Model Theory and Difference Algebra

模型理论与差分代数

• 批准号: 1500976
• 负责人: Alice Medvedev
• 金额: $ 10.34万
• 依托单位: CUNY City College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500976&HistoricalAwards=false
• 关键词: Model; Theory; Difference; Algebra

Logic is the study of formal reasoning rules that rely on the grammatical structure of the statements rather than their content. Revived in the beginning of the twentieth century to deal with some foundational issues, mathematical logic turned out to also be useful for obtaining new results in mathematics. The recent applications of the model theory of difference and differential algebra to algebraic number theory are some of the most exciting examples of this. Difference equations, like differential equations, model real-world processes that change over time. Differential equations describe quantities that vary continuously with time, such as the positions of planets in space, while difference equations describe quantities that are only measured at discrete time intervals, such as the annual GDP of a country. Difference algebra is the abstract setting for studying difference equations; it also has applications to the kind of algebraic number theory that underpins modern cryptography and internet security. Medvedev proposes to study the fine structure of solution sets of difference equations: to find algorithms for computing their dimensions and for identifying the very special cases where it is possible to define some kind of addition and/or multiplication on these sets.A difference field is a field with a distinguished automorphism. The theory of difference closed fields is supersimple, meaning that Lascar rank is a good notion of dimension for complete types. Furthermore, the complete types of Lascar rank 1 satisfy the Zilber Trichotomy: each is nonorthogonal to a definable field, or nonorthogonal to a definable one-based group, or is disintegrated. While the fieldlike case of the trichotomy is relatively easy to identify in explicit examples, the dividing line between the grouplike and the disintegrated cases is much less clear. Similarly, even in the relatively nice case of groups, it is not always easy to determine the Lascar rank of a particular type. Beginning with her PhD thesis, Medvedev has worked on these sorts of problems, and on applications to algebraic dynamics. She is confident that she can generalize the main theorem of her PhD thesis from curves to higher-dimensional algebraic varieties. She has already accomplished the first part of this in far greater generality; the last part of the original proof should generalize easily; and the middle piece is supplied by an observation in a paper by Chatzidakis and Hrushovski. She expects to also obtain concrete results on the Lascar rank of certain groups G defined by systems of polynomial difference equations. This question can be translated to the language of linear algebra over the quasiendomorphism ring of the underlying algebraic group. For example, when G is a subgroup of the multiplicative group of a field in characteristic zero, this question reduces to linear algebra over the field of rational numbers. In addition, Medvedev proposes to continue expanding her foundational notes about difference schemes defined by Hrushovski in his work on the model theory of Frobenius automorphisms, filling in many missing detail, adding enlightening examples, and reorganizing the presentation to make it (more) understandable. Medvedev expects to continue involving students in this work, encouraging logicians and algebraic geometers to learn each other's languages when they are still young.

逻辑是对形式推理规则的研究，这些规则依赖于陈述的语法结构而不是其内容。 数理逻辑在二十世纪初为了解决一些基础问题而复兴，事实证明它对于获得数学新结果也很有用。最近差分模型论和微分代数在代数数论中的应用是这方面最令人兴奋的例子。差分方程与微分方程一样，可以模拟随时间变化的现实世界过程。微分方程描述随时间连续变化的量，例如行星在太空中的位置，而差分方程描述仅在离散时间间隔测量的量，例如一个国家的年GDP。差分代数是研究差分方程的抽象设置；它还应用于支撑现代密码学和互联网安全的代数数论。 梅德韦杰夫建议研究差分方程解集的精细结构：找到计算其维数的算法，并识别可以在这些集合上定义某种加法和/或乘法的非常特殊的情况。差分域是具有显着自同构的域。差分闭域理论非常简单，这意味着拉斯卡等级是完整类型的一个很好的维数概念。此外，Lascar 等级 1 的完整类型满足 Zilber 三分法：每个类型都与可定义的域非正交，或者与可定义的基于 1 的群非正交，或者是分解的。虽然三分法的领域案例在明确的例子中相对容易识别，但群体案例和分解案例之间的分界线却不太清晰。同样，即使在相对较好的群体情况下，确定特定类型的拉斯卡等级也并不总是那么容易。从她的博士论文开始，梅德韦杰夫就致力于研究这类问题以及代数动力学的应用。她有信心能够将博士论文的主要定理从曲线推广到高维代数簇。她已经更普遍地完成了第一部分；原始证明的最后部分应该很容易概括；中间部分是查齐达基斯和赫鲁索夫斯基在一篇论文中的观察结果。她还期望获得由多项式差分方程组定义的某些群 G 的拉斯卡秩的具体结果。这个问题可以转化为基础代数群的拟同态环上的线性代数语言。例如，当 G 是特征零域的乘法群的子群时，这个问题就简化为有理数域上的线性代数。此外，梅德韦杰夫建议继续扩展赫鲁索夫斯基在弗罗贝尼乌斯自同构模型理论工作中定义的差分格式的基础笔记，填补许多缺失的细节，添加启发性的例子，并重新组织演示文稿以使其（更）易于理解。梅德韦杰夫希望继续让学生参与这项工作，鼓励逻辑学家和代数几何学家在年轻时学习彼此的语言。