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

算术和代数微分：维特向量、数论和微分代数

• 批准号: 1502219
• 负责人: Thomas Scanlon
• 金额: $ 3万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-05-01 至 2016-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1502219&HistoricalAwards=false
• 关键词: Arithmetic; algebraic; differentiation; Witt; vectors

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

该奖项在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“算术和代数差异化：Witt vectors，数字理论和差异代数”的研究会议部分支持参加会议。该会议以算术和代数分化为中心，以特殊的角色为中心。这是一种丰富而异国情调的代数结构，起源于20世纪初的代数数理论，并且在近几十年来，它在算术代数的某些最重要的进步中发挥了关键作用。该主题吸引了传统上并行工作的不同领域的数学研究人员的注意。本次会议的主要目标是将这些研究人员从数字理论，代数拓扑和应用模型理论中汇集在一起​​，以通过这种知识分子互相剥夺来推动所有这些领域中WITT载体的研究。会议将集中于与Witt Vectors有关的主题。 Witt载体在算术几何形状中尤其重要，尤其是在P-Adic Hodge理论中。甚至最近，以De Rham-Witt Complex的形式，Witt向量在代数拓扑中具有重要的应用，尤其是在Hesselholt-Madsen及其在代数K理论上的追随者的工作中。同时，Witt向量在数量理论和算术代数几何学上已经进一步发展，也​​许最重要的是Buium的作品。他的关键见解是，Witt向量与差分运算符的某些算术类似物密切相关，然后他能够将经典差分代数几何的大部分延伸到“算术差分”代数几何形状。在这里，我们认真对待一个类似物是普通的分化是正式的力量系列，因为算术差异是对witt向量的。该程序包括，最值得注意的是，将借助问题中的应用程序扩展到功能字段上，将其扩展到数字字段上的此类问题。 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的p差异运算符进入模型理论环境。 Chatzidakis-Hrushovski在差异领域模型理论上的工作提出了算术差分方程理论代数部分的精细结构。 Hrushovski和Scanlon在该理论上的随后应用在二世的几何形状上，证明了其力量和Pink-Rössler对理论的重新加工，然后Rössler仅由Rössler恢复了构想，将这些思想归还给了代数的几何形状。作为数字理论，代数拓扑和应用模型理论传统上是非常独立的领域，对于其中一个领域的Witt矢量和算术分化的专家来说，即使在很大程度上与同一数学对象合作，也不容易保持在其他领域的发展。因此，会议的目的是记住这一点。它将在这些领域的研究人员中汇集研究人员，他们从自己的角度和出于自己的目的研究了对媒介的研究和算术分化。这将使他们能够了解其他领域的最新发展。此外，人们希望以非常不同的方式将研究人员从非常不同的传统中汇集在一起​​，以对相同的数学对象进行思考的方式，将导致所有这些领域的震动。