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Topological Hochschild homology and zeta values
Provider: Faculty of Science

Activity no.: 5594-23-07-31
Enrollment deadline: 30/01/2023
PlaceDepartment of Mathematical Sciences
Date and time30.01.2023, at: 08:00 - 03.02.2023, at: 16:00
Regular seats50
ECTS credits2.50
Contact personNina Weisse    E-mail address: weisse@math.ku.dk
Enrolment Handling/Course OrganiserLars Hesselholt    E-mail address: trk734@ku.dk
Written languageEnglish
Teaching languageEnglish partially in English
Grading scalePassed / Not passed
Course workload
Course workload categoryHours
Preparation / Self-Study20.00
Course hours40.00
Evaluation / reporting5.00
Sum65.00
Content
The course will present the spectacular recent advances in motivic filtrations and their applications. It is well-recognized that every cohomology theory, in arithmetic geometry and elsewhere, should arise as the graded pieces of a "motivic" filtration of some localizing invariant, notably, algebraic K-theory and topological cyclic homology. The definition of the appropriate motivic filtrations, however, was long elusive.
In 2002, Voevodsky was awarded the Fields medal for his definition of a motivic filtration of the algebraic K-theory of smooth varieties. His work uses algebraic cycles, which are notoriously difficult to understand. In 2018, Scholze was awarded the Fields medal, in part, for his definition with Bhatt and Morrow of a motivic filtration of topological cyclic homology. Their definition uses an entirely different approach, which is much simpler and easier to employ elsewhere. This breakthrough has led to many advances. Morin has recently introduced a refined variant of the Bhatt-Morrow-Scholze filtration and proved that the cohomology theory defined by its graded pieces precisely accounts for an archimedean factor in a conjectural formula for the special values of the Hasse-Weil zeta function of a regular scheme, proper over the integers. It is very exciting to see that such quantative archimedean information can be extracted from topological cyclic homology! Antieau and Krause with Nikolaus have employed the new cohomology theory that emerges from topological cyclic homology, prismatic cohomology, to determine the structure of algebraic K-theory in a number of situations, that were hitherto inaccessible. It is fantastic opportunity to have Antieau, Krause, and Morin present this material in Copenhagen!

Lecturers
Professor (David) Benjamin Antieau, Northwestern University
Chargé de Recherche CNRS Baptiste Morin, Institut de Mathématiques de Bordeaux, Université de Bordeaux
Postdoctoral Researcher Achim Krause, Universität Münster, Germany

Remarks

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