Talbot 2025

Many groups and spaces come in families depending on a parameter: configuration spaces depend on the number of points considered, mapping class groups of surfaces on the genus of the surface. For such families, it often happens that the homology stabilizes as this parameter goes to infinity. Moreover, computing the stable homology frequently turns out to be easier because other tools can be used. In recent years, combining homological stability results with stable computations has become a powerful tool in algebraic topology and robust machinery for proving homological stability theorems has been developed. In this workshop we aim to introduce the participants to this circle of ideas.

Outline: This workshop will explain how to prove homological stability results through examples, such as symmetric groups, configuration spaces, mapping class groups, and others, and how to use them in conjunction with stable computations. The homological stability machines that we will cover are Quillen’s classical inductive approach and a more recent approach using Ek-algebras. Both machines have as input connectivity results for simplicial complexes and we will also see how such results are proved.

Background: The workshop will be aimed towards graduate students with a basic understanding of algebraic topology, including spectral sequences and classifying spaces.

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.  Here is some advice on how to give a good Talbot talk. 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. We cover all local expenses, including lodging and food. We also have limited funding available for participant travel costs, but we encourage participants to seek other sources of travel funding if possible.

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 especially encourage women and minorities to apply.

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.