2018 Workshop

Free Loop Spaces with Nathalie Wahl and Alexandru Oancea

July 29 - August 4, 2018

Fischbach bei Dahn, Germany

About

The fourth European Talbot workshop took place in Germany from July 29 to August 4, 2018. The goal was to bring together a group of 30-35 graduate students and post-docs to work on a focused topic under the guidance of two senior mentors.

Most of the talks were given by the participants, with enough free time in the afternoon and evenings for further discussions and interaction. The character of the workshop is expository in nature, starting with the basic ideas and leading to a survey of the most recent developments in the field. Since all participants stayed together at a group house, jointly responsible for cooking and cleaning, we hope to have created an informal and inspiring atmosphere.

Topic

Loop spaces play an important role in many branches of mathematics. Recent years have seen exciting progress on the old dream of using homotopy-theoretic methods on loop spaces to determine geometric properties of manifolds and study moduli spaces. We will in this workshop focus on the homology of free loop spaces on manifolds, considering it through the eyes of Morse theory, Hochschild homology and string topology. We will be particularly interested in applications, older as well as recent, to the study of closed geodesics on manifolds, Lagrangians in symplectic topology, and to the moduli space of Riemann surfaces through topological field theories.

More specifically, we are going to study the following topics:

• Morse theory on the free loop space of manifolds and
• Relationship to closed geodesics, including the Gromoll-Meyer theorem.
• Relationship to Hochschild homology via: the dga of cochains for simply connected manifolds (Jones), iterated integrals (Chen), the dga of chains on the based loop space (Burghelea-Fiedorowicz, Goodwillie)
• Operations on free loop spaces, geometrically and algebraically, i.e., via intersection theory and respectively Hochschild homology: Chas-Sullivan and Goresky-Hingston products. Applications to closed geodesics following Hingston-Rademacher.
• Overview of the relationship to Floer theory on the cotangent bundle of the manifold. Applications in symplectic topology.
• E₂-structure on the chains on free loop spaces from different points of view: geometric, algebraic and also symplectic.

The goal is to cover a good part of the background on the study of free loop spaces on manifolds, reaching to some of the most recent papers on the topic. We will use the rich, yet computable, example of the loop spaces of spheres as a guiding thread through the workshop.

Syllabus

A list of talks can be found here.

A public Dropbox folder with all the publicly available references is available here.

Application

The application deadline has passed.

Poster

The poster for the workshop may be found here.

Funding

The local expenses for all participants will be covered by the workshop, in particular costs for accommodation and meals. Additional funds covering travel expenses might become available, however we strongly encourage participants to look for funding themselves.

Contact

For questions or suggestions, please send an email to the organizers at organisers@europeantalbot.org.

This workshop is organized by Bertram Arnold, Tobias Barthel, Jack Davies, Alice Hedenlund, Magdalena Kedziorek, and Sean Tilson, and its concept has been inspired by the US Talbot workshop. We gratefully acknowledge funding and support from SPP 1786 and SFB 1085.