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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!
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: [https://salhamdani.github.io Summer Al Hamdani] (alhamdani (at) wisc.edu) and [https://sites.google.com/wisc.edu/allisonbyars Allison Byars] (abyars (at) wisc.edu).
We currently meet '''Wednesdays, 1:20pm-2:10pm, in Van Vleck 901'''. Cookies are provided. If you have any questions, please email the organizers: [https://salhamdani.github.io Summer Al Hamdani] (alhamdani (at) wisc.edu) and [https://sites.google.com/wisc.edu/allisonbyars Allison Byars] (abyars (at) wisc.edu).


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


=== Spring 2024 ===
We also loosely coordinate with the graduate analysis seminar (and many students go to both). The graduate analysis seminar meets on Fridays 1:20pm-2:10pm, in Van Vleck B219. For questions on the graduate analysis seminar, please email the organizers: Betsy Stovall (stovall (at) math.wisc.edu) and Lars Niedorf (niedorf (at) wisc.edu).
 
=== Spring 2025 Schedule for GAPS (Wednesdays in 901) and Graduate Analysis Seminar (Fridays in B219) ===
{| class="wikitable"
{| class="wikitable"
|+
|'''Date'''
!Date
|'''Presenter'''
!Speaker
|'''Title'''
!Title
|'''Comments'''
!Comments
|-
|-
|2/26
|January 22
|Organizational Meeting
|Summer Al Hamdani and Allison Byars
|Organizing!
|
|
|-
|January 29
|Gautam Neelakantan
|On Almgren’s frequency function
|
|
|-
|-
|3/4
|January 31
|skip-bc of PLANT
|Sam Craig
|The p-adic Kakeya problem
|
|
|-
|February 5
|Kaiyi Huang
|Brascamp-Lieb Inequalities
|
|
|-
|-
|3/11
|February 7
|Amelia Stokolosa
|Dimas de Albuquerque
|Inverses of product kernels and flag kernels on graded Lie groups
|Sign uncertainty principle for the Fourier transform
|1:20-1:50
|
|-
|-
|3/11
|February 12
|Allison Byars
|Sam Craig
|Wave Packets for DNLS
|An introduction to the Kakeya problem.
|1:55-2:10
|
|-
|-
|3/18
|February 14
|Gautam Neelakantan
|Different Perspectives on Pseudodifferential Operators
|
|-
|February 19
|Mingfeng Chen
|Mingfeng Chen
|Nikodym set vs Local smoothing for wave equation
|Rational points near manifold
|
|-
|February 21
|Mingfeng Chen
|Rational points near planar curves
|
|-
|February 26
|Chiara Paulsen
|Some basics about orthogonal polynomials on the unit circle
|
|-
|February 28
|Chiara Paulsen
|A proof of Szegő’s theorem
|
|-
|March 5
|TBD
|
|
|-
|March 7
|Jia Hao Tan
|Uniqueness of Signal Recovery: Uncertainty Principles, Randomness and Restriction
|
|-
|March 12
|TBD
|
|
|-
|March 14
|Yupeng Zhang
|TBD
|
|-
|March 19
|TBD
|
|
|-
|March 26
|SPRING BREAK
|CANCELLED
|:)
|-
|April 2
|Jiankun Li
|
|
|-
|April 4
|Jiankun Li
|
|
|-
|April 9
|Gustavo Flores
|
|
|-
|April 11
|Gustavo Flores
|
|
|-
|April 16
|Kaiwen Jin
|
|
|
|-
|-
|4/1
|April 18
|Lizhe Wan
|Kaiwen Jin
|Two dimensional deep capillary solitary water waves with constant vorticity
|
|
|
|-
|-
|4/8
|April 23
|Taylor Tan
|Dimas de Albuquerque
|Signal Recovery, Uncertainty Principles, and Restriction
|
|
|
|-
|-
|4/15
|April 25
|Kaiyi Huang
|Summer Al Hamdani
|A proof of ergodic theorem using Ramanujan’s circle method
|
|
|
|-
|-
|4/22
|April 30
|Sam Craig
|Raj Vasu
|Restriction operators for quadratic manifolds are not weak-type bounded at the endpoint
|
|
|
|-
|-
|4/29
|(May 2)
|Allison Byars
|TBD
|Dispersive PDEs and long time behavior of DNLS
|
|
|
|}
|}


==== Spring 2024 Abstracts ====
=== Abstracts ===
 
====== '''January 29. Gautam Neelakantan, ''On Almgren’s frequency function.''''' ======
'''Abstract:''' I will introduce Almgren’s frequency function and discuss its monotonicity property and application to unique continuation problems. If time permits I will also talk about other Almgren type frequency functions and some open problems related to the same.
 
====== '''January 31. Sam Craig, ''The p-adic Kakeya problem''.''' ======
'''Abstract:''' The Kakeya conjecture over the p-adic field $Q_p$ is that the Hausdorff dimension of a set in $Q_p^n$ with a line segment in every direction is always $n$. This was proven in 2021 by Bodan Arsovski. I will present proof.
 
==== '''February 5. Kaiyi Huang, ''Brascamp-Lieb Inequalities.''''' ====
'''Abstract:''' I will sketch the proof of this classical of inequalities done by Bennett, et al (2008).
 
==== '''February 12. Sam Craig,  ''An introduction to the Kakeya problem.''''' ====
'''Abstract:''' I will introduce the Kakeya and Kakeya maximal conjecture, prove that L^p bounds for the Kakeya maximal function imply lower bounds on the dimension of Kakeya sets, and use Wolff’s brush method to prove the dimension of Kakeya sets in R^d is at least (d+2)/2.
 
==== '''February 19. Mingfeng Chen, ''Rational points near manifold.''''' ====
'''Abstract:''' How many rational points are near a compact manifold? This natural problem is closed related to many problems in number theory. In the first talk, I will survey the state of art and give some applications.
 
==== '''February 26. Chiara Paulsen, ''Some basics about orthogonal polynomials on the unit circle.''''' ====
'''Abstract:''' In this talk I will introduce some concepts about orthogonal polynomials associated to a probability measure $\mu$ on the unit circle $\mathbb{T}$. We will look at objects like its Verblunsky coefficients, the Schur parameters of the Schur function associated to $\mu$ or the Toeplitz determinant of $\mu$ and connect all these objects in various ways.
 
==== '''March 5. TBD, ''TBD.''''' ====
'''Abstract:''' TBA.


===== '''[https://sites.google.com/wisc.edu/stokolosa/home Amelia Stokolosa]: Inverses of product kernels and flag kernels on graded Lie groups''' =====
==== '''March 12. TBD, ''TBD.''''' ====
'''''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:''' TBA.


===== '''[https://sites.google.com/wisc.edu/allisonbyars Allison Byars]: Wave Packets for DNLS''' =====
==== '''March 19. TBD, ''TBD.''''' ====
'''''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.
'''Abstract:''' TBA.


===== '''[https://sites.google.com/view/chenmingfeng/home Mingfeng Chen]: Nikodym set vs Local smoothing for wave equation''' =====
==== '''April 2. Jiankun Li, ''TBD.''''' ====
'''''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.
'''Abstract:''' TBA.


===== '''[https://sites.google.com/wisc.edu/lizhewan/ Lizhe Wan]: Two dimensional deep capillary solitary water waves with constant vorticity''' =====
==== '''April 9. Gustavo Flores, ''TBD.''''' ====
'''''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.
'''Abstract:''' TBA.


===== '''Taylor Tan: Signal Recovery, Uncertainty Principles, and Restriction''' =====
==== '''April 16. Kaiwen Jin, ''TBD.''''' ====
'''''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.  
'''Abstract:''' TBA.


===== '''Kaiyi Huang: A proof of ergodic theorem using Ramanujan’s circle method''' =====
==== '''April 23. Dimas de Albuquerque, ''TBD.''''' ====
'''''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.
'''Abstract:''' TBA.


===== '''[https://people.math.wisc.edu/~secraig2/ Sam Craig]:  Restriction operators for quadratic manifolds are not weak-type bounded at the endpoint''' =====
==== '''April 30. Raj Vasu, ''TBD.''''' ====
'''''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.
'''Abstract:''' TBA.


===== '''[https://sites.google.com/wisc.edu/allisonbyars Allison Byars]: Dispersive PDEs and long time behavior of DNLS''' =====
=== Previous Semesters ===
'''''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.
[[GAPS Previous Semesters|Click here]] to view of all previous semesters' speakers and abstracts.

Latest revision as of 16:02, 28 February 2025

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.

We also loosely coordinate with the graduate analysis seminar (and many students go to both). The graduate analysis seminar meets on Fridays 1:20pm-2:10pm, in Van Vleck B219. For questions on the graduate analysis seminar, please email the organizers: Betsy Stovall (stovall (at) math.wisc.edu) and Lars Niedorf (niedorf (at) wisc.edu).

Spring 2025 Schedule for GAPS (Wednesdays in 901) and Graduate Analysis Seminar (Fridays in B219)

Date Presenter Title Comments
January 22 Summer Al Hamdani and Allison Byars Organizing!
January 29 Gautam Neelakantan On Almgren’s frequency function
January 31 Sam Craig The p-adic Kakeya problem
February 5 Kaiyi Huang Brascamp-Lieb Inequalities
February 7 Dimas de Albuquerque Sign uncertainty principle for the Fourier transform
February 12 Sam Craig An introduction to the Kakeya problem.
February 14 Gautam Neelakantan Different Perspectives on Pseudodifferential Operators
February 19 Mingfeng Chen Rational points near manifold
February 21 Mingfeng Chen Rational points near planar curves
February 26 Chiara Paulsen Some basics about orthogonal polynomials on the unit circle
February 28 Chiara Paulsen A proof of Szegő’s theorem
March 5 TBD
March 7 Jia Hao Tan Uniqueness of Signal Recovery: Uncertainty Principles, Randomness and Restriction
March 12 TBD
March 14 Yupeng Zhang TBD
March 19 TBD
March 26 SPRING BREAK CANCELLED :)
April 2 Jiankun Li
April 4 Jiankun Li
April 9 Gustavo Flores
April 11 Gustavo Flores
April 16 Kaiwen Jin
April 18 Kaiwen Jin
April 23 Dimas de Albuquerque
April 25 Summer Al Hamdani
April 30 Raj Vasu
(May 2) TBD

Abstracts

January 29. Gautam Neelakantan, On Almgren’s frequency function.

Abstract: I will introduce Almgren’s frequency function and discuss its monotonicity property and application to unique continuation problems. If time permits I will also talk about other Almgren type frequency functions and some open problems related to the same.

January 31. Sam Craig, The p-adic Kakeya problem.

Abstract: The Kakeya conjecture over the p-adic field $Q_p$ is that the Hausdorff dimension of a set in $Q_p^n$ with a line segment in every direction is always $n$. This was proven in 2021 by Bodan Arsovski. I will present proof.

February 5. Kaiyi Huang, Brascamp-Lieb Inequalities.

Abstract: I will sketch the proof of this classical of inequalities done by Bennett, et al (2008).

February 12. Sam Craig, An introduction to the Kakeya problem.

Abstract: I will introduce the Kakeya and Kakeya maximal conjecture, prove that L^p bounds for the Kakeya maximal function imply lower bounds on the dimension of Kakeya sets, and use Wolff’s brush method to prove the dimension of Kakeya sets in R^d is at least (d+2)/2.

February 19. Mingfeng Chen, Rational points near manifold.

Abstract: How many rational points are near a compact manifold? This natural problem is closed related to many problems in number theory. In the first talk, I will survey the state of art and give some applications.

February 26. Chiara Paulsen, Some basics about orthogonal polynomials on the unit circle.

Abstract: In this talk I will introduce some concepts about orthogonal polynomials associated to a probability measure $\mu$ on the unit circle $\mathbb{T}$. We will look at objects like its Verblunsky coefficients, the Schur parameters of the Schur function associated to $\mu$ or the Toeplitz determinant of $\mu$ and connect all these objects in various ways.

March 5. TBD, TBD.

Abstract: TBA.

March 12. TBD, TBD.

Abstract: TBA.

March 19. TBD, TBD.

Abstract: TBA.

April 2. Jiankun Li, TBD.

Abstract: TBA.

April 9. Gustavo Flores, TBD.

Abstract: TBA.

April 16. Kaiwen Jin, TBD.

Abstract: TBA.

April 23. Dimas de Albuquerque, TBD.

Abstract: TBA.

April 30. Raj Vasu, TBD.

Abstract: TBA.

Previous Semesters

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

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