Fall 2021 and Spring 2022 Analysis Seminars: Difference between revisions

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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 <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 <math>d\geq 5</math>. 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 <math>k\geq4</math>,  and discriminant $D$. For any integer $n$, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> 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 <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.


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Revision as of 18:03, 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

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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Naser Talebizadeh Sardari

Quadratic forms and the semiclassical eigenfunction hypothesis

Let [math]\displaystyle{ Q(X) }[/math] be any integral primitive positive definite quadratic form in [math]\displaystyle{ k }[/math] variables, where [math]\displaystyle{ k\geq4 }[/math], and discriminant $D$. For any integer $n$, we give an upper bound on the number of integral solutions of [math]\displaystyle{ Q(X)=n }[/math] 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 [math]\displaystyle{ d\geq 5 }[/math]. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.

Name

Title

Abstract


Extras

Blank Analysis Seminar Template

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