The **Seminar on diffeology and related topics **will be held monthly, check this web site for the dat. Each talk in the seminar will last approximately 1-1.5 hours, together with 0.5 hour discussion afterwards**.**Β 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

π Hugo Cattarucci BotΓ³s (Max-Planck-Institut fΓΌr Mathematik)**When**: Friday, July 19, 2023 β 12:00 GMT.**Where**: Zoom link**Title**: Orbifolds and orbibundles in complex hyperbolic geometry.**Abstract**: This talk explores diffeology in the study of orbibundles, the natural generalization of fiber bundles over manifolds to the context of orbifolds, and its applications to complex hyperbolic geometry, following the preprint ”Orbifolds and orbibundles in complex hyperbolic geometry” by H. BotΓ³s.

In their paper titled ”Orbifolds as Diffeologies”, P. Iglesias, Y. Karshon, and M. Zadka approach orbifolds from a diffeological perspective. We use diffeology to provide a geometric framework for orbibundles and orbifold covering maps. The introduction of principal and vector orbibundles is discussed as well. This framework is then applied to study circle and disc orbibundles over 2-orbifolds (because, in the study of 4-manifolds, these objects are of interest), and we introduce the notion of Euler number for these orbibundles. Additionally, for good orbibundles, the Chern-Weil approach is provided to calculate the Euler number for vector orbibundles of rank 2.

We examine the extension of several classical results in complex hyperbolic geometry to the context of orbifolds.

### Previous Talks

β Alireza Ahmadi (Yazd University, Iran)**When**: Friday, Jun 14, 2023 β 12:00 GMT.**Where**: Zoom link**Title**: Δech Cohomology in Diffeology from Another Perspective.**Abstract**: Δech cohomology provides a way to get more global information from local data. I discuss the Δech technique to study problems in which data are given over covering generating families. Besides the existing ones, I suggest a new Δech cohomology for diffeological spaces with coefficients in a (pre)sheaf, explore its key properties, and describe some of its applications

β Katsuhiko Kuribayashi (Shinshu University, Japan)**When**: Wednesday, May 17, 2023 β 12:00 GMT.**Where**: Zoom link**Title**: On the cohomology algebras of the free loop space of the real projective space and its diffeological version.**Abstract**: introduce a spectral sequence converging to the cohomology algebra of the classifying space of a category internal to the category of topological spaces. By applying the machinery to a Borel construction, we determine explicitly the rational cohomology algebra of the free loop space of the real projective space. Moreover, we try to represent generators in the singular de Rham cohomology algebra of the diffeological free loop space of a non-simply connected manifold \(M\) with differential forms on the universal cover of \(M\) via Chen’s iterated integral map.

β Norio Iwase (Kyushu University, Japan)**When**: Wednesday, April 19, 2023 β 12:00 GMT.**Where**: Zoom link**Title**: Concatenations and reflexivity.**Abstract**: (This talk is partly a joint work with Yuki Kojima)

Part 1: we show the path space of all smooth paths, which is stable outside \((0,1)\), can be realised as a function space from a diffeological space \(I\) of dimension 1. We then introduce two versions of concatenations on the path space, a usual one and a “stable” one. On a reflexive diffeological space, the usual concatenation is smooth.

Part 2: We first introduce a notion of a \(q\)-cubic set and cubic complex. Using the diffeology on \(I\), we define a diffeology on a \(q\)-cubic set which is of diffeological dimension \(q\). Then we obtain a diffeological realisation of a finite cubic complex. This enables us to define a smooth \(A_\infty\) structure on the path space introduced in Part 1.

Part 3: we introduce a version of smooth CW complex called a fat smooth CW complex, including all manifolds of finite dimension. A “regular” smooth CW complex is shown to be reflexive.

β Jim Stasheff (University of North Carolina, USA)**When**: Friday March 17, 2023 β 13:00 GMT.**Where**: Webex link**Title**: How I became a so-called \(\textit{cohomological physicist}\).**Abstract**: This is a mini-mathematical-autobigraphy with emphasis on those who have most influenced my mathematical career and whence came the accidental flashes of insight that drew me.

β Yael Karshon (Tel-Aviv University and Univ. of Toronto)**When**: Friday February 3, 2023 β 12:00 GMT.**Where**: Zoom link**Title**: Smooth maps on convex sets**Abstract**: There are several notions of a smooth map from a convex set to a cartesian space. Some of these notions coincide, but not all of them do. We construct a real-valued function on a convex subset of the plane that does not extend to a smooth function on any open neighbourhood of the convex set, but that for each \(k\) extends to a \(C^k\) function on an open neighbourhood of the convex set. It follows that the diffeological and Sikorski notions of smoothness on convex sets do not coincide. We show that, for a convex set that is locally closed, these notions do coincide. With the diffeological notion of smoothness for convex sets, we then show that the category of diffeological spaces is isomorphic to the category of so-called exhaustive Chen spaces. This is joint work with Jordan Watts.

β Dmitri Pavlov (Texas Tech. University)**When**: Friday January 6, 2023 β 12:00 GMT.**Where**: Zoom link**Title**: The smooth Oka principle and model structures on diffeological spaces and smooth sets**Abstract**: I will explain why the category of diffeological spaces does not admit a model structure transferred via the smooth singular complex functor from simplicial sets, resolving in the negative a conjecture of Christensen and Wu. Embedding diffeological spaces into sheaves of sets (not necessarily concrete) on the site of smooth manifolds, I will then prove the existence of a proper combinatorial model structure on such sheaves transferred via the smooth singular complex functor from simplicial sets. I will show the resulting model category to be Quillen equivalent to the model category of simplicial sets. The resulting model structure is cartesian, all smooth manifolds are cofibrant, and categories of algebras over operads admit model structures. I will use these results to establish analogous model structures on simplicial presheaves on smooth manifolds, as well as presheaves valued in left proper combinatorial model categories, and prove a generalization of the smooth Oka principle established in arXiv:1912.10544. I will apply the Oka principle to establish classification theorems for differential-geometric objects like closed differential forms, principal bundles with connection, and higher bundle gerbes with connection on arbitrary cofibrant diffeological spaces. No knowledge of model categories will be assumed or required in the talk.

β **Luigi Alfonsi** (University of Hertfordshire)**When**: Friday December 2, 2022 β 12:00 GMT.**Where**: Zoom link**Title**: Classical BFV-BV theory as derived n-plectic geometry**Abstract**: In this talk I will show that a classical BFV-BV theory can be reformulated in terms of derived n-plectic geometry on the derived smooth space of solutions of the Euler-Lagrange equations. In analogy with its ordinary counterpart, a derived n-plectic structure is a proposed generalisation of a derived symplectic structure where a derived closed 2-form is replaced by a derived closed (n+1)-form. In particular, I will be interested in derived n-plectic spaces which are smooth: for this reason, I will consider n-plectic structures on certain derived enhancements of ordinary diffeological spaces. I will argue that this derived n-plectic geometry is to BFV-BV theory as ordinary n-plectic geometry is to ordinary Lagrangian field theory.

β **Urs Schreiber** (New York University, Abu Dhabi)**When**: Friday November 4, 2022 β 12:00 GMT.**Where**: Zoom link**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.

β Severin Bunk (University of Oxford)**When**: Thursday October 6, 2022 β 12:00 GMT.**Where**: Zoom link**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.

β Philip Saville (University of Cambridge)**When**: Thursday September 1, 2022 β 12:00 GMT.**Where**: Zoom link / Slides**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.

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

β **Elisa Prato** (University of Florence)**When**: Thursday July 7, 2022 β 12:00 GMT.**Where**: Zoom link**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.

β **Konrad Waldorf** (University of Greifswald) **When**: Thursday June 2, 2022 β 12:00 GMT.**Where**: Zoom link**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…

β **Hiroshi Kihara** (University of AIZU)**When**: Thursday May 12, 2022 β 12:00 GMT.**Where**: Zoom link**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.

β **Emilio Minichiello** (CUNY Graduate Center)**When**: Thursday April 7th, 2022 β 12:00 GMT.**Where**: Zoom link**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.

β **Serap GΓΌrer** (Galatasaray University)**When**: Thursday March 17th, 2022 β 12:00 GMT.**Where**: Zoom link**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.

β **Jean-Pierre Magnot** (Anger University)**When**: Thursday February 17th, 2022 β 12:00 GMT.**Where**: Zoom link**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.

β **Hong Van Le** (Czech Academy of Sciences)**When**: Thursday January 13th, 2022 β 12:30 GMT.**Where**: Zoom link**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.

β **Ralph Twun** (University of Legon, Ghana)**When**: Thursday December 2nd, 2021 β 12:00 GMT.**Where**: Zoom link**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.

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

β **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…

β **David Miyamoto** (Toronto University, Canada)**When**: Thursday September 2nd, 2021 β 12:00 GMT.**Where**: Zoom link**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.

β **Enxin Wu** (Shantou University, China)**When**: Thursday July 8th, 2021 β 12:00 GMT.**Where**: Zoom link**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.

β **Katsuhiko Kuribayashi** (Shinshu University, Japan),**When**: Thursday June 3rd, 2021 β 12:00 GMT.**Where**: Zoom link, Slides**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.

β **Norio Iwase** (Kyushu University, Japan)**When**: Thursday May 6th, 2021 β 12:00 GMT.**Where**: Zoom link**Title**: *Whitney Approximation sor Smooth CW Complexes*.**Abstract**: In this talk, we introduce a notion of smooth CW complex using disks, while we know there are different definitions using cubes or simplexes instead of disks. It is a kind of future work for us to clarify the relationship among them. With our definition of a smooth CW complex, we show Whitney Approximation for smooth CW complex, which enables us to obtain that any continuous CW complex is continuously homotopy equivalent to a smooth CW complex.

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.

β **Patrick Iglesias-Zemmour** (The Hebrew University of Jerusalem, Israel)**When**: Thursday April 1st, 2021 β 1PM GMT.**Where**: Zoom link / Lecture Notes**Title**: *Orbifolds, Quasifolds as Diffeologies and C*-algebras*.**Abstract**: I will show how to associate to these special diffeological spaces that are orbifolds, and more generally quasifolds, a C*-algebra in a functorial way, in which diffeomorphisms translate into Morita equivalences.