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Eric M Downes (Jun 20 2024 at 12:10):Returning to our normal time next week, we have @Emilio Minichiello
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Title: Introduction to Diffeological Spaces
Abstract: In this talk I will give a friendly introduction to the study of diffeological spaces. A diffeological space consists of a set X together with a collection D of set functions U -> X where U is a Euclidean space, that satisfy three simple axioms. In this talk we will describe how this simple definition provides a new, powerful framework for differential geometry. Namely, every finite dimensional smooth manifold is a diffeological space, as are many infinite dimensional ones, orbifolds, and many other objects of interest in differential geometry. Further, the category of diffeological spaces is much better behaved than the category of finite dimensional smooth manifolds, in a way that we will make precise. Despite the fact that diffeological spaces are much more general than manifolds, many classical constructions in differential geometry still make sense for them, such as tangent spaces, differential forms, homotopy theory and fiber bundles. However, recent results show that many of the cherished and basic theorems of smooth manifold theory fail for general diffeological spaces, but this failure opens up worlds of interesting possibilities. We will review two such results. One being the difference between the internal and external tangent space of a diffeological space, and the obstruction between Cech cohomology and deRham cohomology. If time permits, I will discuss some of the work from my thesis connecting diffeological spaces to methods from higher topos theory.
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Emilio Minichiello (Jun 26 2024 at 13:44):Here are the slides for the talk, see you all soon! Cat-Seminar-2024-Diffeology-Presentation.pdf
Jean-Baptiste Vienney (Jun 26 2024 at 14:00):We're starting in a few minutes
Eric M Downes (Jun 26 2024 at 14:58):Ah! So sorry everyone!
Eric M Downes (Jun 26 2024 at 14:58):I didnt mean to kick you all out! We're in the zoom now if there are more questions?
Eric M Downes (Jun 26 2024 at 15:22):Okay, finished for real; will upload videos later.
Jean-Baptiste Vienney (Jun 26 2024 at 15:26):Sorry, I lost the internet connection when I was on the Zoom 5 minutes ago.
Every smooth manifold is a disjoint union of submanifolds with a well-defined dimension. Is the same true for diffeological spaces?
Jean-Baptiste Vienney (Jun 26 2024 at 15:32):(When I say smooth manifold, I mean “if you define the term smooth manifold such that charts can come from open sets of any )
Emilio Minichiello (Jun 26 2024 at 15:39):Well, depending on what you mean, yeah, but in a trivial way. Like any subset of a diffeological space is a diffeological space, so trivially its a disjoint union of itself. Certainly its not the case that diffeological spaces are disjoint unions of any sort of nice subspaces like unions of manifolds or something, if that's what you meant. There is a notion of dimension for diffeological spaces I should add, which is defined by looking at the plot category, you can find that definition in Patrick's textbook.
Oscar Cunningham (Jun 26 2024 at 15:48):Sorry I didn't see the talk, thank you for the slides! I'm guessing the notion of dimension isn't constant across a diffeological space? For example if I took the union of the y-axis and the upper half-plane as a subset of .
Emilio Minichiello (Jun 26 2024 at 15:56):The way Patrick Iglesias Zemmour defines it in his textbook is by first looking at the supremum of the dimensions of the domains of the plots inside some generating family for the diffeology, and then take the infimum over the generating families. So it is uniform across the space, but ultimately I don't think this is such a great notion. For example in Exercise 51 of the diffeology textbook, you prove that the half-line given the subset diffeology of has infinite dimension.
Emilio Minichiello (Jun 26 2024 at 15:57):Not actually sure of the dimension of the Axes example, maybe it is written down somewhere though.
Eric M Downes (Jun 26 2024 at 16:26):Talk is now uploaded to dropbox for anyone who missed it!
Federica Pasqualone (Jul 01 2024 at 22:35):@Emilio Minichiello First off, thanks for the great talk the other day! This embedding business is quite useful I agree! Q: Is there any particular idea behind the first diff. axiom? Now I am getting curious... Thanks.
Emilio Minichiello (Jul 02 2024 at 17:10):Hi, and thank you! The first axiom for a diffeological space? As in that every parametrization of the form R^0 -> X is a plot? The idea behind the first axiom is that it guarantees that not only is X is a sheaf but that it is also concrete. I didnt go very deep into concreteness in my talk but theres some nice stuff written about it on the nlab
Emilio Minichiello (Jul 02 2024 at 17:12):https://ncatlab.org/nlab/show/concrete+sheaf
Federica Pasqualone (Jul 02 2024 at 19:31):ah ... now we are talking! Thanks for the link. :sunglasses: