Arithmetic and algebraic differentiation: Witt vectors, number theory, and differential algebra

该奖项部分支持参加研究会议“算术和代数微分：维特向量，数论和微分代数”在伯克利举行，加州期间5月6日，2015年5月10日，2015年。 会议以算术和代数微分为主题，特别强调维特向量的作用。这是一个丰富而奇异的代数结构，起源于世纪初的代数数论，在最近几十年里，它在算术代数几何的一些最重要的进展中发挥了关键作用。这个问题吸引了来自不同领域的数学研究人员的注意，他们传统上一直在平行工作。 本次会议的主要目标是汇集这些研究人员从数论，代数拓扑，并应用模型理论，推动维特向量的研究在所有这些领域通过这种智力交叉施肥。会议将集中讨论与维特向量有关的主题。维特向量在算术几何中特别重要，特别是在p进霍奇理论中。 甚至最近，在德拉姆-维特复形的形式下，维特向量在代数拓扑学中有着重要的应用，特别是在赫塞霍尔特-马德森及其追随者关于代数K-理论的工作中。与此同时，维特向量在数论和算术代数几何中得到了进一步的发展，也许最重要的是在Buium的工作中。他的关键见解是，维特向量密切相关的某些算术类似物的微分算子，他然后能够扩大大部分经典微分代数几何的“算术微分”代数几何。在这里，我们认真对待的类比，普通的微分是正式的幂级数算术微分是维特向量。该计划包括，最值得注意的是，扩大应用丢番图问题的功能领域，这些问题的数字领域。这方面的理论，然后拿起，并进行了进一步的应用模型理论家，特别是在工作的贝勒-麦金太尔-斯坎隆，其中阿克斯-科亨-Ershov式定理证明的维特向量被认为是一个一阶结构，在语言的环扩充的维特-弗罗贝纽斯运算符，然后在工作的斯坎隆（和最近的扩展Rideau）把Buium的p-微分算子模型理论的背景下。 工作的Chatzidakis，Hrushovski模型理论的差异领域带来了精细结构的代数部分理论的算术微分方程。 随后应用这一理论的丢番图几何的Hrushovski和Scanlon证明了它的权力和改造的理论由粉红色，罗斯勒，然后罗斯勒单独返回的想法，代数几何正确。由于数论、代数拓扑学和应用模型理论在传统上是相当独立的领域，因此，对于维特向量和算术微分的专家来说，在这些领域中的一个领域中保持对其他领域的发展的领先地位并不容易，即使他们在很大程度上使用相同的数学对象。因此，这次会议的目的就是纠正这一点。它将汇集研究人员在这些领域谁研究维特向量和算术微分从自己的观点和自己的目的。这将使他们了解其他领域的最新发展。此外，人们希望，将来自不同传统、对同一数学对象有不同思考方式的研究人员聚集在一起，将导致所有这些领域的震荡。https://math.berkeley.edu/~scanlon/aad15.html

This award partially supports participation in the research conference "Arithmetic and Algebraic Differentiation: Witt vectors, number theory and differential algebra" held in Berkeley, California during the period May 6th, 2015 to May 10th, 2015. The conference is centered on the topic of arithmetic and algebraic differentiation, with a special emphasis on the role of the Witt vectors. This is a rich and exotic algebraic construction that originated in the early 20th century in algebraic number theory and which, in recent decades, has played a key part in some of the most important advances in arithmetic algebraic geometry. The subject has attracted the attention of mathematical researchers from disparate fields who have traditionally worked in parallel. A principal goal of this conference is to bring together these researchers from number theory, algebraic topology, and applied model theory to propel the study of the Witt vectors in all of these fields through this intellectual cross-fertilization. The conference will concentrate on topics related to Witt vectors. The Witt vectors have been especially crucial in the arithmetic geometry, notably in p-adic Hodge theory. Even more recently, in the form of the de Rham--Witt complex, the Witt vectors have had important applications in algebraic topology, especially in the work of Hesselholt-Madsen and their followers on algebraic K-theory. In the meantime, Witt vectors have grown further in number theory and arithmetic algebraic geometry, perhaps most importantly in Buium's work. His key insight was that Witt vectors are closely related to certain arithmetic analogues of differential operators, and he was then able to extend large parts of classical differential algebraic geometry to an "arithmetic differential" algebraic geometry. Here, we take seriously the analogy that ordinary differentiation is to formal power series as arithmetic differentiation is to the Witt vectors. This program includes, most notably, extending applications in Diophantine questions over function fields to such questions over number fields. This aspect of the theory was then picked up and carried further by applied model theorists especially in the work of Bélair-Macintyre-Scanlon in which Ax-Kochen-Ershov-style theorems are proven for the Witt vectors considered as a first-order structure in the language of rings augmented by the Witt-Frobenius operator and then in the work of Scanlon (and the recent extensions by Rideau) putting Buium's p-differential operators into a model theoretic context. The work of Chatzidakis-Hrushovski on the model theory of difference fields brought out the fine structure of the algebraic part of the theory of arithmetic differential equations. The subsequent applications of this theory to diophantine geometry by Hrushovski and Scanlon demonstrated its power and the reworking of the theory by Pink-Rössler and then by Rössler alone returned the ideas to algebraic geometry proper. As number theory, algebraic topology, and applied model theory have traditionally been quite separate fields, and it has not been easy for experts on Witt vectors and arithmetic differentiation in one of these fields to keep on top of developments in the others, even though they are working with largely the same mathematical objects. The purpose of the conference, then, is to remedy this. It will bring together researchers in these fields who study Witt vectors and arithmetic differentiation from their own points of view and for their own purposes. This will allow them to learn about the latest developments in other fields. Further, one hopes that bringing together researchers from very different traditions with very different ways of thinking about the same mathematical objects will lead to jolts forward in all of these fields.Conference web site: https://math.berkeley.edu/~scanlon/aad15.html