February 27, 2012

Prequantization in Cohesive Homotopy Type Theory

Posted by Urs Schreiber

Chris Rogers and myself are studying structures in higher (meaning: -categorified) geometric (pre)quantization. Later this year I may post something about the higher structures that appear in this context, but right now I will highlight something else.

Our constructions proceed in two steps. First we give a general abstract axiomatic definition of geometric (pre)quantization internal to any cohesive ∞-topos. In the next step we pass to suitable models of the axioms and work out how there this reproduces the traditional notions as well as various generalizations of these that have been proposed, and usefully generalizes all this in various way.

In other words, we formulate geometric prequantization in cohesive homotopy type theory and then study its models. Since here in the $n$Café we had, in the last months, discussed most of the relevant ingredients before, I thought it would be fun to highlight just this step.

See Prequantum physics in a cohesive ∞-topos for more exposition.

Technical details on what I say below are in section 2.3.24 (general definitions) and section 3.3.17 (models in smooth cohesion).

Posted at 6:43 PM UTC | Permalink | Followups (14)

February 26, 2012

Types, Homotopy and Univalent Foundations: Special Issue

Posted by Tom Leinster

Been thinking about the interaction between type theory and homotopy theory? Got a paper you want to write? Already written your masterpiece, but looking for somewhere to submit it?

Here’s news from Nicola Gambino of a special issue of Mathematical Structures in Computer Science devoted to such things. Details below the fold.

(And in case you’re wondering, that’s not an Elsevier journal: it’s published by those nice people at Cambridge University Press.)

February 22, 2012

ESI Program K-Theory and Quantum Fields

Posted by Urs Schreiber

This summer in Vienna at ESI takes place a program titled

K-Theory and Quantum Fields

May 21 - July 27, 2012 (webpage)

organized by Matt Ando, Alan Carey, Harald Grosse and Jouko Mickelsson.

It starts out with

• introductory instructional lectures in the week May 28-June 1;

• advanced instructional lectures in the week June 4- June 8.

Details should appear by beginning of March on the webpage. As far as I am aware, the following speakers and topics are expected for the instructional lectures.

• John Francis: factorization algebras/homology;
• Dan Freed (probably on June 7th and 8th, topic to be announced);
• Mathai Varghese: T-duality and K-theory;
• Frederic Rochon: introduction to K-theory;
• myself: twisted differential structures in string theory;
• Danny Stevenson: twisted K-theory, higher structures;
• Bai-Ling Wang: K-theory, K-homology and twisted geometric cycles.

Posted at 8:44 PM UTC | Permalink | Followups (5)

February 21, 2012

In Göttingen

Posted by Urs Schreiber

Currently the higher order structures group in Göttingen is running their reading seminar on our article L-infinity connections with Hisham Sati and Jim Stasheff. This week I am visiting Göttingen, by kind invitation by Chenchang Zhu, in order to answer questions and to speak about related structures.

Today we had a long session working our way towards the basics of synthetic differential infinity-groupoids, a context where $L_\infty$-algebras exist alongside genuine smooth $\infty$-groupoids as those who have first order infinitesimal spaces of $k$-morphisms for all $k$.

Tomorrow I’ll try to explain how using this we construct, with Domenico Fiorenza, the smooth moduli 2-stack of string 2-connections and the smooth moduli 6-stack of fivebrane 6-connections by exponentiating those big double square diagrams of $L_\infty$-algebras discussed in the article.

Today I learned from the participants that interest in the nonabelian 2-form on the 5-brane has been a motivation to pick this reading seminar topic, so I will close with a remark on the basic idea in our recent article on that application of the machinery.

Meanwhile, I am glad to have had a chance to chat with Chris Rogers about our project of studying the higher Atiyah/Courant algebroids of circle $n$-bundles for all $n$.

So it’s busy enough. Just thought I’d drop a note on what I am up to this week.

February 16, 2012

Math 2.0

Posted by John Baez

Building on the Elsevier boycott, a lot of people are working on positive steps to make expensive journals obsolete. My email is flooded with discussions, different groups making different plans.

Email is great, but not for everything. So Andrew Stacey (the technical mastermind behind the nLab, Azimuth Wiki and Azimuth Forum) and Scott Morrison (one of the brains behind MathOverflow) have started a forum to talk about the many issues involved:

Math 2.0.

Check it out.

Posted at 3:43 AM UTC | Permalink | Followups (1)

February 15, 2012

Workshop on Formal Topology, Higher Dimensional Algebra, Categories and Types

Posted by Urs Schreiber

This June takes place the Fourth Workshop on Formal Topology (and related topics, including constructive and computable topology, point-free topology, and other forms of non-classical topology) in Ljubljana, Slovenia.

Keynote speakers are Per Martin-Löf, Ieke Moerdijk and Vladimir Voevodsky.

See the workshop webpage for details.

Adjoined to that, on the last day, is a Workshop on Higher Dimensional Algebra, Categories and Types, which I am helping to organize a bit.

This should consist of three talks, roughly one on type theory, one on type theory and homotopy theory and one on higher category theory.

Two confirmed of three speakers are Thorsten Altenkirch and Steve Awodey.

For more details on this see the HDACT Workshop Webpage.

Posted at 5:19 PM UTC | Permalink | Followups (1)

February 14, 2012

Modern Perspectives in Homotopy Theory

Posted by Urs Schreiber

This April (10th - 13th) takes place a school titled

Modern Perspectives in Homotopy Theory

subtitled

∞-Categories, ∞-Operads and Homotopy Type Theory

in Swansea, UK.

See the school’s website.

David Gepner will speak about $\infty$-categories, Ieke Moerdijk about $\infty$-operads (and dendroidal sets), and Mike Shulman about $\infty$-type theory.

February 9, 2012

The Moduli 3-Stack of the C-Field

Posted by Urs Schreiber

We are in the process of finalizing a little article

Domenico Fiorenza, Hisham Sati, U.S., The $E_8$ moduli 3-stack of the C-field in M-theory

Abstract The higher gauge field in 11-dimensional supergravity – the C-field – is constrained by quantum effects to be a cocycle in some twisted version of ordinary differential cohomology. We argue that it should indeed be a cocycle in a certain twisted nonabelian differential cohomology. We give a simple and natural characterization of the full smooth moduli 3-stack of configurations of the C-field, the field of gravity and the (auxiliary) E8-Yang-Mills field. We show that the truncation of this moduli 3-stack to a bare 1-groupoid of field configurations reproduces the differential integral Wu structures that Hopkins-Singer had shown (HS02) to formalize Witten’s argument (Wi96) on the nature of the C-field. Finally we give a similarly simple and natural characterization of the moduli 2-stack of boundary C-field configurations and show that it is equivalent to the smooth moduli 2-stack of anomaly free heterotic supergravity field configurations (SSS12).

This may be read as a companion to the article that I mentioned last time, at Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

$\,$

A pdf of the article is behind the above link. Any comment you might have would be most welcome.

Posted at 9:11 AM UTC | Permalink | Followups (11)

The Cost of Knowledge

Posted by John Baez

As of this moment, 4760 scholars have joined a boycott of the publishing company Elsevier. Of these, only 20% are mathematicians. But since the boycott was started by a mathematician, 34 of us wrote and signed statement explaining the boycott. Here it is.

Posted at 1:51 AM UTC | Permalink | Followups (1)

February 8, 2012

Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory

Posted by Alexander Hoffnung

This past October Alistair Savage and I organized a workshop on Category Theoretic Methods in Representation Theory at the University of Ottawa. The event was generously supported by the Fields Institute.

Following the success of the October workshop, Oded Yacobi, Chris Dodd, and I decided to hold another workshop, this time with a focus on researchers who are still very early on in their careers. The Fields Institute has again offered funding and this time will host the event as well.

We would like to draw your attention to the upcoming Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory to be held May 7-9, 2012 at the Fields Institute in Toronto.

Keep reading below the fold and see the workshop website for more on the content of the workshop, registration information, and applications for financial support.

Posted at 7:28 PM UTC | Permalink | Followups (9)

February 7, 2012

Good News

Posted by John Baez

I’m in Sydney talking to Ross Street and other category theorists at Macquarie University. Once I’d expressed worries about the fate of Australian category theory now that Street has retired. I’m happy to say that we can put those worries to rest.

First of all, while Street has formally retired, he still comes to work every day and is very active. Second, Steve Lack has moved from Sydney University to Macquarie, so he can now have graduate students. On top of that, he’s gotten a 4-year grant that’ll let him spend most of his time on research! Richard Garner is here too, and has a 5-year research-only postdoc. Also in the math department are Michael Batanin, Alexei Davydov and Mark Weber. And if that weren’t enough, Dominic Verity has joined Mike Johnson over in Macquarie’s computing department.

So, it’s a great place to go if you want to talk to category theorists! I apologize to anyone whom I left out.

And on a more personal note…

Posted at 12:03 AM UTC | Permalink | Followups (21)