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

该奖项部分支持参加2015年5月6日至2015年5月10日在加利福尼亚州伯克利举行的研究会议《算术和代数微分：维特向量、数论和微分代数》。会议集中在算术和代数微分的主题上，特别强调维特向量的作用。这是一种丰富而奇特的代数结构，起源于20世纪初的代数数论，在近几十年来，它在算术代数几何的一些最重要的进步中发挥了关键作用。这个课题引起了来自不同领域的数学研究人员的注意，他们传统上是并行工作的。这次会议的一个主要目标是将来自数论、代数拓扑学和应用模型理论的研究人员聚集在一起，通过这种智能交叉促进所有这些领域的Witt矢量的研究。会议将集中讨论与Witt向量有关的主题。Witt向量在算术几何中尤其重要，特别是在p-进Hodge理论中。最近，以de Rham-Witt复形的形式，Witt向量在代数拓扑学中有重要的应用，特别是在Hesselholt-Madsen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发展，也许最重要的是在布伊姆的工作中。他的主要见解是，Witt向量与微分算子的某些算术类似物密切相关，于是他能够将经典微分代数几何的很大一部分扩展到“算术微分”代数几何。在这里，我们认真地类比，普通微分对于形式幂函数级数，就像算术微分对于Witt向量一样。该程序包括，最值得注意的是，将函数域上丢番图问题的应用扩展到数域上的此类问题。该理论的这一方面随后被应用模型理论家们进一步继承和发展，特别是在Bélair-Macintyre-Scanlon的工作中，其中Ax-Kochen-Ershov式的定理被证明为被Witt-Frobenius算子扩充的环的语言中的一阶结构的Witt向量，然后在Scanlon的工作中(以及Rideau最近的扩展)将Buium的p-微分算子放入模型理论的背景中。Chatzidakis-Hrushovski关于差分场模型理论的工作揭示了算术微分方程组理论的代数部分的精细结构。随后Hrushovski和Scanlon将这一理论应用于丢番图几何，证明了它的力量，并证明了Pink-Rössler和Rössler对该理论的重新设计，然后仅由Rössler将该思想返回到真正的代数几何。由于数论、代数拓扑学和应用模型理论传统上是完全独立的领域，这些领域中的Witt向量和算术微分方面的专家要跟上其他领域的发展并非易事，尽管它们的工作对象基本上相同。那么，这次会议的目的就是纠正这一点。它将把这些领域的研究人员聚集在一起，他们从自己的角度和为了自己的目的研究维特向量和算术微分。这将使他们了解其他领域的最新发展。此外，人们希望，将来自非常不同传统、对相同数学对象有非常不同思维方式的研究人员聚集在一起，将在所有这些领域取得重大进展。会议网站：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