Model Theory and Difference Algebra

逻辑学研究的是形式推理规则，这些规则依赖于语句的语法结构而不是它们的内容。 在世纪初，数理逻辑重新兴起，用来处理一些基础性问题，结果证明，它对数学中获得新的结果也很有用。最近的应用模型理论的差异和微分代数数论是一些最令人兴奋的例子。差分方程，像微分方程一样，模拟随时间变化的真实世界过程。微分方程描述的是随时间连续变化的量，例如行星在空间中的位置，而差分方程描述的是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背景;它也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，并确定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。差域是具有杰出自同构的域。差分闭域的理论是超级简单的，这意味着Lascar秩是完备类型的维数的一个很好的概念。此外，Lascar秩1的完全类型满足Zilber三分法：每一个都与可定义的域不正交，或与可定义的基于1的群不正交，或被分解。虽然在明确的例子中比较容易识别出类似场的情况，但类群情况和分解情况之间的分界线就不那么清楚了。类似地，即使在相对较好的群体情况下，也不总是容易确定特定类型的Lascar等级。从她的博士论文开始，梅德韦杰夫一直致力于解决这类问题，并将其应用于代数动力学。她有信心，她可以推广她的博士论文的主要定理从曲线到高维代数簇。她已经完成了第一部分，这在更大的一般性;最后一部分的原始证明应该很容易推广;和中间的一块是提供了一个观察在一份文件中的Chatzidakis和Hrushovski。她预计也将获得具体的结果对拉斯卡秩的某些群体G定义的系统多项式差分方程。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关于Hrushovski在他关于Frobenius自同构模型理论的工作中定义的差分方案的基础注释，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，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。

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.