NSCPHD1112  Witt vectors, lambda-rings, and arithmetic jet spaces

Volume 2015/2016
Content

 

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This class will be an introduction to the theory of Witt vectors, lambda-rings, and arithmetic jet spaces, which are in the end different points of view on the same mathematics. We will spend most of the time on the usual 'big' and 'p-typical' theory, but I hope to spend some time on a new 'elliptic' analogue of the usual theory.

In the first lecture, I willl give a transparent and conceptual definition of the p-typical Witt vector functor which is not very well known. The lectures after that will be on the following basic topics: the connection between Witt vectors and lambda-rings, variants (big, ramified, relative to number fields, etc), and algebraic-geometric properties which allow one to extend Witt vectors and lambda-structures from rings to schemes. The second part of the course will be devoted to further topics, such as the (relative) de Rham-Witt complex, symmetric functions and positivity, 'plethories' and possible classification conjectures, lambda-schemes and the hypothetical field with one element, shtukas and the function-field theory, and most importantly elliptic Witt vectors and elliptic de Rham-Witt cohomology.

To follow the class, you should be comfortable with category theory and basic commutative algebra. It would be helpful to have some familiarity with scheme theory and algebraic number theory, but it's not essential.

Lectures: Three times 45 minutes per week.
Recitations: Three times 45 minutes per week.
Credit
7,5 ECTS
Type of assessment
Continuous assessment
  • Category
  • Hours
  • Lectures
  • 21
  • Theory exercises
  • 21
  • Preparation
  • 164
  • Total
  • 206