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January 26, 2012
Banning Elsevier
Posted by John Baez
Please take the pledge not to do business with Elsevier. 402 scientists have done it so far:
You can separately say you 1) won’t publish with them, 2) won’t referee for them, and/or 3) won’t do editorial work. At least do number 2): activism is rarely so little work, but when a huge corporation relies so heavily on nasty monopolistic practices and unpaid volunteer labor, they leave themselves open to this.
January 21, 2012
Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory
Posted by Urs Schreiber
We are in the process of finalizing a little article:
Domenico Fiorenza, Hisham Sati, Urs Schreiber,
Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory
Below is the abstract, and, below the fold, the beginning of the introduction. A pdf with the current version is behind the above link.
We would be grateful for comments.
The article is written in a non-formal style with an audience of certain physicists in mind, but ample pointers are given to places where all the ingredients are spelled out in detail and precisely.
Abstract
The worldvolume theory of coincident M5-branes is expected to contain a nonabelian 2-form/nonabelian gerbe gauge theory that is a higher analog of self-dual Yang-Mills theory. But the precise details – in particular the global moduli / instanton / magnetic charge structure – have remained elusive.
Here we argue that the holographic dual of this nonabelian 2-form field, under duality, can be deduced from anomaly cancellation.
We find this way a 7-dimensional nonabelian Chern-Simons theory of twisted String 2-connection fields, which, in a certain higher gauge, are given locally by non-abelian 2-forms with values in a Kac-Moody loop Lie algebra. We construct the corresponding action functional on the entire smooth moduli 2-stack of field configurations, thereby defining the theory globally, at all levels and with the full instanton structure, which is nontrivial due to the twists imposed by the quantum corrections. Along the way we explain some general phenomena of higher nonabelian gauge theory that we need.
January 15, 2012
Vorsicht! Funktor!
Posted by Tom Leinster
Constanze Roitzheim pointed me to this excellent photo by Gerd Laures:
Translation: Caution! Functor! Keep 2m back.
Click the photo for the full size version.
January 13, 2012
Logic as Invariant-Theory
Posted by David Corfield
Greetings from Oberwolfach, where I’m attending a workshop on Explicit Versus Tacit Knowledge in Mathematics. ‘Tacit knowledge’ is a term we owe to Michael Polanyi, though we haven’t heard all that much about his views this week. (In fact, the attentive Café visitor may have seen more of Polanyi here over the years, here, here and here.)
But there are always plenty of reasons to hope to gain from a workshop, and I was particularly interested by a talk by Christophe Eckes on the debt Hermann Weyl owed to Felix Klein and his group-theoretic approach. I was already alerted to the theme earlier in the week, having found out about a paper An Extension of Klein’s Erlanger Program: Logic as Invariant-Theory in which F. I. Mautner had attempted to carry over Weyl’s treatment of the Erlanger Program from Classical Groups to logic. Essentially we’re seeing the action of the symmetric group, , on, a set of elements and on powers of . Invariants of the full symmetric group are logical constructions. We even get to see a tensor notation in whose terms propositions may be expressed.
January 12, 2012
Propositions as Some Types and Algebraic Nonalgebraicity
Posted by Mike Shulman
Perhaps the aspect of homotopy type theory which causes the most confusion for newcomers (at least, those without a background in type theory) is its treatment of logic. With that in mind, I want to share some general thoughts about all the ways in which foundational systems can treat logic, and why I think HoTT makes the choice that it does. This even turns out to have interesting implications for the difference between “algebraic” and “non-algebraic” structures.
January 11, 2012
Mathematical Aspects of String and M-Theory in Cambridge
Posted by Urs Schreiber
In Cambridge is currently is taking place the workshop Mathematical Aspects of String and M-theory in the context of the program Mathematics and Applications of Branes in String and M-theory.
I am not there (I am at Higher structures along the lower Rhine), but my co-author Hisham Sati is. Hisham is speaking about our upcoming article with Domenico Fiorenza, on the 7-dimensional Chern-Simons QFT of String 2-connections dual to M5-branes. That’s the one I had briefly mentioned in my talk in Göttingen a few weeks back. A nice application of higher cohesive topos theory, I think.
While, judging from the workshop abstracts, this seems the only higher category theory that is explicitly mentioned in Oxford, of course there is secretly higher category theory all over the place in this subject. For instance the two talks on generalized complex geometry are secretly about the geometry of Courant Lie 2-algebroids.
I’ll post more details a little later.
January 3, 2012
A Semigroup Approach to Finite Markov Chains
Posted by Tom Leinster
Guest post by Benjamin Steinberg
A finite Markov chain is a very simple kind of transition system:
It can be encoded as a square matrix of nonnegative reals, each of whose rows sums to 1 (a ‘stochastic matrix’). Here you’ll meet the fundamental results on finite Markov chains, proved in a way that appears to be new. — TL
I’m going to prove all the standard finite Markov chain convergence results using the theory of compact semigroups. This approach may be known, but I couldn’t find it. I don’t obtain any of the bounds on convergence, although I suppose one can obtain them via this method with a little work.
December 30, 2011
The Café Will Be Closed 2 Days in January
Posted by John Baez
The -Category Café will be closed for 48 hours starting Friday, January 6th, 2012 at 6 pm CST. My calculations say this is midnight Greenwich Mean Time, the end of January 6th and beginning of January 7th.
They are temporarily shutting off all power and internet to the building at the University of Texas at which the Café secretly resides.
December 25, 2011
The Eventual Image, Part 2
Posted by Tom Leinster
We all love a good universal property. Objects with a simple universal property are usually important. So you might guess (mightn’t you?) that objects with two simple universal properties are more important still.
Perhaps the most famous example is direct sum. The direct sum of of two vector spaces, modules, etc., is both the product and the coproduct of and . And the importance of direct sums is written all over homological algebra.
In that example, the two universal properties are dual to one another. This is often the way. A less obvious example, but one whose importance becomes more and more apparent the deeper you dig into category theory, is the splitting of idempotents. This process can be viewed as either a limit or a colimit.
Objects with two universal properties are extra special, then. When the universe hands you one, it’s a treat.
Now: it’s Christmas. Ordinary people give each other socks. But since you already have enough socks, I give you instead: a machine for producing objects with two dual universal properties. Merry Christmas!
December 20, 2011
On the Law of Large Numbers (Such As 60)
Posted by Tom Leinster
I spent yesterday morning in the computer science department at Strathclyde, for the 60th birthday celebrations of Peter Hancock. Often these events are months away from the person’s actual birthday, but in this case it was only one day off. Happy birthday, Hank!
One of the talks I went to was by Alex Simpson, the title of which was too good not to use as the title of this post. Simpson’s recent work has re-examined foundational questions about probability and randomness. It’s also the most exciting use of locales I think I’ve ever encountered. I want to tell you something about it here.
December 19, 2011
What Do You Think of EPSRC Policy?
Posted by Tom Leinster
This year has seen a dramatic deterioration in the relationship between mathematicians working in the UK and our main funding body, the Engineering and Physical Sciences Research Council (EPSRC). Maybe you’ve read about it on Tim Gowers’s blog. Or maybe you’ve seen Burt Totaro’s roundup of strongly worded letters to the EPSRC and the government, from probably every learned society that has anything to do with British mathematics.
It’s depressing stuff, even if you’re not in the UK. Hopefully we’ll serve as an example to others, but there’s always the suspicion that policy changes in one country are part of a worldwide trend. Indeed, there seem to be resemblances to the situation in Canada — see, for instance, these posts by Nassif Ghoussoub (and the splendidly titled “UK mathematicians unload on intransigent patronizing bureaucracy”).
Back from NIPS 2011
Posted by David Corfield
Somehow 5 years have slipped by since my post Back from NIPS 2006. These NIPS (Neural Information Processing Systems) conferences bring together the machine learning community every December for a main conference in a city, followed by two days of workshops in a ski resort nearby. This time it was the turn of Granada and Sierra Nevada.
I could only manage time off for the workshops, where I was invited to participate in the Machine Learning and Philosophy session. The main thrust of my talk was to indicate that philosophy of science has shown that there’s much more to learning than what can be achieved by classifying and regression algorithms. There’s a tendency to believe that results in formal learning theory have a great deal to say about learning in general. It seems to me rather like believing that theorems in proof theory are the final word on what is philosophically interesting about mathematics. I have always believed instead that concept formation in science and mathematics is of the utmost importance.
December 18, 2011
4th Odense Winter School on Geometry and Mathematical Physics
Posted by Urs Schreiber
I have just arrived in Denmark, where I am attending the
4th Odense Winter School on Geometry and Theoretical Physics (website, program ).
I’ll try to report in the comment section below on what’s going on.
Myself, I’ll be lecturing on Stacks, differential geometry and action functionals, following section 1.1 “Motivation” with technical details sprinkled in from section 1.3 “Models and applications” of the cohesive document.
I had given similar lectures (with roughly twice the time, though) last weekend in Warsaw, at the
Polish Seminar on Category Theory and its Applications (website)
with an accompanying homotopy type theoretic comment in the Warsaw category theory seminar on the geometric and the homotopical interval/circle in homotopy type theory (see here for what this is about).
I am trying to use these events to fine-tune the section 1. Introduction further (not that there is not lots of room to improve it). I am aware of a bunch of deficiencies of the exposition, that I am trying to find the time to fix. But there must be also a bunch of deficiencies that I am not yet aware of. If you tell me about those that bug you, I’d be very grateful. (Many thanks at this point to John Huerta, who has recently given me a list of useful feedback on this.)
December 16, 2011
Categorifying Fractional Euler Characteristics
Posted by John Baez
Mathematicians should know how to count. Most of us do. But still there are some mysteries left when we try to count things in a way that gives a negative, fractional, irrational or complex answer.
Luckily, we’ve been making lots of progress. The Euler characteristic of a space can be a negative integer. The cardinality of a groupoid can be a nonnegative real number. The Euler characteristic of a category can even be a negative real number! And here’s yet another approach:
- Igor Frenkel, Catharina Stroppel, Joshua Sussan, Categorifying fractional Euler characteristics, the Jones-Wenzl projector and 3j-symbols.
December 12, 2011
What Might Be Done About High Prices of Journals?
Posted by John Baez
The International Mathematical Union (IMU) and the International Council for Industrial and Applied Mathematics (ICIAM) are important math organizations. For example, the IMU runs the big prestigious International Congress of Mathematicians. Now they have a blog, and they want to know our ideas on what to do about high-priced journals:
- Blog on Mathematical Journals, What might be done about high prices of journals?, December 7th, 2011.
I’ll disable comments here — I want you to comment over there! But I’ll tell you what I said.
(It takes a while for comments to appear, so you may not see what I said over there yet.)