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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 ===
=== Fall 2024 Schedule ===
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{| class="wikitable"
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!Comments
!Comments
|-
|-
|2/26
|September 4
|Organizational Meeting
|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
|
|
|-
|-
|3/4
|October 2
|skip-bc of PLANT
|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!
|
|
|-
|-
|3/11
|October 23
|Amelia Stokolosa
|Amelia Stokolosa & Allison Byars
|Inverses of product kernels and flag kernels on graded Lie groups
|Practicing our talks for the AMS sectional (see titles below)
|1:20-1:50
|
|-
|-
|3/11
|October 30
|Allison Byars
|Multiple
|Wave Packets for DNLS
|''Elevator Pitches''
|1:55-2:10
|
|-
|-
|3/18
|November 6
|Mingfeng Chen
|Amelia Stokolosa
|Nikodym set vs Local smoothing for wave equation
|''Theory and Applications of the Nash-Moser inverse function theorem by Hamilton''
|
|
|-
|-
|4/1
|November 13
|Lizhe Wan
|Summer Al Hamdani
|Two dimensional deep capillary solitary water waves with constant vorticity
|On the ball multiplier theorem
|
|
|-
|-
|4/8
|November 20
|Taylor Tan
|Adrian Calderon
|Signal Recovery, Uncertainty Principles, and Restriction
|On the comparison principle and doubling variables method in the viscosity solution theory for Hamilton-Jacobi Equations
|
|
|-
|-
|4/15
|November 27
|Kaiyi Huang
|CANCELLED
|A proof of ergodic theorem using Ramanujan’s circle method
|CANCELLED
|
|Day before Thanksgiving!
|-
|-
|4/22
|December 4
|Sam Craig
|Gustavo Flores
|Restriction operators for quadratic manifolds are not weak-type bounded at the endpoint
|TBA
|
|
|-
|-
|4/29
|December 11
|Allison Byars
|Dimas de Albuquerque
|TBD
|TBA
|
|
|}
|}


==== Spring 2024 Abstracts ====
==== September 11. [https://sites.google.com/view/jakefiedler 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. [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.'' ====
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. [https://sites.google.com/wisc.edu/stokolosa/home 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.


===== '''[https://sites.google.com/wisc.edu/stokolosa/home Amelia Stokolosa]: Inverses of product kernels and flag kernels on graded Lie groups''' =====
==== November 13. Summer Al Hamdani, ''On the ball multiplier theorem, pt. I.'' ====
'''''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: 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."


===== '''[https://sites.google.com/wisc.edu/allisonbyars Allison Byars]: Wave Packets for DNLS''' =====
The second part of this talk will happen on November 15th in the Graduate Analysis Seminar:
'''''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. 


===== '''[https://sites.google.com/view/chenmingfeng/home Mingfeng Chen]: Nikodym set vs Local smoothing for wave equation''' =====
'''Title:''' On the ball multiplier theorem, pt. II
'''''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.


===== '''[https://sites.google.com/wisc.edu/lizhewan/ Lizhe Wan]: Two dimensional deep capillary solitary water waves with constant vorticity''' =====
'''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."
'''''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''' =====
==== November 20. Adrian Calderon, On the comparison principle and doubling variables method in the viscosity solution theory for Hamilton-Jacobi Equations''.'' ====
'''''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: 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.   


===== '''Kaiyi Huang: A proof of ergodic theorem using Ramanujan’s circle method''' =====
==== December 4. Gustavo Flores, ''TBA.'' ====
'''''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''' =====
==== December 11. Dimas de Albuquerque, ''TBA.'' ====
'''''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]: TBD''' =====
=== Previous Semesters ===
'''''Abstract.'''''
[[GAPS Previous Semesters|Click here]] to view of all previous semesters' speakers and abstracts.

Revision as of 19:05, 15 November 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 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 TBA
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, TBA.

Abstract: TBA.

December 11. Dimas de Albuquerque, TBA.

Abstract: TBA.

Previous Semesters

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

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