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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.  


=== Fall 2024 Schedule ===
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
|-
|-
|September 4
|January 22
|Summer & Allison
|Summer Al Hamdani and Allison Byars
|Planning / Social!
|Organizing!
|
|
|-
|-
|September 11
|January 29
|Jake Fiedler
|Gautam Neelakantan
|Projection theorems in geometric measure theory
|On Almgren’s frequency function
|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
|January 31
|Sam Craig
|Sam Craig
|Structural properties of sticky Kakeya sets
|The p-adic Kakeya problem
|A continuation of this talk will happen on September 20th in the Graduate Analysis Seminar.
|
|-
|-
|September 25
|February 5
|Kaiyi Huang
|Kaiyi Huang
|A fast algorithm to solve the discrete integrable NLS
|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
|
|
|-
|-
|October 2
|February 19
|Kaiwen Jin
|Mingfeng Chen
|$L^2$ Schrödinger maximal function estimate via fractal extension estimate
|Rational points near manifold
|A continuation of this talk (title TBA) will happen on October 4th in the Graduate Analysis Seminar.
|
|-
|February 21
|Mingfeng Chen
|Rational points near planar curves
|
|-
|-
|October 9
|February 26
|Chiara Paulsen
|Chiara Paulsen
|Norm convergence of ergodic averages
|TBD
|
|
|-
|-
|October 16
|February 28
|CANCELLED
|Chiara Paulsen
|good luck with grading midterms!
|TBD
|
|
|-
|-
|October 23
|March 5
|Amelia Stokolosa & Allison Byars
|TBD
|Practicing our talks for the AMS sectional (see titles below)
|
|
|
|-
|-
|October 30
|March 7
|Multiple
|Jia Hao Tan
|''Elevator Pitches''
|TBD
|
|
|-
|-
|November 6
|March 12
|Amelia Stokolosa
|TBD
|TBA
|
|
|
|-
|-
|November 13
|March 14
|Summer Al Hamdani
|Yupeng Zhang
|On the ball multiplier theorem
|TBD
|
|
|-
|-
|November 20
|March 19
|Adrian Calderon
|TBD
|TBA
|
|
|
|-
|-
|November 27
|March 26
|SPRING BREAK
|CANCELLED
|CANCELLED
|CANCELLED
|:)
|Day before Thanksgiving!
|-
|April 2
|Jiankun Li
|
|
|-
|April 4
|Jiankun Li
|
|
|-
|April 9
|Gustavo Flores
|
|
|-
|-
|December 4
|April 11
|Gustavo Flores
|Gustavo Flores
|TBA
|
|
|-
|April 16
|Kaiwen Jin
|
|
|
|-
|-
|December 11
|April 18
|Kaiwen Jin
|
|
|-
|April 23
|Dimas de Albuquerque
|Dimas de Albuquerque
|TBA
|
|
|-
|April 25
|Summer Al Hamdani
|
|
|-
|April 30
|Raj Vasu
|
|
|-
|(May 2)
|TBD
|
|
|
|}
|}


==== September 11. [https://sites.google.com/view/jakefiedler Jake Fiedler], ''Projection theorems in geometric measure theory.'' ====
=== Abstracts ===
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. [https://people.math.wisc.edu/~secraig2/ 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.'' ====
====== '''January 29. Gautam Neelakantan, ''On Almgren’s frequency function.''''' ======
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.
'''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.


==== October 2. Kaiwen Jin, ''L^2 Schrödinger maximal function estimate via fractal extension estimate.'' ====
====== '''January 31. Sam Craig, ''The p-adic Kakeya problem''.''' ======
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.
'''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.


==== October 9. Chiara Paulsen, ''Norm convergence of ergodic averages.'' ====
==== '''February 5. Kaiyi Huang, ''Brascamp-Lieb Inequalities.''''' ====
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..
'''Abstract:''' I will sketch the proof of this classical of inequalities done by Bennett, et al (2008).


==== October 16. CANCELLED. ====
==== '''February 12. Sam Craig,  ''An introduction to the Kakeya problem.''''' ====
Abstract: good luck with grading! :)
'''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.


==== October 23. Amelia Stokolosa & Allison Byars, ''Practicing our talks for the AMS sectional.'' ====
==== '''February 19. Mingfeng Chen, ''Rational points near manifold.''''' ====
Allison's title: Global Dynamics of small data solutions to the Derivative Nonlinear Schrödinger equation
'''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.


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.  
==== '''February 26. Chiara Paulsen, ''TBD.''''' ====
'''Abstract:''' TBA.


Amelia's title: Inverses and tame estimates for product kernels and flag kernels on graded Lie groups
==== '''March 5. TBD, ''TBD.''''' ====
'''Abstract:''' TBA.


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.
==== '''March 12. TBD, ''TBD.''''' ====
'''Abstract:''' TBA.


==== October 30. Multiple, ''Elevator Pitches.'' ====
==== '''March 19. TBD, ''TBD.''''' ====
Abstract: Elevator pitches from several members of the analysis and PDEs groups.
'''Abstract:''' TBA.


==== November 6. Amelia Stokolosa, TBA ====
==== '''April 2. Jiankun Li, ''TBD.''''' ====
Abstract: TBA.
'''Abstract:''' TBA.


==== November 13. Summer Al Hamdani, ''On the ball multiplier theorem.'' ====
==== '''April 9. Gustavo Flores, ''TBD.''''' ====
Abstract: TBA.
'''Abstract:''' TBA.


==== November 20. Adrian Calderon, ''TBA.'' ====
==== '''April 16. Kaiwen Jin, ''TBD.''''' ====
Abstract: TBA.
'''Abstract:''' TBA.


==== December 4. Gustavo Flores, ''TBA.'' ====
==== '''April 23. Dimas de Albuquerque, ''TBD.''''' ====
Abstract: TBA.
'''Abstract:''' TBA.


==== December 11. Dimas de Albuquerque, ''TBA.'' ====
==== '''April 30. Raj Vasu, ''TBD.''''' ====
Abstract: TBA.
'''Abstract:''' TBA.


=== Previous Semesters ===
=== Previous Semesters ===
[[GAPS Previous Semesters|Click here]] to view of all previous semesters' speakers and abstracts.
[[GAPS Previous Semesters|Click here]] to view of all previous semesters' speakers and abstracts.

Latest revision as of 15:07, 17 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 TBD
February 28 Chiara Paulsen TBD
March 5 TBD
March 7 Jia Hao Tan TBD
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, TBD.

Abstract: TBA.

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.

Retrieved from "https://wiki.math.wisc.edu/index.php?title=GAPS&oldid=28123"