Talbot 2019: Moduli Spaces of Manifolds

Mentored by Soren Galatius and Oscar Randal-Williams

April 7-13, 2019
McCullough County, TX



Topic: The 2019 workshop, aimed toward graduate students and other young researchers, will give an introduction to classical and recent tools for analysing moduli spaces of manifolds. These are classifying spaces for families of smooth manifolds, and their homotopy type can be described in many ways: for example as classifying spaces of groups of diffeomorphisms, or as spaces of compact submanifolds of infinite-dimensional euclidean space of a fixed diffeomorphism type. It is a basic problem in algebraic topology to describe the cohomology of these classifying spaces, or equivalently to describe the rings of characteristic classes for smooth fibre bundles.

There is no compact manifold of dimension > 3 for which a complete answer is known (even with rational coefficients). Nonetheless, many partial answers are available, and these moduli spaces are an active topic of current research. This workshop will survey the classical approach to analysing moduli spaces of manifolds (via surgery theory and pseudoisotopy theory), but will focus on the more recent approach via cobordism categories, fibrewise surgery, and homological stability. It will also touch on a method recently introduced by Weiss exploiting embedding calculus.

Suggested prerequisites:

Basic homotopy theory, including simplicial sets and some exposure to spectra.

Proof of the h-cobordism theorem. (See Milnor's "Lectures on the h-cobordism theorem" or Lück's "A basic introduction to surgery theory.")

Vector bundles and their characteristic classes. (Parts of Milnor--Stasheff's "Characteristic classes.")

Pontrjagin--Thom theory. (See section 7 of Milnor's "Topology from the differentiable viewpoint", or section 18 of Milnor--Stasheff.)

Mentors: The 2019 Talbot workshop will be mentored by Prof. Soren Galatius of the University of Copenhagen and Prof. Oscar Randal-Williams of the University of Cambridge.

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

Timeline: The 2019 Talbot workshop will take place from April 7-13, 2019. The appliction will be open soon, and is due on February 10, 2019 at 11:59 pm EST.

Funding: We cover all local expenses including lodging and food. We also have limited funding available for participants' 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.