Analytic problems around automorphic forms and L-functions

围绕自守形式和 L 函数的分析问题

基本信息

  • 批准号:
    2302210
  • 负责人:
  • 金额:
    $ 24.56万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2023
  • 资助国家:
    美国
  • 起止时间:
    2023-09-01 至 2026-08-31
  • 项目状态:
    未结题

项目摘要

One of the main tools for understanding the distribution of prime numbers is through properties of the Riemann zeta function. The zeta function is the most fundamental example of an L-function, which is a mathematical construction that combines arithmetical information about all the primes at once. More general L-functions, such as Dirichlet L-functions, are useful for understanding primes in arithmetic progressions. One of the main ways that L-functions are studied is by placing them into families, such as the family of all Dirichlet L-functions, and viewing their properties through this framework. Much of the proposed work in this proposal concerns the development of properties of new families of L-functions. One of the main goals is to better-understand the size of these L-functions, especially in certain ranges that have been inaccessible using previous methods. The PI will continue to mentor and collaborate with undergraduate students, particularly through the Texas A&M REU. Such opportunities are important for preparing students for graduate studies, particularly for undergraduate students from non-PhD granting institutions as well as from population groups underrepresented in STEM fields. The PI will also continue to advise PhD students to work on problems related to families of L-functions and their moments.The PI will study new families of automorphic forms and their associated L-functions, especially via moments of L-functions and large sieve inequalities. The PI plans to study high moments of L-functions in order to make progress on the challenging but important L-functions in conductor-dropping families. The proposer will also study narrower families of L-functions through the use of new versions of the relative trace formula that isolate small families based on their local behavior. In a related vein, the proposer will study large sieve inequalities for families of automorphic forms, with two main goals. One objective is to establish large sieve bounds in some of the new, narrow families. A second goal is to develop heuristics for conjecturing the size of a large sieve bound for more general families. The PI will mentor PhD students on problems on moments of L-functions for both narrow families and for higher degree L-functions. The proposer will study newform Dedekind sums with his undergraduate students. The methods employed will be techniques from analytic number theory such as functional equations, exponential sums and integrals, and the spectral theory of automorphic forms, including the Arthur-Selberg trace formula and the relative trace formula.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
One of the main tools for understanding the distribution of prime numbers is through properties of the Riemann zeta function. The zeta function is the most fundamental example of an L-function, which is a mathematical construction that combines arithmetical information about all the primes at once. More general L-functions, such as Dirichlet L-functions, are useful for understanding primes in arithmetic progressions. One of the main ways that L-functions are studied is by placing them into families, such as the family of all Dirichlet L-functions, and viewing their properties through this framework. Much of the proposed work in this proposal concerns the development of properties of new families of L-functions. One of the main goals is to better-understand the size of these L-functions, especially in certain ranges that have been inaccessible using previous methods. The PI will continue to mentor and collaborate with undergraduate students, particularly through the Texas A&M REU. Such opportunities are important for preparing students for graduate studies, particularly for undergraduate students from non-PhD granting institutions as well as from population groups underrepresented in STEM fields. The PI will also continue to advise PhD students to work on problems related to families of L-functions and their moments.The PI will study new families of automorphic forms and their associated L-functions, especially via moments of L-functions and large sieve inequalities. The PI plans to study high moments of L-functions in order to make progress on the challenging but important L-functions in conductor-dropping families. The proposer will also study narrower families of L-functions through the use of new versions of the relative trace formula that isolate small families based on their local behavior. In a related vein, the proposer will study large sieve inequalities for families of automorphic forms, with two main goals. One objective is to establish large sieve bounds in some of the new, narrow families. A second goal is to develop heuristics for conjecturing the size of a large sieve bound for more general families. The PI will mentor PhD students on problems on moments of L-functions for both narrow families and for higher degree L-functions. The proposer will study newform Dedekind sums with his undergraduate students. The methods employed will be techniques from analytic number theory such as functional equations, exponential sums and integrals, and the spectral theory of automorphic forms, including the Arthur-Selberg trace formula and the relative trace formula.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

项目成果

期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)

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Matthew Young其他文献

Development of an autonomous robotic system for terrain mapping
开发用于地形测绘的自主机器人系统
Spontaneous, Intrasphenoidal Rupture of Ecchordosis Physaliphora with Pneumocephalus Captured During Serial Imaging and Clinical Follow-Up: Pathoanatomic Features and Management
  • DOI:
    10.1016/j.wneu.2020.05.220
  • 发表时间:
    2020-09-01
  • 期刊:
  • 影响因子:
  • 作者:
    Ahrya Derakhshani;Stephanie Livingston;Christopher William;Seth Lieberman;Matthew Young;Donato Pacione;Seena Dehkharghani
  • 通讯作者:
    Seena Dehkharghani
The Future of Front-line Metastatic Bladder Cancer is Platinum-free
晚期转移性膀胱癌的未来是无铂治疗
  • DOI:
    10.1016/j.euf.2024.04.005
  • 发表时间:
    2024-03-01
  • 期刊:
  • 影响因子:
    5.600
  • 作者:
    Matthew Young
  • 通讯作者:
    Matthew Young
2204 IS MRI WITH DIFFUSION WEIGHTED IMAGING EFFECTIVE IN DETECTING PROSTATE CANCER IN MEN WITH PREVIOUS NEGATIVE BIOPSIES?
  • DOI:
    10.1016/j.juro.2013.02.2113
  • 发表时间:
    2013-04-01
  • 期刊:
  • 影响因子:
  • 作者:
    Matthew Young;Ryan Levey;James Rosoff;Josh Smith;George Ghareeb;Brian Lane;Andrew Hardie;Thomas Keane;Stephen Savage
  • 通讯作者:
    Stephen Savage
Ocean connectivity drives trophic support for consumers in an intermittently closed coastal lagoon
  • DOI:
    10.1016/j.ecss.2021.107665
  • 发表时间:
    2022-01-05
  • 期刊:
  • 影响因子:
  • 作者:
    Matthew Young;Frederick Feyrer;Darren Fong;Rachel Johnson;Tamara Kraus;Veronica Larwood;Elizabeth Stumpner;Megan Young
  • 通讯作者:
    Megan Young

Matthew Young的其他文献

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{{ truncateString('Matthew Young', 18)}}的其他基金

Representation theory in unoriented and non-semisimple physics
无向和非半简单物理中的表示论
  • 批准号:
    2302363
  • 财政年份:
    2023
  • 资助金额:
    $ 24.56万
  • 项目类别:
    Standard Grant
Families of L-Functions and Analytic Number Theory
L 函数族和解析数论
  • 批准号:
    2001306
  • 财政年份:
    2020
  • 资助金额:
    $ 24.56万
  • 项目类别:
    Standard Grant
Automorphic Forms and L-Functions
自守形式和 L 函数
  • 批准号:
    1702221
  • 财政年份:
    2017
  • 资助金额:
    $ 24.56万
  • 项目类别:
    Standard Grant
Analytic theory of automorphic forms
自守形式的解析理论
  • 批准号:
    1401008
  • 财政年份:
    2014
  • 资助金额:
    $ 24.56万
  • 项目类别:
    Standard Grant
Families of L-functions and automorphic forms
L 函数族和自守形式
  • 批准号:
    1101261
  • 财政年份:
    2011
  • 资助金额:
    $ 24.56万
  • 项目类别:
    Standard Grant
Mean values f L-functions
L 函数平均值
  • 批准号:
    0758235
  • 财政年份:
    2008
  • 资助金额:
    $ 24.56万
  • 项目类别:
    Continuing Grant
PostDoctoral Research Fellowship
博士后研究奖学金
  • 批准号:
    0402999
  • 财政年份:
    2004
  • 资助金额:
    $ 24.56万
  • 项目类别:
    Fellowship

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复杂图像处理中的自由非连续问题及其水平集方法研究
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