Difference between revisions of "Analysis Seminar"
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Quadratic forms and the semiclassical eigenfunction hypothesis | Quadratic forms and the semiclassical eigenfunction hypothesis | ||
− | Let <math>$Q(X)$</math> be any integral primitive positive definite quadratic form in $k$ variables, where $k\geq4$, and discriminant $D$. For any integer $n$, we give an upper bound on the number of integral solutions of $Q(X)=n$ in terms of $n$, $k$, and $D$. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus $\mathbb{T}^d$ for $d\geq 5$. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis. | + | Let <math>$Q(X)$</math> be any integral primitive positive definite quadratic form in <math>$k$</math> variables, where $k\geq4$, and discriminant $D$. For any integer $n$, we give an upper bound on the number of integral solutions of $Q(X)=n$ in terms of $n$, $k$, and $D$. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus $\mathbb{T}^d$ for $d\geq 5$. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis. |
===Name=== | ===Name=== |
Revision as of 13:02, 8 September 2017
Analysis Seminar
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.
If you wish to invite a speaker please contact Betsy at stovall(at)math
Contents
Previous Analysis seminars
Summer/Fall 2017 Analysis Seminar Schedule
date | speaker | institution | title | host(s) |
---|---|---|---|---|
September 8 in B239 | Tess Anderson | UW Madison | Title | |
September 12 | Title | |||
September 19 | Brian Street | UW Madison | Title | Betsy |
September 26 | Hiroyoshi Mitake | Hiroshima University | Title | Hung |
October 3 | Joris Roos | UW Madison | Title | Betsy |
October 10 | Michael Greenblatt | UI Chicago | Title | Andreas |
October 17 | David Beltran | Bilbao | Title | Andreas |
October 24 | Xiaochun Li | UIUC | Title | Betsy |
Thursday, October 26 | Fedya Nazarov | Kent State University | Title | Betsy, Andreas |
Friday, October 27 in B239 | Stefanie Petermichl | University of Toulouse | Title | Betsy, Andreas |
November 14 | Naser Talebizadeh Sardari | UW Madison | Title | Betsy |
November 28 | Xianghong Chen | UW Milwaukee | Title | Betsy |
December 5 | Title | |||
December 12 | Alex Stokolos | GA Southern | Title | Andreas |
Abstracts
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Name
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Naser Talebizadeh Sardari
Quadratic forms and the semiclassical eigenfunction hypothesis
Let be any integral primitive positive definite quadratic form in variables, where $k\geq4$, and discriminant $D$. For any integer $n$, we give an upper bound on the number of integral solutions of $Q(X)=n$ in terms of $n$, $k$, and $D$. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus $\mathbb{T}^d$ for $d\geq 5$. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.
Name
Title
Abstract