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Urs Schreiber
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I am writing a textbook Introduction to Topology. The first six chapters are done.

With Firefox, you get the hyptertext version here:
http://ncatlab.org/nlab/print/Introduction+to+Topology+--+1

Otherwise there is the stand-alone pdf available here:
http://ncatlab.org/nlab/print/Introduction+to+Topology

The book exposes the classical theory by classical means, assuming no background except the usual informal familarity with sets.

Special aspects are:

(A) A gentle introduction to the language of category theory is made in passing, in informal remarks each time we encounter various constructions in topology.

(B) Accordingly, universal constructions of topological spaces (the weak/strong topologies!) are explained without being shy about explaining their universal properties, which is useful, but while keeping the category-theoretic jargon away.

(C) Besides the classical separation axioms, soberity of topological spaces is introduced and explained. While this important concept easily fits into an elementary classical discussion, previous textbook accounts of this have been well buried beneath formal logic and frame-, locale- and topos-theory.

Here the opposite perspective is offered: The concept of sober space is in fact immediate from the traditional classical perspective, moreover it is relevant in hands-on examples (the Zariski topology is sober but not Hausdorff!), and it serves to motivate and explain why one might want to re-do classical topology by doing away with the underlying point sets and retaining only the logic of space.

(D) Following Todd Trimble, the book gives an easy proof of the characterization of compact spaces by closedness of the maps projecting them out (secretly the result of some nice category theory happening under the hood, but broken down to an easy classical proof). This allows for instance to give an easy elementary proof of the general Tychonoff theorem.





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Here are the slides for +Felix Wellen's talk in Nantes last week [1]:
Aspects of Differential Geometry in Homotopy Type Theory
https://ncatlab.org/schreiber/files/wellenDGinHoTT.pdf

All details will be in his thesis, due out later this summer:
https://ncatlab.org/schreiber/show/Formalizing+Cartan+Geometry+in+Modal+HoTT#WellenThesis

[1] https://plus.google.com/+UrsSchreiber/posts/X8Wm6yjWw81

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Next week in Nantes [1], +Felix Wellen will be speaking about Formalizing aspects of differential Cohesion in Homotopy Type Theory.

This is about the results from his PhD thesis [2], due out later this summer

Abstract. The categories of smooth manifolds or schemes may be faithfully embedded in categories of (higher) sheaves on some site. Statements proven in Homotopy Type Theory may be transferred to such categories of sheaves.
We demonstrate how a modality in the type theory may be used to access some aspects of the differential geometric structure of the sheaves, given by manifolds or schemes, and provide some basic results concerning a generalized version of the tangent bundle.

[1] https://fpfm.github.io/

[2] https://ncatlab.org/schreiber/show/Formalizing+Cartan+Geometry+in+Modal+HoTT
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With Marco Benini and +Alexander Schenkel : arxiv.org/abs/1704.01378 .

A description of gauge field theory such as Yang-Mills theory (e.g. QED, QCD) which is both non-perturbative and local necessarily needs to promote the spaces of field configurations to "higher spaces", called "stacks", see e.g. [1] for exposition. Indeed the axioms of algebraic quantum field theory [2] on curved spacetimes [3] break for gauge theories, unless one refines them in a stacky way, a project that Marco and Alexander are working on [4].

But the stacks of gauge fields on a given spacetime region, hence the stacks of Yang-Mills fields on Lorentzian manifolds need special care, because they are not exactly the mapping stacks of that spacetime region into the "universal moduli stack", but are the "differential concretification" thereof.

I had introduced that concept of differential concretification of moduli stacks of (higher) gauge fields in [5]. Unfortunately, as Marco and Alexander thankfully noticed, the formula I gave contained a mistake. In this article we state the correct formula for 1-stacks, and in a followup we will discuss the correct differential concretification for higher stacks of higher gauge fields.

After constructing the stack of Yang-Mills fields on Lorentzian manifolds in great detail, this article ends with observing a crucial consequence of refining the description of gauge fields this way: the initial value problem for Cauchy problems of gauge fields changes somewhat in its dependence on parameters compared to the traditional description. Of course solving that Cauchy problem is a key step in quantizing systems of stacks of Yang-Mills fields to a local net of quantum observables (a kind of co-stack now). The article does not solve this new problem in PDE theory, but it ends by working out exactly what the stacky Cachy problem is, that needs to be solved next.

[1] https://ncatlab.org/schreiber/show/Higher+field+bundles+for+gauge+fields

[2] https://ncatlab.org/nlab/show/AQFT

[3] https://ncatlab.org/nlab/show/AQFT+on+curved+spacetimes

[4] https://ncatlab.org/nlab/show/gauge+theory#ReferencesInAQFT

[5] https://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos
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Here are the slides for my talk on Super topological T-duality
https://ncatlab.org/schreiber/print/Super+topological+T-Duality
later today in Regensburg [1]. Comments are welcome.

[1] http://www-cgi.uni-regensburg.de/Fakultaeten/MAT/sfb-higher-invariants/index.php/SpringSchool2017
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Next week in Regensburg takes place a Conference on invertible objects and duality in derived algebraic geometry and homotopy theory
http://www-cgi.uni-regensburg.de/Fakultaeten/MAT/sfb-higher-invariants/index.php/SpringSchool2017

Myself I will be speaking about "Super Topological T-Duality"
https://ncatlab.org/schreiber/show/Super+topological+T-Duality


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The rules of "topological T-dualty" [1] are mathematical conditions that ensure that certain pull-push integral transforms in twisted K-theory (i.e. RR-fields) are isomorphisms. The name suggests that these rules follow the physics of T-duality in string theory, but the evidence provided for this in the original article Bouwknegt-Evslin-Mathai-03 [2] was somewhat hand-wavy. Many string theorists do not know the rules of "topologial T-duality".

In [3] we have a systematic derivation of the rules, including fine detail added later by Bunke et al., from first principles. We are not starting from the perturbative string, but starting from the non-perturbative formulation where all p-branes involved are treated on the same footing via the super-cocycles that define their Green-Schwarz action functionals. We start from scratch and derive for instance the Buscher rules for RR-fields. Derivation of this "Hori's formula", shown below, will mark the end tomorrow of my lecture series on fundamental super p-branes [4] that I was giving in Alberto Cattaneo's group in Zurich this month.


[1] https://ncatlab.org/nlab/show/topological+T-duality
[2] https://ncatlab.org/nlab/show/topological+T-duality#BouwknegtEvslinMathai04
[3] https://ncatlab.org/schreiber/show/T-Duality+from+super+Lie+n-algebra+cocycles+for+super+p-branes
[4] https://ncatlab.org/nlab/show/geometry+of+physics+--+fundamental+super+p-branes
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For me this note [1] was part of the motivation for developing synthetic variational calculus in infinity-toposes with +Igor Khavkine [2]. Of course there are many other motivations for that, too, but the superalgbra extensions obtained from the Noether charges of the super p-branes are a particularly striking example of the higher Noether theorem. This is discussed a bit in the talk slides at [3].

[1] https://ncatlab.org/schreiber/show/Lie+n-algebras+of+BPS+charges
[2] https://ncatlab.org/schreiber/show/Synthetic+variational+calculus
[3] https://ncatlab.org/schreiber/show/Obstruction+theory+for+parameterized+higher+WZW+terms
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Introduction to rational homotopy theory [1] the video recording of my 7th lecture in the super-p-brane seminar [2] is now available here:

http://www.karlin.mff.cuni.cz/~soucek/zaznamy_prednasek/Urs%20Schreiber/7_seminar/index.html


[1] https://ncatlab.org/nlab/show/rational+homotopy+theory
[2] https://ncatlab.org/schreiber/show/From+the+Superpoint+to+T-Folds

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Yesterday in Prague +Igor Khavkine gave a beautiful exposition of synthetic PDE theory. The video recording is now available here: http://www.karlin.mff.cuni.cz/~soucek/zaznamy_prednasek/SDG%20PDE/index.html
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