A first encounter with diffeology

Speaker: David Miyamoto

Queen's University, Kingston, Canada

About the Series

This lecture series, as the title indicates, will for many in the audience be a first encounter with diffeology. Diffeological spaces generalize smooth manifolds, in such a way that arbitrary subsets, quotients, and mapping spaces of manifolds, all of which are rarely manifolds, carry natural diffeological structures. We will explore how diffeological structures extend the tools and notions from smooth manifolds to singular or infinite-dimensional spaces, and highlight recent progress in the application of diffeology to differential geometry.

Schedule

Diffeology: definitions and examples

Part 1 of the series

Tuesday, March 10, 2026

04:00 PM IST

Abstract: We will introduce the category of diffeological spaces, and give several examples: the cusp \(x^2 = y^3\), as a subset of \(\mathbb{R}^2\); the irrational tori \(\mathbb{R}/(\mathbb{Z} + \alpha\mathbb{Z})\), for irrational \(\alpha\); and the diffeomorphism group of the real line, \(\operatorname{Diff}(\mathbb{R})\). These will illustrate how the category of diffeological spaces is closed under taking, respectively, subsets, quotients, and mapping spaces, and help give a sense of the data contained within a diffeological structure.

Post-Lecture Resources:

Watch Recording Lecture Notes

Talk Concluded

This talk has ended — see the recording.

Higher and infinite-dimensional structures in geometry

Part 2 of the series

Tuesday, March 17, 2026

04:00 PM IST

Abstract: We will introduce two other approaches to singular and infinite-dimensional spaces, namely Lie groupoids (a step towards differentiable stacks), and convenient infinite-dimensional manifold structures. We will then explore how relevant diffeological spaces appear in both contexts.

Post-Lecture Resources:

Watch Recording Lecture Notes

Talk Concluded

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Diffeological geometry

Part 3 of the series

Tuesday, March 24, 2026

04:00 PM IST

Abstract: Having completed the introductions, we will summarize some recent progress in using diffeology to understand geometry. Among these will include: a generalization of the Serre-Swan theorem using diffeological vector bundles; an integration of Lie algebroids to diffeological groupoids, even when no integrating Lie groupoid exists; and a diffeological approach to the question of geometric quantization.

Post-Lecture Resources:

Watch Recording Lecture Notes

Talk Concluded

This talk has ended — see the recording.

References

Books

About the Speaker

David Miyamoto

Queen's University, Kingston, Canada

David Miyamoto completed his PhD in 2023 at the University of Toronto under the supervision of Yael Karshon. He went on to a postdoctoral research position in Christian Blohmann's group at Max Planck Institute, Bonn, Germany (2023–2025), and is currently a Coleman Postdoctoral Fellow at Queen's University, Kingston, Canada. His research interests include symplectic and Poisson geometry, Lie groupoids, foliations, and diffeology.

Email: d.miyamoto@queensu.ca
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