Talbot 2023: Computations in stable motivic homotopy theory

Mentored by Oliver Röndigs and Paul Arne Østvær

Date: June 4-10, 2023
Location: McGrath, MN

Topic: Motivic homotopy theory involves the study of algebraic varieties and schemes using methods not only from algebraic geometry, but also from algebraic topology. A basic idea is to use techniques from homotopy theory, such as the construction of homotopy groups and the use of homotopical invariants, to investigate topological and geometric properties of these varieties. It was developed in the late 20th century as a way to extend classical homotopy theory, which studies topological spaces up to continuous deformations, to the realm of algebraic geometry.

One of the fundamental problems in motivic homotopy theory over F is to calculate the homotopy groups πt,w(1) of the motivic sphere spectrum 1. Here the integers t, w ∈ Z refer to the topological degree and the weight, respectively. In a precise sense, these groups are the universal motivic invariants because the motivic sphere is the unit for the tensor product on motivic spectra. All the relations witnessed in the graded ring π*,*1 hold in every other theory representable in the stable motivic homotopy category, such as algebraic cobordism, algebraic and hermitian K-theory, motivic cohomology, and higher Witt theory.

The goal of the workshop is to discuss “Why, what, and how” on specific computations in stable motivic homotopy theory.

Please see the preliminary syllabus for more detailed information and references.

Suggested prerequisites:

Mentors: The 2023 Talbot workshop will be mentored by Prof. Oliver Röndigs of the Osnabrück University and Paul Arne Østvær of University of Milan, Italy and University of Oslo, Norway.

Format: The workshop discussions will have an expository character and a majority of the talks will be given by participants. Breaks will be built into the schedule for informal discussions and collaborations. The workshop will take place in a communal setting, with participants sharing living space and cooking and cleaning responsibilities.

The workshop will be held entirely in-person. Participants should expect to follow mitigation measures such as masking in indoor common spaces and eating meals in a socially distanced/outdoor setting.

Timeline: The 2023 Talbot workshop will take place June 4-10, 2023.

Funding: We cover all local expenses, including lodging and food. We also have limited funding available for participant travel costs.

Who should apply: Talbot is meant to encourage collaboration among young researchers, particularly graduate students. To this end, the workshop aims to gather participants with a diverse array of knowledge and interests, so applicants need not be an expert in the field. In particular, students at all levels of graduate education are encouraged to apply. Our decisions are based not on applicants' credentials but on our assessment of how much they would benefit from the workshop. As we are committed to promoting diversity in mathematics, we also especially encourage women and minorities to apply.

Inclusiveness statement: In accordance with the Statement of Inclusiveness, this workshop will be open to everybody, regardless of race, sex, religion, national origin, sexual orientation, gender identity, disability, age, pregnancy, immigration status, or any other aspect of identity. We are committed to ensuring that the Talbot Workshop is a supportive, inclusive, and safe environment for all participants, and that all participants are treated with dignity and respect.

Contact Information: Please email the organizers at talbotworkshop(at)gmail.com if you have any questions.