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Arithmetic and Homotopy Theory
Provider: Faculty of Science

Activity no.: 5568-25-07-31
Enrollment deadline: 15/05/2025
PlaceDepartment of Mathematical Sciences
Universitetsparken 5, 2100 København Ø
Date and time23.06.2025, at: 09:00 - 27.06.2025, at: 16:00
Regular seats100
ECTS credits2.50
Contact personNina Weisse    E-mail address: weisse@math.ku.dk
Enrolment Handling/Course OrganiserJesper Grodal    E-mail address: jg@math.ku.dk
Written languageEnglish
Teaching languageEnglish
Exam formCourse participation
Exam detailsAttending the lecture series, active participation in the exercise sessions.
Course workload
Course workload categoryHours
Preparation / Self-Study34.00
Lectures15.00
Practical exercises10.00
Theory exercises10.00
Sum69.00
Aim and content
The course aims to introduce participants to the rich and multifaceted ways in which stable homotopy theory and arithmetic geometry interact.

The guest lectures are each experts in their areas. Weinstein will give a series of lectures on his recent work computing the rationalization of the K(n)-local sphere by studying the pro-etale cohomology the Drinfeld and Lubin-Tate tower. Hahn will give a series of lectures on the even filtration and the associated prismatic and syntomic stacks. Morrow will give a series of lectures on his recent definition of a well-behaved theory of motivic cohomology for singular, equi-characteristic schemes.

Formel requirements
Recommended Academic Qualifications: Knowledge about number theory, algebraic topology and basics of homotopy theory.

Learning outcome
Knowledge:
• Participants will understand how to use p-adic geometry to study rationalizations of K(n)-local objects.
• Participants will understand how to import homotopy theoretic ideas to construct cohomology theories for arithmetic schemes.

Skills:
• To be able to use acquired knowledge within the participants own research.
• To be able to compute pro-etale cohomology.
• To be able to compute syntomicizations of simple examples of p-adic formal schemes.

Competences:
• To be able to produce independent proofs of results in homotopy theory using tools from arithmetic geometry.
• To be able to produce independent proofs of results in arithmetic geometry using tools from homotopy theory.

Literature
Information will follow.

Target group
The course is designed for PhD students and early career researchers who want to learn about the latest developments at the intersection of stable homotopy theory and arithmetic geometry. Participants should have some basic knowledge of homotopy theory and K-theory.

Teaching and learning methods
The masterclass will consist of three lecture series each highlighting a different subject at the boundary of homotopy theory and arithmetic geometry.

This will be accompanied by exercise sessions and informal discussion sessions in which the participants can practice the computational methods which are discussed theoretically in the lectures.

Lecturers
Guest lecturers:
Jared Weinstein, Boston University
Matthew Morrow, CNRS, Orsay
Jeremy Hahn, MIT

Remarks
Registration will open in the autumn of 2024.

Deadline for registration:
For funding: 15 April 2025
without funding: 15 May 2025

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