# Seminar

The Seminar on diffeology and related topics will be held monthly, (hopefully) on the first Thursday of each month in a regular basis. Each talk in the seminar will last approximately 1-1.5 hours, together with 0.5 hour discussion afterwards. A reminder will be sent to everybody who asks to be on the mailing list (send us an email), one or two day before the seminar. The time is announced in Greenwich Mean Time, use the GMT tool to convert it to your time zone.

### Next Talks

🔊 Severin Bunk (University of Oxford)
When: Thursday October 6, 2022 — 12:00 GMT.
Title: On homotopy types of smooth spaces
Abstract: There are several ways of associating a homotopy type to a diffeological space, such as via the diffeological topology or via the smooth singular complex. Kihara has shown that these are equivalent and embedded them into a model categorical framework. In this talk I will examine the smooth singular complex further, showing that it extends beyond the realm of diffeological spaces, to simplicial presheaves. I will explain that this induces another, equivalent, homotopy theory of smooth spaces and, if time permits, discuss examples..

🔊 Urs Schreiber (New York University, Abu Dhabi)
When: Thursday November 3, 2022 — 12:00 GMT.
Title: Twisted equivariant differential (TED) K-theory via diffeological stacks
Abstract: Twisted equivariant K-theory turns out to have a neat, transparent formulation in terms of diffeological stacks, which naturally lends itself to differential refinement. The talk gives an exposition of the main constructions and a brief indication of a remarkably rich application to the quantum physics of “anyons” and its role in topological quantum computation.

### Previous Talks

Philip Saville (University of Cambridge)
When: Thursday September 1, 2022 — 12:00 GMT.
Title: Diffeological spaces as a model for differentiable programs: a tutorial
Abstract: In programming language theory, the field of _semantics_ answers questions such as ‘when can two programs be considered equivalent?’ or ‘how does adding richer features affect which programs are expressible?’ by looking at the “meaning” of programs. This can be done in two broad ways, corresponding to the difference between syntax and semantics in logic. Operational semantics uses the relation generated by the steps a program makes while running. Denotational semantics, meanwhile, assigns a mathematical “denotation” to each program and uses the resulting mathematical model to compare programs. Such models make precise our intuition that a program is a special kind of function (more generally: morphism in a category) taking input values to output values.
_Differentiable_ programming languages have recently been of increasing importance in scientific computing and machine learning. To study such languages, several authors have used diffeological spaces as a denotational model. In this talk I will introduce the key ideas of denotational semantics, explain how diffeological spaces come into the picture, and point to recent constructions and applications which are denotationally natural but which may be unusual to specialists in diffeological spaces.
No prior knowledge of programming or programming language theory will be assumed.

Open bar
When: Thursday August 18, 2022 — 12:00 GMT.
Abstract: We invite everyone who wants to share questions, problems, comments in a special seance of open discussion.

Elisa Prato (University of Florence)
When: Thursday July 7, 2022 — 12:00 GMT.
Title: Symplectic toric quasifolds
Abstract: The Delzant construction assigns a symplectic toric manifold to each smooth convex polytope. If the polytope is simple and rational, but not smooth, the Delzant construction yields symplectic toric orbifolds. Symplectic toric quasifolds, on the other hand, were introduced in order to extend this construction to any simple convex polytope. If the polytopes are nonrational, these spaces are highly singular, but still retain a lot of the properties of their smooth/orbifold counterparts. In particular, they possess a beautiful atlas, whose charts are given by quotients of open subsets of $${\bf R}^{2n}$$ modulo the action of countable subgroups of the standard torus $$T^n$$. In this talk, we begin by recalling the Delzant construction. We then outline the extension of this construction to the nonrational case. We discuss several examples and, if time allows, we describe the symplectic reduction and the symplectic cutting operations in this setting.

When: Thursday June 2, 2022 — 12:00 GMT.
Title: Diffeology as an extension of topology by geometry
Abstract: I will review the functors from manifolds to diffeological spaces, and
from diffeological spaces to topological spaces, and describe the
importance of these transitions for applications of diffeology. I will
then talk about Grothendieck topologies on all three categories, and
about the interaction between stacks on manifolds, on diffeological
spaces, and on topological spaces / Slides…

Hiroshi Kihara (University of AIZU)
When: Thursday May 12, 2022 — 12:00 GMT.
Title: $$C^\infty-$$manifolds with skeletal diffeology
Abstract: We formulate and study the notion of $$d$$-skeletal diffeology, which generalizes that of wire diffeology, introducing the dual notion of $$d$$-coskeletal diffeology. We first show that paracompact finite-dimensional $$C^\infty-$$manidolds $$M_d$$ with skeletal diffeology inherit good topologies and smooth paracompactness from $$M$$. We then study the pathology of $$M_d$$. Above all, we prove the following: For $$d<{\rm dim}\ M$$, every immersion $$f:M\longrightarrow N$$ is isolated in the diffeological mapping space between $$M_d$$ and $$N_d$$ and the $$d$$-dimensional smooth homotopy group of $$M_d$$ is uncountable.

When: Thursday April 7th, 2022 — 12:00 GMT.
Title: Diffeological Principal Bundles and Principal Infinity Bundles
Abstract: This talk will discuss my paper on the arxiv of the same name. In particular, I will explain how to think about diffeological spaces as sheaves, and in particular simplicial presheaves. The category of simplicial presheaves over cartesian spaces has a model structure, and cofibrantly replacing a diffeological space in this model structure reproduces a construction by Iglesias-Zemmour. Using this cofibrant replacement we can show that if G is a diffeological group, then G-principal infinity bundles as defined by Schreiber et al on a diffeological space X coincide with diffeological principal G-bundles over X as they are normally defined in diffeology. Using this I will discuss how this allows us to connect the framework of higher generalized smooth spaces (higher topos theory over cartesian spaces) with “classical” constructions in diffeology. One such result of this connection shows that the Cech cohomology of a diffeological space defined automatically by higher topos theory coincides with Iglesias-Zemmour’s Cech cohomology.

Serap Gürer (Galatasaray University)
When: Thursday March 17th, 2022 — 12:00 GMT.
Title: Orbifolds as Stratified Diffeologies
Abstract: Every diffeological space is naturally stratified by the action of its diffeomorphisms: the Klein Stratification. I will describe this stratification for (diffeological) orbifolds in the general context of formal and geometric stratifications in diffeology.

Jean-Pierre Magnot (Anger University)
When: Thursday February 17th, 2022 — 12:00 GMT.
Title: Topics on (pseudo-)differential operators for diffeologists
Abstract: In this talk, I will present for diffeologists, who are a priori non-specialists of pseudo-differential operators, renormalization and integrable systems, specific features in these topics where diffeologies impose their useful presence.

Hong Van Le (Czech Academy of Sciences)
When: Thursday January 13th, 2022 — 12:30 GMT.
Title: Diffeological   statistical models  and    Machine  Learning
Abstract: In Machine Learning we develop mathematical methods for modeling data structures, which express the dependency between observables, and design efficient learning algorithms for estimation of such dependency. The most advanced part of Machine Learning is statistical learning theory that takes into account our incomplete information of observables, using probability theory, or preferably, using measure theory and functional analysis. In this way we not only unveil hidden structure of data but also make a prediction for the future. In my lecture I shall consider some important problems in statistical learning theory and demonstrate the efficiency of categorical language, manifested, in particular, in terms of probabilistic morphisms, and the use of natural geometric constructions, e..g. diffeological Fisher metric, in solving these problems.

Ralph Twun (University of Legon, Ghana)
When: Thursday December 2nd, 2021 — 12:00 GMT.
Title: Diffeologial categories and induced representations
Abstract: In this talk I will discuss the properties of the category of Diffeological Spaces which make it suitable as an enriching category. Using the concepts of indexed colimits and their relation to Kan extensions, I will discuss induced representations for diffeological groups and related ideas.

Jordan Watts (Central Michigan University)
When: Thursday November 4th, 2021 — 12:00 GMT.
Title: Sheaves for Diffeological Spaces.
Abstract: We will define sheaves for diffeological spaces and give a construction of their Čech cohomology.  As an application, given an abelian diffeological group $$G$$, we will show that the first degree Čech cohomology classes for the sheaf of smooth $$G$$-valued functioned classify diffeological principal $$G$$-bundles..

Patrick Iglesias-Zemmour (The Hebrew University of Jerusalem, Israel)
When: Thursday October 7th, 2021.
Where: Zoom link / paper / Slides
Title: Why irrational tori are important…
Abstract: I will comment two constructions / theorems in symplectic diffeology that exist only because diffeology gives us a non trivial access to quotients of the type $${\bf R}/\Gamma$$, where $$\Gamma$$ is any subgroup. In particular, I will show how “Every symplectic manifold is a (linear) coadjoint orbit”. In other words: coadjoint orbits are the universal model of symplectic manifolds / Slides…

When: Thursday September 2nd, 2021 — 12:00 GMT.
Title: The basic forms of a singular foliation.
Abstract: A singular foliation F gives a partition of a manifold M into leaves whose dimension may vary. Associated to a singular foliation are two complexes, that of the diffeological differential forms on the leaf space M/F, and that of the basic differential forms on M. We prove the pullback by the quotient map provides an isomorphism of these complexes in the following cases:
– when F is a regular foliation,
– when points in the leaves of the same dimension assemble into an embedded (more generally, diffeological) submanifold of,
– and, as a special case of the latter, when F is induced by a linearizable Lie groupoid.

Enxin Wu (Shantou University, China)
When: Thursday July 8th, 2021 — 12:00 GMT.
Title: Diffeological vector spaces.
Abstract: Diffeological vector spaces appear in various places in diffeology. In this talk, I will give a detailed discussion of many important classes of them. Many open questions will be posted.

Katsuhiko Kuribayashi (Shinshu University, Japan),
When: Thursday June 3rd, 2021 — 12:00 GMT.
Title: A singular de Rham algebra and spectral sequences in diffeology.
Abstract: In this talk, I will introduce a singular de Rham algebra under which
the de Rham theorem holds for every diffeological space.
The Leray-Serre spectral sequence and the Eilenberg-Moore spectral sequence
are also discussed in diffeology.

Norio Iwase (Kyushu University, Japan)
When: Thursday May 6th, 2021 — 12:00 GMT.
We observe also that a smooth CW complex has enough many functions, i.e. it has an open base of the form $$\phi^{-1}(]0,1[)$$. Furthermore, it follows that, for any D-open covering of a smooth CW complex, there exists a partition of unity subordinate to the covering.