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

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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