What is a Witt vector?


Russell Love Theatre
Peter Hall
Monash Road


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The Witt vector construction puts an exotic addition law and multiplication law on vectors. It had a continuing presence in algebraic number theory beginning with Witt's discovery in the 1930s. Starting in the 1980s, it took on an even bigger role with the rise of p-adic Hodge theory.

These addition and multiplication laws have always been regarded as somewhat mysterious, even though they involve nothing more than polynomials and elementary modular arithmetic and could be presented in an introductory abstract-algebra class. In this talk, I will explain another way setting up the theory, first discovered by Joyal, which is not very well known but is completely natural. This point of view on Witt vectors has risen in importance in the past year with the appearance of prismatic cohomology, due to Bhatt-Scholze, and Drinfeld's reformulation of it. It is also related to Buium's work on what might be called derivatives of integers with respect to prime numbers.


  • Dr James Borger
    Dr James Borger, Australian National University