausdorff dimension. We will introduce some of the main ideas of GMT, discuss Marstrand's projection theorem and other projection results, and begin to explore some new tools that have enabled recent progress in this area. This is the first of two talks.

The second talk will happen on September 13th, at 1:20pm-2:10pm in VV B235 during the Graduate Analysis Seminar:

Title: Universal sets for projections

Abstract: In this talk, we will consider certain variants of Marstrand's projection theorem that hold for classes of sets in the plane. In particular, we will examine the class of sets with optimal oracles, the class of weakly regular sets, and the class of Ahlfors-David regular sets. This is the second of two talks and is based on joint work with Don Stull.

September 18. Sam Craig, Structural properties of sticky Kakeya sets.

Abstract: We heard last week about the Kakeya set conjecture, that a set in $\mathbb{R}^n$ with a line segment in every direction has Hausdorff dimension $n$. A 2022 paper by Hong Wang and Josh Zahl proves this in $\mathbb{R}^3$ for sticky Kakeya sets, which have an additional structural property called stickiness. I will outline how sticky Kakeya sets with near-minimal dimension must have additional structural properties Wang and Zahl call local and global grains and how these properties, along with previously known sum-product estimates, lead to a contradiction. This talk will be followed by a talk on Friday giving more details on how Wang and Zahl prove the existence of local and global grains.

September 25. Kaiyi Huang, A fast algorithm to solve the discrete integrable NLS.

Abstract: We study the discrete integrable nonlinear Schrödinger equation (aka. Ablowitz—Ladik equation) on the integer lattice with l^2 initial data. Thanks to the stability results of Schur’s algorithm and nonlinear Fourier transform properties, there is a fast algorithm to solve the equation of high accuracy.

October 2. Kaiwen Jin, L^2 Schrödinger maximal function estimate via fractal extension estimate.

Abstract: I plan to present the paper Sharp $L^2$ estimate of Schrödinger maximal function in higher dimensions by Xiumin Du and Ruixiang Zhang (2019) in two consecutive talks. In the GAPS seminar, the main focus will be how we can deduce the Schrödinger maximal function estimate, which in turn will imply pointwise convergence of the free Schrödinger equation, from the fractal extension estimate using a localization argument, parabolic rescaling, and locally constant property. If time permits, I will also give a proof sketch of the main theorem in the paper about fractal extension estimate. In the second talk on Friday, I will give a more detailed sketch of the proof of the fractal extension estimate.

October 9. Chiara Paulsen, Norm convergence of ergodic averages.

Abstract: We will look at the norm convergence of ergodic averages of the form $\frac{1}{N}\sum_{n=0}^{N-1}T^nf_1T^{2n}f_2...T^{kn}f_k$ where $T$ is the dynamic of an ergodic system and $f_1,...,f_k\in L^\infty$ using the method of characteristic factors..

October 16. CANCELLED.

Abstract: good luck with grading! :)

October 23. Amelia Stokolosa & Allison Byars, Practicing our talks for the AMS sectional.

Allison's title: Global Dynamics of small data solutions to the Derivative Nonlinear Schrödinger equation

Abstract: $L^2$-well-posedness for the derivative nonlinear Schrödinger 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 or does it admit solitons? 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, assuming small and localized data, we will explore how this method can be applied to the DNLS equation to prove a global in time dispersive estimate.

Amelia's title: Inverses and tame estimates for product kernels and flag kernels on graded Lie groups

Abstract: We obtain an inverse theorem for a class of left-invariant multi-parameter singular integral operators on graded Lie groups. Our result extends the work by Christ, Geller, Głowacki, and Polin on single-parameter homogeneous kernels to almost homogeneous kernels with respect to multi-parameter dilations, namely product kernels and flag kernels. In the non-commutative setting of graded Lie groups, we cannot make use of the Fourier transform to study our operators. Instead, we present two proofs: one relying on PDE tools and the other on Banach-algebraic tools.

October 30. Multiple, Elevator Pitches.

Abstract: Elevator pitches from several members of the analysis and PDEs groups.

November 6. Amelia Stokolosa, Theory and Applications of the Nash-Moser inverse function theorem by Hamilton.

Abstract: In this talk, we will examine the Nash-Moser Inverse Function Theorem by Hamilton. This methodology presents an extension of the Inverse Function Theorem on Banach spaces to a subclass of Fréchet spaces. The Nash-Moser Inverse Function Theorem turns out to be particularly useful in the study of certain nonlinear PDEs.

November 13. Summer Al Hamdani, On the ball multiplier theorem, pt. I.

Abstract: We introduce and provide some historical context on the disc conjecture, which states that the operator $T$ defined on $L^p(\mathbb{R}^n)$ by $\widehat{Tf}(x) = \chi_B(x) \hat{f}(x)$ (where $\chi_B$ is the characteristic function of the unit ball) is bounded on all of $L^p(\mathbb{R}^n)$ with $2n/(n+1) < p < 2n/(n-1)$. In this talk, we will see a disproof of the disc conjecture, which follows due to a lemma by Yves Meyer.  This talk is based on Charles Fefferman's 1971 paper, "The Multiplier Problem for the Ball."

The second part of this talk will happen on November 15th in the Graduate Analysis Seminar:

Title: On the ball multiplier theorem, pt. II

Abstract: In the second part of this talk, we briefly recap the premise and disproof of the disc conjecture. Then, we construct an explicit counterexample in $\R^2$ using a variant of Besicovitch's construction for the Kakeya needle problem. This talk is based on Charles Fefferman's 1971 paper, "The Multiplier Problem for the Ball."

November 20. Adrian Calderon, On the comparison principle and doubling variables method in the viscosity solution theory for Hamilton-Jacobi Equations.

Abstract: In this talk, we will discuss an important aspect of well-posedness theory, namely uniqueness, for first order Hamilton-Jacobi equations. We will provide a proof of the standard comparison principle and showcase the main tool: the doubling variables method, which is a robust tool used for many applications in viscosity solution theory. This talk aims to be accessible for those not directly in this sub-field of mathematics.   

December 4. Gustavo Flores, The Haar Measure.

Abstract: The Lebesgue measure, among other things, is convenient because it is translation-invariant: the measure of a set does not depend on “where” on the real line it is. More generally, if G is a locally compact Hausdorff (LCH) group, it turns out that G admits such a translation-invariant measure, called a Haar measure. In this talk, we sketch a proof of the existence of Haar measures on LCH groups and discuss some properties of Haar measures.

December 11. Dimas de Albuquerque, TBA.

Abstract: TBA.

Previous Semesters

Click here to view of all previous semesters' speakers and abstracts.