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Urs Schreiber
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Urs Schreiber

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This Saturday morning, at "Foundations of Maths" in Bielefeld http://fomus.weebly.com/ I give a presentation "Modern Physics Formalized in Modal Homotopy Type theory". I will follow the chapter

"Higher Prequantum Geometry"
in G. Catren, M. Anel (eds.)
New Spaces for Mathematics and Physics
IHP Paris.
https://ncatlab.org/schreiber/show/Higher+Prequantum+Geometry

with emphasis on its part 3:
"Abstract prequantum field theory".
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+David Corfield I can't see it - one needs a FB account :-(
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The satellite meeting of the 7th European Congress of Mathematics on Classical and quantum symmetries in mathematics and physics takes place next week in Jena.

The full program is here

  https://dl.dropboxusercontent.com/u/12630719/CQSYMP16.pdf

Highlights include Martin Schnabl giving a plenary talk on "Topological defects and open string field theory".

Myself, I will speak on the open problem of deriving the coefficients of the generalized cohomology of M2/M5-brane charges:

https://ncatlab.org/schreiber/show/Generalized+cohomology+of+M2/M5-branes
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For the moment I am done with the last part of a textbook Introduction to Stable homotopy theory, the last part being about the Adams spectral sequence.

With Firefox go to

https://ncatlab.org/nlab/show/Introduction+to+Stable+homotopy+theory+--+2

otherwise use the pdf

https://dl.dropboxusercontent.com/u/12630719/StableHomotopyTheory-2.pdf

The first part, in turn, of this last part, up to the discussion of convergence,  provides full details with full proofs. The section on convergence presently just states Bousfield's convergence theorems, after introducing all the infrastructure needed to state and parse them. Maybe later there'll be an occasion that I add an exposition of these proofs, too.

Then the last part of the last part is a walk through the detailed computation of the classical Adams spectral sequence for the computation of the stable homotopy groups of spheres in low degrees.  I spell out all the lemmas required for running the May spectral sequence for the second page and give detailed examples of the kind of computations that one needs to do, enough that the interested reader should see how to proceed.

Beware that this last part is not completely finalized towards the end, but it should be well readable already. I will get back to this later.

For the main document see

https://ncatlab.org/nlab/show/Introduction+to+Stable+homotopy+theory
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Fernando Yamauti's profile photoUrs Schreiber's profile photo
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+Fernando Yamauti I am serious that I will sell bound copies if there is demand. You'll realize that the format of the book is that of a hypertext web document which does not fit the established venues that you point to.  But if you find my offer too expensive, why don't you just print and bind the document via any cheaper service that you may find?
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I am writing a textbook: Introduction to Stable homotopy theory. The first two chapters of four are done,  [1] and [2] below.

The main page is here: https://ncatlab.org/nlab/show/Introduction+to+Stable+homotopy+theory

Unique feature:

* no tricks, full proofs, self-contained

* state of the art, via stable model categories of structured spectra

About the topic: last week Graeme Segal gave the Kan memorial lecture in Utrecht on The ubiquity of homotopy

http://www.uu.nl/en/events/the-utrecht-mathematical-colloquium-graeme-segal-university-of-oxford

and announced it like so:

"Much of mathematics is about discovering robust kinds of structure which organize and illuminate large areas of the subject. Perhaps the most basic organizing concept of our thought is space. It leads us to the homotopy category, which captures many of our geometric intuitions but also arises unexpectedly in contexts far from ordinary spaces. Still more is this true of the `stable homotopy' category, which sits midway between geometry and algebra.
The theme of my lectures is the strangeness and the ubiquity of the homotopy and stable homotopy categories, and how they give us new ideas of what a space is, and why manifolds and spaces with algebraic structure play such a special role."



[1]
Chapter Classical homotopy theory

With a MathML browser (Firefox or Iceweasel) see

https://ncatlab.org/nlab/show/Introduction+to+Stable+homotopy+theory+--+P

otherwise there is the pdf:

https://dl.dropboxusercontent.com/u/12630719/StableHomotopyTheory-P.pdf


[2]
Chapter Stable homotopy theory -- Sequential spectra

With a MathML browser (Firefox or Iceweasel) see

https://ncatlab.org/nlab/show/Introduction+to+stable+homotopy+theory+--+1-1

otherwise there is the pdf

https://dl.dropboxusercontent.com/u/12630719/StableHomotopyTheory-1-1.pdf
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Trent Knebel's profile photoUrs Schreiber's profile photo
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Following +Trent Knebel 's suggestion, I have sent a link to MathOverflow here: http://mathoverflow.net/a/244707/381
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Rational sphere valued supercocycles in M-theory.  An article that we are preparing with +Domenico Fiorenza and +hisham sati:

https://ncatlab.org/schreiber/print/Rational+sphere+valued+supercocycles+in+M-theory

We show that supercocycles on super L-infinity algebras capture, at the rational level, the twisted cohomological charge structure of the fields of M-theory and of type IIA string theory. We show that rational 4-sphere-valued supercocycles for M-branes in M-theory descend to supercocycles in type IIA string theory to yield the Ramond-Ramond fields predicted by the rational image of twisted K-theory, with the twist given by the B-field. In particular, we derive the correspondence

   M2/M5 ↔ F1/Dp/NS5

via dimensional reduction of sphere-valued L-∞ supercocycles in rational homotopy theory.
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David Roberts's profile photoZoran Škoda's profile photoUrs Schreiber's profile photo
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+Zoran Škoda: Earlier this year I had been invited by Roger Picken for one month, to give a lecture series [1]. We ended up splitting this into a sequence of separate visits, each for one week.

[1] https://plus.google.com/+UrsSchreiber/posts/1CovBAaechy
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More on the higher differential geometry of M5-branes, by +Branislav Jurco , +Christian Saemann and +Martin Wolf :

Higher Groupoid Bundles, Higher Spaces, and Self-Dual Tensor Field Equations
https://arxiv.org/abs/1604.01639

I'll say more  about this end of July, when Christian and myself will be talking about the higher structures in M5-branes at the String Math meeting of the 7th European Congress of Mathematics in Jena https://www.tpi.uni-jena.de/tiki-index.php?page=MathPhys16
Abstract: We develop a description of higher gauge theory with higher groupoids as gauge structure from first principles. This approach captures ordinary gauge theories and gauged sigma models as well as their categorifications on a very general class of (higher) spaces comprising presentable ...
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+Urs Schreiber​ yes, I know :-) 
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The nLab has now almost 12.000 entries (as of today: 11.939) At the Foundations of Mathematics conference in Bielefeld this week [1], I will be giving a presentation of the nLab wiki [2]. It's not a talk, but an informal session. Bring your questions.

What is the nLab?
https://ncatlab.org/schreiber/show/What+is...+the+nLab

Connect the relevant information. We all waste too much time with searching for mathematical information that is already out there. As a student, before the dawn of the internet, I wasted days in the library, on chasing references to the secrets of the universe. Now the internet exists, but we still waste time searching randomly. Things have not been connected. The nLab means to connect the dots. The idea is that you stop searching randomly and just follow the links. Hypertext. That was the original vision of the web. We need more research-level maths hypertext.

Show the big picture. We are in an age where in theoretical physics we are supposed to work on quantum gravity and unification, needing the very latest of the developments in mathematics. At the same time we still bring up students with old textbooks. This way even the best of them at the end of their study can only grasp a tiny fraction of the big picture, because the knowledge is so scattered in tiny sub-expert communities. This is insane and unnecessary. The nLab means to connect the dots and show the big picture. That’s why it’s organized by higher category theory. This is the structure that helps organize things and bring them together conceptually. (While of course many specific entries need not be category theoretic at all).

Tap the power of the swarm. Record. Provide a place where all the notes that we all make merge and become better. Sometimes I see people proudly show me their private maths notebooks. Too bad that only one single person is profiting from it. We are an army of wheel-reinventers. (That’s necessary to some extent for personal exercise, but we’ll get nowhere if every single person retraces every single step. That has ended being sensible several hundred years ago.) At the same time people are eager to throw around information, such as on MathOverflow. The nLab is to collect the stable bits, organized, hyperlinked, usefully.

Stop duplicating answers. We all exchange lots of questions and answers on research over coffee, in emails etc. When the next person comes and asks the same question we repeat it. And next time again. That’s a waste of energy in a finite life. If you have an answer and you put it into the nLab, next time you just link to it. For instance, whenever I want to say anything substantial on MathOverflow, I first put it on the nLab (if it’s not already there) and then link to that in my answer.

(Which points to the main technical problem the nLab has: it does not bathe your ego in +1s and badges.)

 

[1] http://fomus.weebly.com/schedule.html
[2] https://ncatlab.org/nlab/show/HomePage
1
David Corfield's profile photoUrs Schreiber's profile photoDavid Roberts's profile photo
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I'd be interested to hear if Ladyman gives more in his talk than in is the already available preprint(s) covering the topic.
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Connes' spectral triples from 2d superconformal field theories

Promted by the discussion of the relation of Connes's "spectral triples" to what the physics community already knows and loves, from here:

https://fmoldove.blogspot.de/2016/07/what-is-noncommutative-geometry.html?showComment=1468934322483#c6802080472818966598

I'll recall the following neat facts, which ought to be known much more widely:

Connes' spectral triples arise as the point particle limit of 2d (super-)conformal field theories. A commented list of references on this relation is here:

 ncatlab.org/nlab/show/2-spectral+triple#References

Under this relation, the way spectral triples encode an effective target space geometry as seen by a quantum particle is analogous to how a 2d (S)CFT encodes an effective target space geomtry as seen by a quantum (super-)string.

The first rigorous account of this is due to

D. Roggenkamp, K. Wendland, "Limits and Degenerations of Unitary Conformal Field Theories" arXiv:hep-th/0308143

summarized in

D. Roggenkamp, K. Wendland, "Decoding the geometry of conformal field theories" arXiv:0803.0657

Yan Soibelman used this relation of 2d SCFT to Connes spectral triple in order to approach the analysis of aspects of the landscape of string vacua:

Y. Soibelman, "Collapsing CFTs, spaces with non-negative Ricci curvature and nc-geometry" , in H. Sati, U. Schreiber al. (eds.), "Mathematical Foundations of Quantum Field and Perturbative String Theory", Proceedings of Symposia in Pure Mathematics, AMS (2001)

https://ncatlab.org/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory#ContributionSoibelman

For a kind of reverse construction, in

S. Carpi, R. Hillier, Y. Kawahigashi, R. Longo, "Spectral triples and the super-Virasoro algebra" (arXiv:0811.4128)

the authors realize 2d SCFTs essentially as local nets (in the sense of AQFT) of spectral triples.

Lifting a spectral triple to a 2d CFT means imposing stronger constriants since there is much more data in the 2d CFT than just its point particle limit reflected in the spectral triple.

This is a version of the fact that there are much stronger constraints on a string background to be consistent (anomaly free) than on a random QFT.

The article by Soibelman referenced above means to make use of this for saying something about the landscape of perturbative string vacua. A perturbative string vacuum is a 2d SCFT of central charge 15, which in addition satisfies modularity and sewing constraints. While the moduli space of all 2d SCFTs is hard to analyze, the shadow that it throws, via the point particle limit, in the space of spectral triples is more tractable. And Soibelman's article analyzes this shadow space.

In this context one cannot help but notice the following coincidence:

The spectral triples arising from 2d SCFTs of central charge 15 in string theory have, famously, KO-dimension 4+6 (mod 8). Now this is precisely the KO dimension that Connes claims in

A. Connes, "Noncommutative Geometry and the standard model with neutrino mixing", JHEP0611:081 (hep-th/0608226)

is necessary to get a viable standard model-like effective theory from a spectral triple.

Maybe it's a coincidence. Or maybe it points to something deep.
4
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I am writing a textbook: Introduction to Stable homotopy theory. The first three chapters of four are done.  After parts [1] and [2]  constructed the stable homotopy category, part [3] exhibits its "tensor triangulated structure" via a "derived symmetric monoidal smash product of spectra".

This makes the stable homotopy category be the complete higher analog of the category of abelian groups, and makes algebra in stable homtopy theory (rings and modules) be higher algebra ("brave new algebra") of ring spectra and ring modules.


Unique feature:

* no tricks, full proofs, self-contained

* state of the art, via stable model categories of structured spectra


The main page is here: https://ncatlab.org/nlab/show/Introduction+to+Stable+homotopy+theory


[3]
Chapter Stable homotopy theory -- Structured ring spectra

With Firefox or Iceweasel see

https://ncatlab.org/nlab/show/Introduction+to+Stable+homotopy+theory+--+1-2

otherwise there is a pdf:

 https://dl.dropboxusercontent.com/u/12630719/StableHomotopyTheory-1-2.pdf



[1]
Chapter Classical homotopy theory

With a MathML browser (Firefox or Iceweasel) see

https://ncatlab.org/nlab/show/Introduction+to+Stable+homotopy+theory+--+P

otherwise there is the pdf:

https://dl.dropboxusercontent.com/u/12630719/StableHomotopyTheory-P.pdf


[2]
Chapter Stable homotopy theory -- Sequential spectra

With a MathML browser (Firefox or Iceweasel) see

https://ncatlab.org/nlab/show/Introduction+to+stable+homotopy+theory+--+1-1

otherwise there is the pdf

https://dl.dropboxusercontent.com/u/12630719/StableHomotopyTheory-1-1.pdf
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Linas Vepstas's profile photo
 
Haven't read this yet, but I like the general direction you've been working in, Urs! Neat stuff!
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New post! "Some remarks on dialectics in mathematics"

http://www.jonmsterling.com/posts/2016-06-22-dialectics-in-mathematics.html
Some Remarks on Dialectics in Mathematics. Dialectical materialism, far from being merely the definitive theory of political and historical movement, also serves as the guiding light for the progressive mathematician in its manifestation as the science of adjoint functors in category theory.
2 comments on original post
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Tomorrow in Lisbon, +hisham sati speaks about aspects of our work on the super-Lie-n-algebraic derivation of key aspects of fundamental (instead of black) M2/M5-branes [1]. This is in continuation of where I had left off a few weeks back [2].

[1] https://ncatlab.org/schreiber/print/Rational+sphere+valued+supercocycles+in+M-theory

[2] https://plus.google.com/+UrsSchreiber/posts/1CovBAaechy
6
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At the Foundations of Mathematics conference in Bielefeld [1], I will be giving a presentation of the nLab wiki [2]. It's not a talk, but an informal session. Bring your questions.

What is the nLab?
https://ncatlab.org/schreiber/show/What+is...+the+nLab

Connect the relevant information. We all waste too much time with searching for mathematical information that is already out there. As a student, before the dawn of the internet, I wasted days in the library, on chasing references to the secrets of the universe. Now the internet exists, but we still waste time searching randomly. Things have not been connected. The nLab means to connect the dots. The idea is that you stop searching randomly and just follow the links. Hypertext. That was the original vision of the web. We need more research-level maths hypertext.

Show the big picture. We are in an age where in theoretical physics we are supposed to work on quantum gravity and unification, needing the very latest of the developments in mathematics. At the same time we still bring up students with old textbooks. This way even the best of them at the end of their study can only grasp a tiny fraction of the big picture, because the knowledge is so scattered in tiny sub-expert communities. This is insane and unnecessary. The nLab means to connect the dots and show the big picture. That’s why it’s organized by higher category theory. This is the structure that helps organize things and bring them together conceptually. (While of course many specific entries need not be category theoretic at all).

Tap the power of the swarm. Record. Provide a place where all the notes that we all make merge and become better. Sometimes I see people proudly show me their private maths notebooks. Too bad that only one single person is profiting from it. We are an army of wheel-reinventers. (That’s necessary to some extent for personal exercise, but we’ll get nowhere if every single person retraces every single step. That has ended being sensible several hundred years ago.) At the same time people are eager to throw around information, such as on MathOverflow. The nLab is to collect the stable bits, organized, hyperlinked, usefully.

Stop duplicating answers. We all exchange lots of questions and answers on research over coffee, in emails etc. When the next person comes and asks the same question we repeat it. And next time again. That’s a waste of energy in a finite life. If you have an answer and you put it into the nLab, next time you just link to it. For instance, whenever I want to say anything substantial on MathOverflow, I first put it on the nLab (if it’s not already there) and then link to that in my answer.

(Which points to the main technical problem the nLab has: it does not bathe your ego in +1s and badges.)

 

[1] http://fomus.weebly.com/schedule.html
[2] https://ncatlab.org/nlab/show/HomePage
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Fouche Ehcuof's profile photoDavid Corfield's profile photo冯悦's profile photoUrs Schreiber's profile photo
6 comments
 
+David Corfield that's a good idea to use that article as an example of the use of the nLab. I only wish it were true that there are "many contributors of the nLab". One thing I will probably highlight at FOMUS is that in fact there are a tiny number of contributors. Probably the totality of entries they used for that article have no more than three genuine authors.
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