Model Theory: Connecting Algebraic, Analytic, and Diophantine Geometry Through Definability

模型理论：通过可定义性连接代数、解析和丢番图几何

• 批准号: 1800492
• 负责人: Thomas Scanlon
• 金额: $ 22万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800492&HistoricalAwards=false
• 关键词: Model; Theory; Connecting; Algebraic; Analytic

Important mathematical structures can often be understood from very different perspectives, either by analyzing them with algebraic or analytic formulas or by regarding them geometrically. This dual algebraic/geometric view is applied in many branches of mathematics, especially for the study of algebraic equations involving numbers (under the name of Diophantine geometry), the study of solutions to polynomial equations (under the name of algebraic geometry), or the study of the possible algebraic relations among solutions to systems of differential or difference equations (under the names of differential algebraic geometry or difference algebraic geometry, respectively), among others. In practice, some of the questions considered in these areas suffer from extreme complexity inherited from their connection to number theory or to even more complicated domains. Work in mathematical logic has elucidated the boundary between those complicated theories and those admitting a tame, geometric theory. This research project aims to extend the class of theories for which a tame geometry can be established and to use these results from mathematical logic to answer questions from the target domains. This research project studies the connections between geometries of various kinds, including algebraic, differential, Diophantine, and analytic geometries, through the model-theoretic lens of definability in suitable theories. Mathematical theories as diverse as those of partial differential equations, difference equations, perfectoid spaces, formal geometry, algebraic dynamics, and homogenous dynamics will be studied. Technically, methods including geometric stability theory as applied to differentially closed fields, o-minimality, and quantifier elimination for valued differential fields and analytic difference rings will be applied for the purpose of answering questions internal to model theory (such as proving or refuting the trichotomy principle for regular types in differentially closed fields with several commuting derivations) and for applications to problems in functional transcendence, dynamics, and Diophantine geometry. It is anticipated that the work will have applications to the structure of algebraic differential equations, the arithmetic of dynamical systems, and mathematical physics, as well as in other areas.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.

通常，可以通过以代数或分析公式分析它们，或通过几何形式分析它们，从非常不同的角度来理解重要的数学结构。 这种双重代数/几何观点适用于数学的许多分支，尤其是用于研究涉及数字的代数方程（以毒液几何的名义）的研究，对多项式方程的解决方案的研究（根据代数几何形状的名称）或在溶液中的区别不同的范围的差异范围（根据代数几何学的名称）的研究（均不相同），或者是区别的差异。代数几何形状或差异几何形状）等。在实践中，这些领域中考虑的一些问题遭受了与数字理论的联系或更复杂的领域所继承的极端复杂性。数学逻辑中的工作阐明了那些复杂的理论与承认温和的几何理论的界限。该研究项目旨在扩展可以建立驯服几何形状的理论类别，并使用数学逻辑中的这些结果来回答目标领域的问题。该研究项目通过合适理论中的模型理论镜头研究了各种几何形状之间的几何形状之间的联系，包括代数，差异，二苯胺和分析几何形状。 将研究将研究与部分微分方程，差异方程，完美的空间，形式几何形状，代数动力学和同质动力学的数学理论。从技术上讲，将应用于差异差异差异领域和分析差环的差异封闭场，O最小性和消除量化量的方法，将应用于回答模型理论内部的问题（例如证明或驳斥了几项近距离范围内的差异性范围，以及用于衍生功能的差异性类型）的目的，以解决模型理论的内部问题（例如证明或驳斥三分法原则）养生几何形状。可以预料，这项工作将在代数微分方程，动力系统和数学物理学以及其他领域的结构上应用。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响审查的评估来通过评估来支持的。

项目成果

A variant of the Mordell–Lang conjecture

莫德尔·朗猜想的一种变体

• DOI: 10.4310/mrl.2019.v26.n5.a7
• 发表时间: 2019
• 期刊: Mathematical Research Letters
• 影响因子: 1
• 作者: Ghioca, Dragos;Hu, Fei;Scanlon, Thomas;Zannier, Umberto
• 通讯作者: Zannier, Umberto

The Logical Complexity of Finitely Generated Commutative Rings

有限生成交换环的逻辑复杂性

• DOI: 10.1093/imrn/rny023
• 发表时间: 2018
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Aschenbrenner, Matthias;Khélif, Anatole;Naziazeno, Eudes;Scanlon, Thomas
• 通讯作者: Scanlon, Thomas

Elimination of unknowns for systems of algebraic differential-difference equations

• DOI: 10.1090/tran/8219
• 发表时间: 2018-12
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Wei Li;A. Ovchinnikov;G. Pogudin;T. Scanlon

SOLVING DIFFERENCE EQUATIONS IN SEQUENCES: UNIVERSALITY AND UNDECIDABILITY

• DOI: 10.1017/fms.2020.14
• 发表时间: 2019-09
• 期刊: Forum of Mathematics, Sigma
• 作者: G. Pogudin;T. Scanlon;M. Wibmer

Effective difference elimination and Nullstellensatz

• DOI: 10.4171/jems/968
• 发表时间: 2017-12
• 期刊: arXiv: Commutative Algebra
• 作者: A. Ovchinnikov;G. Pogudin;T. Scanlon