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

本奖项部分支持参加2015年5月6日至5月10日在加州伯克利举行的“算术与代数微分：威特向量，数论与微分代数”研究会议。会议的主题是算术和代数微分，特别强调了维特向量的作用。这是一个丰富而奇特的代数构造，起源于20世纪初的代数数论，近几十年来，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。这个主题吸引了来自不同领域的数学研究人员的注意，他们传统上是并行工作的。本次会议的主要目标是将数论、代数拓扑和应用模型理论的研究人员聚集在一起，通过这种智力的交流，推动维特向量在所有这些领域的研究。会议将集中讨论与威特载体有关的主题。威特向量在算术几何中，特别是在p进霍奇理论中，起着特别重要的作用。甚至在最近，以de Rham—Witt复合体的形式，Witt向量在代数拓扑中有重要的应用，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，威特向量在数论和算术代数几何中得到了进一步的发展，这在布厄姆的工作中可能是最重要的。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，然后他能够将经典微分代数几何的大部分扩展到“算术微分”代数几何。在这里，我们认真地把常微分与形式幂级数类比为等差微分与维特向量的类比。这个程序包括，最值得注意的是，扩展应用程序在丢番图问题在函数字段，以这样的问题在数字字段。理论的这一方面随后被应用模型理论家所接受并进一步发展，特别是在b<s:1> -麦金蒂尔-斯坎伦的工作中，其中ax - kochen - ershov式定理证明了Witt向量被认为是环语言中的一阶结构，由维特-弗罗本尼乌斯算子增广，然后在斯坎伦的工作中（以及最近由里多的扩展）将Buium的p微分算子纳入模型理论的背景下。查茨达克斯-赫鲁晓夫斯基关于差分场模型理论的工作，揭示了算术微分方程理论中代数部分的精细结构。随后赫鲁晓夫斯基和斯坎伦将这一理论应用于丢番图几何，证明了它的力量，并通过Pink-Rössler和Rössler对该理论进行了重新设计，然后仅通过Rössler就将其思想回归到代数几何本身。由于数论、代数拓扑学和应用模型论传统上是完全独立的领域，对于其中一个领域的维特向量和算术微分专家来说，要保持在其他领域的发展是不容易的，即使他们在很大程度上研究相同的数学对象。因此，这次会议的目的就是要纠正这一点。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自己的目的来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，将来自不同传统、用不同方式思考相同数学对象的研究人员聚集在一起，将在所有这些领域取得突破性进展。会议网址：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