Fall 2021 and Spring 2022 Analysis Seminars: Difference between revisions
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Since Bourgain and Demeter's seminal 2017 decoupling result for nondegenerate hypersurfaces, several attempts have been made to extend the theory to degenerate hypersurfaces $M$. In this talk, we will discuss using surfaces derived from the local Taylor expansions of $M$ in order to obtain "approximate" decoupling results. By themselves, these approximate decouplings do not avail much. However, upon considerate iteration, for a specifically chosen $M$, they culminate in a decoupling partition of $M$ into caps small enough either as originally desired or otherwise genuinely nondegenerate at the local scale. A key feature that will be discussed is the notion of approximating a non-convex hypersurface $M$ by convex hypersurfaces at various scales. In this manner, contrary to initial intuition, non-trivial $\ell^2$ decoupling results will be obtained for $M$. | Since Bourgain and Demeter's seminal 2017 decoupling result for nondegenerate hypersurfaces, several attempts have been made to extend the theory to degenerate hypersurfaces $M$. In this talk, we will discuss using surfaces derived from the local Taylor expansions of $M$ in order to obtain "approximate" decoupling results. By themselves, these approximate decouplings do not avail much. However, upon considerate iteration, for a specifically chosen $M$, they culminate in a decoupling partition of $M$ into caps small enough either as originally desired or otherwise genuinely nondegenerate at the local scale. A key feature that will be discussed is the notion of approximating a non-convex hypersurface $M$ by convex hypersurfaces at various scales. In this manner, contrary to initial intuition, non-trivial $\ell^2$ decoupling results will be obtained for $M$. | ||
=== | ===Jack Burkart=== | ||
Transcendental Julia Sets with Fractional Packing Dimension | |||
If f is an entire function, the Julia set of f is the set of all points such that f and its iterates locally do not form a normal family; nearby points have very different orbits under iteration by f. A topic of interest in complex dynamics is studying the fractal geometry of the Julia set. | |||
In this talk, we will discuss my thesis result where I construct non-polynomial (transcendental) entire functions whose Julia set has packing dimension strictly between (1,2). We will introduce various notions of dimension and basic objects in complex dynamics, and discuss a history of dimension results in complex dynamics. We will discuss some key aspects of the proof, which include a use of Whitney decompositions of domains as a tool to calculate the packing dimension, and some open questions I am thinking about. | |||
===Name=== | ===Name=== | ||
Revision as of 18:01, 24 September 2021
The 2021-2022 Analysis Seminar will be organized by David Beltran and Andreas Seeger. Some of the talks will be in person (room Van Vleck B139) and some will be online. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate speakers).
Zoom links will be sent to those who have signed up for the Analysis Seminar List. If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+join (at) g-groups (dot) wisc (dot) edu. If you are from an institution different than UW-Madison, please send as well as an additional email to David and Andreas (dbeltran, seeger at math (dot) wisc (dot) edu) to notify the request.
If you'd like to suggest speakers for the fall semester please contact David and Andreas.
Analysis Seminar Schedule
| date | speaker | institution | title | host(s) |
|---|---|---|---|---|
| September 21, VV B139 | Dóminique Kemp | UW-Madison | Decoupling by way of approximation | |
| September 28, VV B139 | UW-Madison | Jack Burkart | Transcendental Julia Sets with Fractional Packing Dimension | |
| October 5, Online | Giuseppe Negro | University of Birmingham | Stability of sharp Fourier restriction to spheres | |
| October 12, VV B139 | Rajula Srivastava, | UW Madison | Title | |
| October 19, Online | Itamar Oliveira, | Cornell University | Title | |
| October 26, VV B139 | Changkeun Oh | UW Madison | Title | |
| November 2, VV B139 | Liding Yao | UW Madison | Title | |
| November 9, VV B139 | Lingxiao Zhang | UW Madison | Title | |
| November 16, VV B139 | Rahul Parhi | UW Madison (EE) | Title | |
| November 30, VV B139 | Alexei Poltoratski | UW Madison | Title | |
| December 7 | Person | Institution | Title | |
| December 14 | Person | Institution | Title | |
| Date | Person | Institution | Title |
Abstracts
Dóminique Kemp
Decoupling by way of approximation
Since Bourgain and Demeter's seminal 2017 decoupling result for nondegenerate hypersurfaces, several attempts have been made to extend the theory to degenerate hypersurfaces $M$. In this talk, we will discuss using surfaces derived from the local Taylor expansions of $M$ in order to obtain "approximate" decoupling results. By themselves, these approximate decouplings do not avail much. However, upon considerate iteration, for a specifically chosen $M$, they culminate in a decoupling partition of $M$ into caps small enough either as originally desired or otherwise genuinely nondegenerate at the local scale. A key feature that will be discussed is the notion of approximating a non-convex hypersurface $M$ by convex hypersurfaces at various scales. In this manner, contrary to initial intuition, non-trivial $\ell^2$ decoupling results will be obtained for $M$.
Jack Burkart
Transcendental Julia Sets with Fractional Packing Dimension
If f is an entire function, the Julia set of f is the set of all points such that f and its iterates locally do not form a normal family; nearby points have very different orbits under iteration by f. A topic of interest in complex dynamics is studying the fractal geometry of the Julia set.
In this talk, we will discuss my thesis result where I construct non-polynomial (transcendental) entire functions whose Julia set has packing dimension strictly between (1,2). We will introduce various notions of dimension and basic objects in complex dynamics, and discuss a history of dimension results in complex dynamics. We will discuss some key aspects of the proof, which include a use of Whitney decompositions of domains as a tool to calculate the packing dimension, and some open questions I am thinking about.
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Previous_Analysis_seminars
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars
Extras
Blank Analysis Seminar Template
Graduate Student Seminar:
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html