Model Theory and Difference Algebra

逻辑学是对形式推理规则的研究，它依赖于语句的语法结构而不是其内容。数学逻辑在二十世纪初被复兴，用来处理一些基础问题，结果证明它对于获得数学中的新结果也很有用。最近差分和微分代数的模型理论在代数数论中的应用就是这方面一些最令人兴奋的例子。差分方程和微分方程一样，模拟了随时间变化的现实世界过程。微分方程描述的是随时间连续变化的量，比如行星在空间中的位置，而差分方程描述的是只能在离散时间间隔内测量的量，比如一个国家的年度GDP。差分代数是研究差分方程的抽象集合；它也应用于支撑现代密码学和互联网安全的代数数论。Medvedev建议研究差分方程解集的精细结构：找到计算它们的维数的算法，并确定可能在这些集上定义某种加法和/或乘法的非常特殊的情况。差域是具有不同自同构的域。差分闭场理论是超简单的，这意味着拉斯卡秩是完备类型的一个很好的维数概念。此外，Lascar秩1的完备型满足Zilber三分法：每个完备型与可定义的域不正交，或与可定义的一基群不正交，或分解。虽然三分法的场状案例在明确的例子中相对容易识别，但群状案例和解体案例之间的分界线要模糊得多。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，以及代数动力学的应用。她有信心将博士论文中的主要定理从曲线推广到高维代数变量。她已经在很大程度上完成了第一部分；原始证明的最后部分要易于推广；中间部分是由查茨达基斯和赫鲁晓夫斯基在一篇论文中的观察提供的。她还希望得到多项式差分方程组所定义的某些群G的拉斯卡秩的具体结果。这个问题可以翻译成线性代数的语言，在基础代数群的拟自同态环上。例如，当G是特征为0的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她关于赫鲁晓夫斯基在他关于Frobenius自同构模型理论的著作中定义的不同方案的基础笔记，填补许多缺失的细节，增加启蒙性的例子，并重新组织演示以使其更容易理解。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他们还年轻的时候学习彼此的语言。

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.