Model Theory and Difference Algebra

逻辑是对依赖于语句的语法结构而不是其内容的形式推理规则的研究。数理逻辑在二十世纪初复兴起来，用来处理一些基本问题，结果证明，它也有助于在数学中获得新的结果。最近，差分和微分代数模型理论在代数数论中的应用就是其中一些最令人兴奋的例子。差分方程式和微分方程式一样，都是模拟随时间变化的真实世界过程。微分方程式描述的是随时间连续变化的量，如行星在空间中的位置，而差分方程式描述的是仅以离散时间间隔测量的量，如一个国家的年度GDP。差分代数是研究差分方程式的抽象背景；它也可以应用于支撑现代密码学和网络安全的那种代数论。梅德韦杰夫建议研究差分方程解集合的精细结构：找到算法来计算它们的维度，并识别在这些集合上可以定义某种加法和/或乘法的非常特殊的情况。差场是具有显著自同构的场。差闭域的理论是超简单的，这意味着Lascar秩是完备类型的一个很好的维度概念。此外，Lascar秩1的完备类型满足Zilber三分法：每种类型都与可定域非正交，或与可定义的一基群非正交，或可分解。虽然在明确的例子中比较容易识别三分式的田野案例，但群体案例和解体案例之间的分界线就不那么清楚了。同样，即使在相对较好的群体情况下，确定特定类型的Lascar排名也并不总是容易的。从她的博士论文开始，梅德韦杰夫一直致力于这类问题，以及在代数动力学中的应用。她相信自己可以将博士论文的主要定理从曲线推广到更高维的代数变体。她已经在更大的一般性上完成了这一点的第一部分；原始证明的最后一部分应该很容易推广；中间部分是由Chatzidakis和Hrushovski在一篇论文中的观察提供的。她还期望得到关于由多项式差分方程组定义的某些群G的Lascar秩的具体结果。这个问题可以转化为基础代数群的拟自同态环上的线性代数语言。例如，当G是特征为零的域的乘法群的子群时，这个问题就归结为有理数域上的线性代数。此外，梅德韦杰夫建议继续扩展她对赫鲁晓夫斯基在他关于弗罗贝尼乌斯自同构模型理论的著作中定义的差分方案的基本注释，填补许多缺失的细节，增加启发性的例子，并重新组织演示文稿，使其(更)易于理解。梅德韦杰夫希望继续让学生参与到这项工作中来，鼓励逻辑学家和代数几何学家在还很年轻的时候就相互学习对方的语言。

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.