CAREER: Undecidability in Number Theory and Applications of Arithmetic Geometry

职业:数论中的不可判定性和算术几何的应用

基本信息

项目摘要

The investigator will study several questions related to diophantine equations. One focus is on generalizations of Hilbert's Tenth Problem. Hilbert's Tenth Problem over the rational numbers and over number fields in general is still open while undecidability is known for several subrings of the rational numbers. One goal is to investigate for which subrings of number fields the problem is undecidable. Another is to extend the currently known undecidability results for function fields. Research on Hilbert's Tenth Problem has led to new areas of interaction between number theory, arithmetic geometry and logic. These will be further explored in this project. Another focus is the study of computational problems in arithmetic geometry that have applications to cryptography. One goal is to construct curves of small genus that are suitable for cryptographic applications. Another goal is to generalize the quantum algorithms for number fields to function fields.The investigator proposes several research projects that involve studying the solutions to multivariable polynomial equations. Looking for solutions to such equations over the integers or rational numbers is one of the fundamental problems in number theory. It has a long history that goes back to ancient Greece. For the first project the investigator will study the fundamental question of whether it is possible to find a procedure that determines whether an arbitrary multivariable polynomial equation has a solution in a given number system. The second project focuses on computational aspects of certain special classes of equations that have applications to cryptography. This area of mathematics is very well-suited for motivating young students to study mathematics. The investigator will teach middle school and high school girls about cryptography and its mathematical background. There will be two yearly workshops which will involve hands-on experiments in the computer lab. There will also be professional development workshops for local mathematics teachers.
研究人员将研究与丢番图方程有关的几个问题。其中一个焦点是希尔伯特第十问题的推广。一般情况下,Hilbert关于有理数域和过数域的第十个问题仍然是公开的,而对于有理数的几个子环是已知的不可判定的。一个目标是调查数域的哪些子环的问题是不可判定的。另一种方法是推广目前已知的函数域的不可判定性结果。对希尔伯特第十问题的研究开创了数论、算术几何和逻辑相互作用的新领域。这些都将在本项目中进一步探索。另一个焦点是研究算术几何中应用于密码学的计算问题。一个目标是构造适合于密码学应用的小亏格曲线。另一个目标是将数域的量子算法推广到函数域。研究人员提出了几个研究项目,涉及研究多变量多项式方程的解。在整数或有理数上求这类方程的解是数论的基本问题之一。它有着悠久的历史,可以追溯到古希腊。对于第一个项目,研究人员将研究一个基本问题,即是否有可能找到一个程序来确定一个任意的多变量多项式方程在给定的数系中是否有解。第二个项目集中在某些特殊类型的方程的计算方面,这些方程在密码学中有应用。这一数学领域非常适合激励年轻学生学习数学。这位调查员将向初中和高中的女孩传授密码学及其数学背景。每年将有两次工作坊,其中包括在计算机实验室进行动手实验。此外,亦会为本地数学教师举办专业发展工作坊。

项目成果

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Kirsten Eisentraeger其他文献

Hilbert's Tenth Problem for function fields of varieties over C
C 上簇函数域的希尔伯特第十问题
  • DOI:
  • 发表时间:
    2006
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Kirsten Eisentraeger
  • 通讯作者:
    Kirsten Eisentraeger
Descent on elliptic curves and Hilbert’s tenth problem
椭圆曲线下降和希尔伯特第十问题
  • DOI:
  • 发表时间:
    2007
  • 期刊:
  • 影响因子:
    0
  • 作者:
    G. Everest;Kirsten Eisentraeger
  • 通讯作者:
    Kirsten Eisentraeger
Constructing Picard curves with complex multiplication using the Chinese remainder theorem
使用中国剩余定理通过复数乘法构造皮卡德曲线
  • DOI:
    10.2140/obs.2019.2.21
  • 发表时间:
    2018
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Sonny Arora;Kirsten Eisentraeger
  • 通讯作者:
    Kirsten Eisentraeger
Hilbert’s Tenth Problem for function fields of varieties over algebraically closed fields of positive characteristic
正特征代数闭域上簇函数域的希尔伯特第十问题
  • DOI:
    10.1007/s00605-011-0364-7
  • 发表时间:
    2011
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Kirsten Eisentraeger
  • 通讯作者:
    Kirsten Eisentraeger
Hilbert’s tenth problem for algebraic function fields of characteristic 2
特征 2 的代数函数域的希尔伯特第十问题
  • DOI:
    10.2140/pjm.2003.210.261
  • 发表时间:
    2002
  • 期刊:
  • 影响因子:
    0.6
  • 作者:
    Kirsten Eisentraeger
  • 通讯作者:
    Kirsten Eisentraeger

Kirsten Eisentraeger的其他文献

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{{ truncateString('Kirsten Eisentraeger', 18)}}的其他基金

SaTC: CORE: Small: Classical and quantum algorithms for number-theoretic problems arising in cryptography
SaTC:核心:小:密码学中出现的数论问题的经典和量子算法
  • 批准号:
    2001470
  • 财政年份:
    2020
  • 资助金额:
    $ 41.14万
  • 项目类别:
    Standard Grant
TWC: Small: Algorithms for Number-Theoretic Problems Arising in Cryptography
TWC:小:密码学中出现的数论问题的算法
  • 批准号:
    1617802
  • 财政年份:
    2016
  • 资助金额:
    $ 41.14万
  • 项目类别:
    Standard Grant
Extensions of Hilbert's Tenth Problem and Computational Aspects of Arithmetic Geometry
希尔伯特第十个问题的推广和算术几何的计算方面
  • 批准号:
    0801123
  • 财政年份:
    2008
  • 资助金额:
    $ 41.14万
  • 项目类别:
    Standard Grant
PostDoctoral Research Fellowship in the Mathematical Sciences
数学科学博士后研究奖学金
  • 批准号:
    0503132
  • 财政年份:
    2005
  • 资助金额:
    $ 41.14万
  • 项目类别:
    Fellowship Award

相似海外基金

On decidability and undecidability of type-related problems of lambda-calculi
关于 lambda 演算类型相关问题的可判定性和不可判定性
  • 批准号:
    25400192
  • 财政年份:
    2013
  • 资助金额:
    $ 41.14万
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)
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