Fall 2025 Analysis Seminar: Difference between revisions
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|Jaehyeon Ryu | |Jaehyeon Ryu | ||
|EWHA Womans University | |EWHA Womans University | ||
| | |Restriction estimates for spectral projections | ||
|Andreas, Lars | |Andreas, Lars | ||
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Abstract: We present recent results giving both upper and lower bounds for the number of integral points lying near smooth nondegenerate curves in $\mathbb{R}^n$. These estimates are new in dimensions $n\geq 4$, and in the case $n\geq 3$ they recover an earlier result of J. J. Huang. Our approach differs significantly from previous work: instead of relying on sharp planar counting results, we employ Fourier analytic methods. A key ingredient is an Arkhipov–Chubarikov–Karatsuba-type oscillatory integral estimate. This is joint work with Jonathan Hickman (Edinburgh). | Abstract: We present recent results giving both upper and lower bounds for the number of integral points lying near smooth nondegenerate curves in $\mathbb{R}^n$. These estimates are new in dimensions $n\geq 4$, and in the case $n\geq 3$ they recover an earlier result of J. J. Huang. Our approach differs significantly from previous work: instead of relying on sharp planar counting results, we employ Fourier analytic methods. A key ingredient is an Arkhipov–Chubarikov–Karatsuba-type oscillatory integral estimate. This is joint work with Jonathan Hickman (Edinburgh). | ||
===Ben Johnsrude=== | ===Ben Johnsrude=== | ||
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We survey some outstanding problems in restriction theory and the current state of knowledge therein. We discuss Guth's argument that decoupling is insufficient to solve Mizohata--Takeuchi. We then introduce a novel tool that allows one to go beyond the limits of the usual decoupling framework in the topics surveyed. | We survey some outstanding problems in restriction theory and the current state of knowledge therein. We discuss Guth's argument that decoupling is insufficient to solve Mizohata--Takeuchi. We then introduce a novel tool that allows one to go beyond the limits of the usual decoupling framework in the topics surveyed. | ||
===Lars Niedorf=== | ===Lars Niedorf=== | ||
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Abstract: We discuss recent sharp Bochner-Riesz estimates for Kohn-Laplacians on Heisenberg groups. Since favorable restriction estimates fail in this setting, we developed an approach that relies on square function bounds for the wave equation associated with the Kohn-Laplacian. This is joint work with Detlef Müller, Andreas Seeger, and Betsy Stovall. | Abstract: We discuss recent sharp Bochner-Riesz estimates for Kohn-Laplacians on Heisenberg groups. Since favorable restriction estimates fail in this setting, we developed an approach that relies on square function bounds for the wave equation associated with the Kohn-Laplacian. This is joint work with Detlef Müller, Andreas Seeger, and Betsy Stovall. | ||
===Hong Wang=== | ===Hong Wang=== | ||
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Abstract: Restriction theory studies functions whose Fourier transforms are supported on some curved manifold in R^n (for example, solutions to the linear Schrodinger equation or to the wave equation). Projection theorems study the Hausdorff dimension of fractal sets under orthogonal projections from R^n to its subspaces. We will survey some recent works in both fields and discuss their interactions. | Abstract: Restriction theory studies functions whose Fourier transforms are supported on some curved manifold in R^n (for example, solutions to the linear Schrodinger equation or to the wave equation). Projection theorems study the Hausdorff dimension of fractal sets under orthogonal projections from R^n to its subspaces. We will survey some recent works in both fields and discuss their interactions. | ||
===Jaehyeon Ryu=== | |||
Title: Restriction estimates for spectral projections | |||
Abstract: Restriction estimates for spectral projections have been widely studied since the work of Burq, Gérard, and Tzvetkov as a method for investigating eigenfunction concentration. The problem of establishing the optimal $L^p$ bounds for the restriction of Laplace-Beltrami eigenfunctions remains open, particularly when the restriction submanifold is of codimension 1 or 2. This talk will explore how these optimal bounds are characterized by the geometry of the underlying manifold $M$ and its submanifold $H$, focusing on the case where $M$ is two-dimensional and $H$ is a smooth curve in $M$. | |||
Revision as of 10:40, 26 September 2025
Organizers: Lars Niedorf and Andreas Seeger
Emails:
- Lars Niedorf: niedorf at math dot wisc dot edu
- Andreas Seeger: seeger at math dot wisc dot edu
Time and Room: Wednesdays, 3:30--4:30 p.m., Van Vleck B235
In some cases the seminar may be scheduled at different time to accommodate speakers.
If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+subscribe (at) g-groups (dot) wisc (dot) edu
| Date | Speaker | Institution | Title | Host(s) | Notes (e.g. unusual room/day/time) |
|---|---|---|---|---|---|
| September 10 | Giorgio Young | UW Madison | Wavepacket spread for almost periodic and partially periodic Schrödinger operators | ||
| September 17 | Rajula Srivastava | UW Madison | A Fourier Analytic Approach to Count Integer Points near Space Curves | ||
| September 24 | Ben Johnsrude | UW Madison | Restriction theory, p-adic analysis, and the decoupling bargain | ||
| October 1 | Lars Niedorf | UW Madison | Spectral multipliers for the Kohn-Laplacian on the Heisenberg group | ||
| Friday/Saturday October 3/4 | Harmonic Analysis Workshop. | Workshop Titles and Abstracts. | Andreas, Betsy, Brian | Talks in VV 911 or B239 (see the linked schedule) | |
| Friday, October 3, 4:00 p.m. | Hong Wang | NYU & IHES | Restriction theory and projection theorems | Andreas, Betsy, Brian | Colloquium integrated in the workshop, VV B239 |
| October 8 | Jaehyeon Ryu | EWHA Womans University | Restriction estimates for spectral projections | Andreas, Lars | |
| October 15 | John Treuer | UC San Diego | Xianghong | ||
| October 22 | Burak Hatinoglu | Michigan State | Alexei | ||
| October 29 | Niclas Technau | UW Madison | |||
| November 5 | Marco Fraccaroli | UMass Lowell | Andreas | ||
| November 12, 4.00 pm. | Rachel Greenfeld | Northwestern | Andreas, Betsy | Wednesday Colloquium in VV B239. | |
| November 19 | Sarah Tammen | UW Madison | |||
| November 26 | --- | No seminar | |||
| December 3 | Terry Harris | UW Madison |
Abstracts
Giorgio Young
Title: Wavepacket spread for almost periodic and partially periodic Schr\"odinger operators
Abstract: In this talk, we will provide an introduction to quantum dynamics, and introduce some notions of wavepacket spread, including ballistic transport and dispersion. We will then present some results showing ballistic transport for one dimensional continuum Schr\"odinger operators with limit-periodic potentials, as well as ``directional ballistic transport" for higher dimensional discrete and continuum Schr\"odinger operators with potentials supported, and periodic, on a strip. Finally, we will describe a recent result finding dispersion at the optimal rate for periodic discrete Schr\"odinger operators. This talk contains joint work with Adam Black, David Damanik, Jake Fillman, and Tal Malinovitch.
Rajula Srivastava
Title: A Fourier Analytic Approach to Count Integer Points near Space Curves
Abstract: We present recent results giving both upper and lower bounds for the number of integral points lying near smooth nondegenerate curves in $\mathbb{R}^n$. These estimates are new in dimensions $n\geq 4$, and in the case $n\geq 3$ they recover an earlier result of J. J. Huang. Our approach differs significantly from previous work: instead of relying on sharp planar counting results, we employ Fourier analytic methods. A key ingredient is an Arkhipov–Chubarikov–Karatsuba-type oscillatory integral estimate. This is joint work with Jonathan Hickman (Edinburgh).
Ben Johnsrude
Title: Restriction theory, p-adic analysis, and the decoupling bargain
Abstract: Restriction theory is concerned with understanding delicate properties of functions with Fourier support contained in a submanifold of space. The major advancement in the subject of the last ten years has been the development of decoupling theory; the latter has led to the solution of several problems in restriction theory and geometric measure theory, and made substantial quantitative progress in others. On the other hand, the framework of decoupling imposes certain limits on the analysis, which in particular prevents one from solving problems such as Mizohata--Takeuchi without first exiting the framework.
We survey some outstanding problems in restriction theory and the current state of knowledge therein. We discuss Guth's argument that decoupling is insufficient to solve Mizohata--Takeuchi. We then introduce a novel tool that allows one to go beyond the limits of the usual decoupling framework in the topics surveyed.
Lars Niedorf
Title: Spectral multipliers for the Kohn-Laplacian on the Heisenberg group
Abstract: We discuss recent sharp Bochner-Riesz estimates for Kohn-Laplacians on Heisenberg groups. Since favorable restriction estimates fail in this setting, we developed an approach that relies on square function bounds for the wave equation associated with the Kohn-Laplacian. This is joint work with Detlef Müller, Andreas Seeger, and Betsy Stovall.
Hong Wang
Title: Restriction theory and projection theorems
Abstract: Restriction theory studies functions whose Fourier transforms are supported on some curved manifold in R^n (for example, solutions to the linear Schrodinger equation or to the wave equation). Projection theorems study the Hausdorff dimension of fractal sets under orthogonal projections from R^n to its subspaces. We will survey some recent works in both fields and discuss their interactions.
Jaehyeon Ryu
Title: Restriction estimates for spectral projections
Abstract: Restriction estimates for spectral projections have been widely studied since the work of Burq, Gérard, and Tzvetkov as a method for investigating eigenfunction concentration. The problem of establishing the optimal $L^p$ bounds for the restriction of Laplace-Beltrami eigenfunctions remains open, particularly when the restriction submanifold is of codimension 1 or 2. This talk will explore how these optimal bounds are characterized by the geometry of the underlying manifold $M$ and its submanifold $H$, focusing on the case where $M$ is two-dimensional and $H$ is a smooth curve in $M$.