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May 31, 2009

The Mathematics of Music at Chicago

Posted by John Baez

As a card-carrying Pythagorean, I’m fascinated by the mathematics of music… even though I’ve never studied it very deeply. So, my fascination was piqued when I learned a bit of ‘neo-Riemannian theory’ from Tom Fiore, a topology postdoc who works on double categories at the University of Chicago.

Neo-Riemannian theory is not an updated version of Riemannian geometry… it goes back to the work of the musicologist Hugo Riemann. The basic idea is that it’s fun to consider things like the 24-element group generated by transpositions (music jargon for what mathematicians call translations in /12\mathbb{Z}/12) and inversion (music jargon for negation in /12\mathbb{Z}/12). And then it’s fun to study operations on triads that commute with transposition and inversion. These operations are generated by three musically significant ones called P, L, and R. Even better, these operations form a 24-element group in their own right! I explained why in week234 of This Week’s Finds. For more details try this:

Yes, that’s my student Alissa Crans, of Lie 2-algebra fame!

Posted at 8:58 PM UTC | Permalink | Followups (22)

May 28, 2009

Quantum Gravity and Quantum Geometry in Corfu

Posted by John Baez

This September there will be a physics ‘summer school’ covering loop quantum gravity, spin networks, renormalization and higher gauge theory:

I look forward to seeing my quantum gravity friends Abhay Ashtekar, John Barrett and Carlo Rovelli again — it’s been a while. It’s sad how changing one’s research focus can mean you don’t see friends you used to meet automatically at conferences.

I’m also eager to meet Vincent Rivasseau, who is a real expert on renormalization and constructive quantum field theory! His book From Perturbative to Constructive Renormalization is very impressive. I had a brief and unsuccessful fling with constructive quantum field theory as a grad student, so it’ll be nice (but a bit scary) to meet someone who’s made real progress in this tough subject.

Posted at 4:22 PM UTC | Permalink | Followups (5)

Metric Coinduction

Posted by David Corfield

Dexter Kozen and Nicholas Ruozzi have a paper Applications of Metric Coinduction which begins

Mathematical induction is firmly entrenched as a fundamental and ubiquitous proof principle for proving properties of inductively defined objects. Mathematics and computer science abound with such objects, and mathematical induction is certainly one of the most important tools, if not the most important, at our disposal.

Perhaps less well entrenched is the notion of coinduction. Despite recent interest, coinduction is still not fully established in our collective mathematical consciousness. A contributing factor is that coinduction is often presented in a relatively restricted form. Coinduction is often considered synonymous with bisimulation and is used to establish equality or other relations on infinite data objects such as streams or recursive types.

In reality, coinduction is far more general. For example, it has been recently been observed that coinductive reasoning can be used to avoid complicated ϵδ\epsilon-\delta arguments involving the limiting behavior of a stochastic process, replacing them with simpler algebraic arguments that establish a coinduction hypothesis as an invariant of the process, then automatically deriving the property in the limit by application of a coinduction principle. The notion of bisimulation is a special case of this: establishing that a certain relation is a bisimulation is tantamount to showing that a certain coinduction hypothesis is an invariant of some process.

Posted at 9:34 AM UTC | Permalink | Followups (9)

May 26, 2009

Alm on Quantization as a Kan Extension

Posted by Urs Schreiber

Recently I was contacted by Johan Alm, a beginning PhD student at Stockholm University, Sweden, with Prof. Merkulov.

He wrote that he had thought about formalizing and proving aspects of the idea that appeared as the The nn-Café Quantum Conjecture about the nature of [[path integral quantization ]].

After a bit of discussion of his work, we thought it would be nice to post some of his notes here:

Johan Alm, Quantization as a Kan extension (lab)

nnCafé-regulars may be pleased to meet some old friends in there, such as the [[Leinster measure]] starring in its role as a canonical path integral measure.

Posted at 7:21 PM UTC | Permalink | Followups (45)

May 24, 2009

Elsevier Journal Prices

Posted by John Baez

Do you have data about Elsevier’s journal prices compared to other journals? If so, let me know! Before we launch the revolution, we need to get our facts straight.

My friend the physicist Ted Jacobson wants such data. With the help of a librarian, he has compared the prices of Elsevier’s physics journals to other physics journals subscribed to by his university…

Posted at 9:47 PM UTC | Permalink | Followups (10)

May 22, 2009

Charles Wells’ Blog

Posted by John Baez

Charles Wells is perhaps most famous for this book on topoi, monads and the category-theoretic formulation of universal algebra using things like ‘algebraic theories’ and ‘sketches’:

It’s free online! Snag a copy and learn some cool stuff. But I’ll warn you — it’s a fairly demanding tome.

Luckily, Charles Wells now has a blog! And I’d like to draw your attention to two entries: one on sketches, and one on the evil influence of the widespread attitude that ‘the philosophy of math is the philosophy of logic’.

Posted at 5:43 PM UTC | Permalink | Followups (12)

May 19, 2009

Where is the Philosophy of Physics?

Posted by David Corfield

As the subtitle of this blog says, we run ‘A group blog on math, physics and philosophy’. To what extent, though, do we cover all the interfaces of this triad? Well, we do some philosophy of mathematics here, and we certainly do some mathematical physics. But the question I’ve been wondering about recently is whether we should be doing more philosophy of physics.

If we followed the position that physics is the search for more and more adequate mathematical structures to describe the world, perhaps we needn’t take the philosophy of physics to be anything more than a philosophy of mathematics along with an account of how the structures which are most promising for physics are chosen. But this view of physics would be controversial.

Posted at 2:01 PM UTC | Permalink | Followups (45)

TFT at Northwestern

Posted by Urs Schreiber

Quite unfortunately I couldn’t make it to this event that started yesterday:

Topological Field Theories at Northwestern University
Workshop: May 18-22, 2009
Conference: May 25-29, 2009
(website, Titles and abstracts)

An impressive concentration of extended TFT expertise.

But with a little luck nnCafé regulars who are there will provide the regrettable rest of us with reports about the highlights and other lights

In fact, Alex Hoffnung already sent me typed notes that he had taken in talks! That’s really nice of him. I am starting to collect this and other material at

[[Northwestern TFT Conference 2009]]

Posted at 1:57 PM UTC | Permalink | Followups (16)

May 18, 2009

A Prehistory of n-Categorical Physics

Posted by John Baez

I’m valiantly struggling to finish this paper:

Perhaps blogging about it will help…

Posted at 8:58 PM UTC | Permalink | Followups (68)

Higher Structures in Göttingen III

Posted by John Baez

Göttingen was famous as a center of mathematics during the days of Gauss, Riemann, Dirichlet, Klein, Minkowksi, Hilbert, Weyl and Courant. One of the founders of category theory, Saunders Mac Lane, studied there! He wrote:

In 1931, after graduating from Yale and spending a vaguely disappointing year of graduate study at Chicago, I was searching for a really first-class mathematics department which would also include mathematical logic. I found both in Göttingen.

It’s worth reading Mac Lane’s story of how the Nazis eviscerated this noble institution.

But now, thanks to the Courant Research Centre on Higher-Order Structures, Göttingen is gaining fame as a center of research on higher structures (like nn-categories and nn-stacks) and their applications to geometry, topology and physics! They’re having another workshop soon:

Posted at 5:41 PM UTC | Permalink | Followups (2)

Journal Club – Geometric Infinity-Function Theory – Week 4

Posted by Urs Schreiber

In our journal club on [[geometric \infty-function theory]] this week Chris Brav talks about chapter 4 of Integral Transforms:

Tensor products and integral transforms.

This is about tensoring and pull-pushing (,1)(\infty,1)-categories of quasi-coherent sheaves on perfect stacks.

Luckily, Chris has added his nice discussion right into the wiki entry, so that we could already work a bit on things like further links, etc. together. Please see section 4 here.

Discussion on previous weeks can be found here:

week 1: Alex Hoffnung on Introduction

week 2: myself on Preliminaries

week 3: Bruce Bartlett Perfect stacks

Posted at 7:08 AM UTC | Permalink | Followups (6)

May 15, 2009

The Relevance of Predicativity

Posted by David Corfield

If I get around to writing a second book in philosophy of mathematics, one thing I’ll probably need to retract is the ill-advised claim made in the first book that the notion of predicativity is irrelevant to mainstream mathematics.

Here’s a passage which goes directly against such a thought, from Nik Weaver’s Is set theory indispensable?

Posted at 4:53 PM UTC | Permalink | Followups (12)

May 11, 2009

Journal Club – Geometric Infinity-Function Theory – Week 3

Posted by Urs Schreiber

This week in our Journal Club on [[geometric \infty-function theory]] Bruce Bartlett talks about section 3 of “Integral Transforms”: perfect stacks.

So far we had

Week 1: Alex Hoffnung on Introduction

Week 2, myself on Preliminaries

See here for our further schedule. We are still looking for volunteers who’d like to chat about section 5 and 6.

Posted at 11:04 PM UTC | Permalink | Followups (31)

May 9, 2009

Smooth Structures in Ottawa II

Posted by John Baez

guest post by Alex Hoffnung

Hi everyone,

I am going to even further neglect my duties to the journal club and take a moment to report on the Fields Workshop on Smooth Structures in Logic, Category Theory and Physics which took place this past weekend at the University of Ottawa. The organizers put together a great series of talks giving an overview of the past and current trends and applications in smooth structures. I should right away try to put the idea of smooth structures in some context. Further, I should warn you that I may do this with some amount of bias.

Posted at 7:57 PM UTC | Permalink | Followups (50)

May 8, 2009

In Search of Terminal Coalgebras

Posted by David Corfield

Tom Leinster has put up the slides for his joint talk – Terminal coalgebras via modules – with Apostolos Matzaris at PSSL 88.

It’s all about establishing the existence of, and constructing, terminal coalgebras in certain situations. I realise though looking through the slides that I never fully got on top of the flatness idea, and nLab is a little reluctant to help at the moment (except for flat module).

So perhaps someone could help me understand the scope of the result, maybe via an example. Say I take the polynomial endofunctor

Φ(X)=1+X+X 2. \Phi(X) = 1 + X + X^2.

Given that terminal coalgebras can be said to have cardinality ii, in which categories will I find such a thing?

Posted at 10:07 AM UTC | Permalink | Followups (20)

May 7, 2009

Odd Currency Puzzle

Posted by John Baez

Sorry to be posting so much light, frothy stuff lately — but since it’s an odd day, I can’t resist another puzzle.

What’s the oddest currency ever used in America?

Of course this is a subjective question, so I’d be interested to hear your opinion…

Posted at 7:34 PM UTC | Permalink | Followups (49)

May 6, 2009

nLab - More General Discussion

Posted by David Corfield

With the previous thread on nLab reaching 343 comments, it’s probably time for a new one.

Let me begin discussions by asking whether it is settled that distributor be the term preferred over profunctor. I ask since it would be good to have an entry on the 2-category of small categories, profunctors and natural transformations. Should it be DistDist or ProfProf?

Posted at 9:06 AM UTC | Permalink | Followups (94)

May 5, 2009


Posted by David Corfield

If we were to have a page at nLab on things to be categorified should it be titled categorifAcienda, categorifIcienda or something else?

My suggestions are based on the gerundives formed from verbs such as agenda and Miranda. Concerning verbs more closely resembling ‘categorify’ we have

  • Satisfacio (satisfy) - satisfaciendus
  • Efficio (bring to pass) - efficiendus

Unfortunately, categorify is a hybrid word, with Greek stem and Latin suffix. I suppose categorize was out of the question.

Posted at 4:26 PM UTC | Permalink | Followups (14)

Journal Club – Geometric Infinity-Function Theory – Week 2

Posted by Urs Schreiber

This week it is my turn to talk in our Journal Club on Geometric \infty-Function Theory about section 2, “Preliminaries” of “Integral Transforms”.

Previous week we had Alex Hoffnung on section 1, Introduction.

For the further schedule see the Journal Club’s nnLab page. We are still looking for volunteers for sections 5 and 6.

Posted at 3:18 PM UTC | Permalink | Followups (7)

May 4, 2009

The Foibles of Science Publishing

Posted by John Baez

The latest news about Elsevier journals and Scientific American.

Posted at 1:17 AM UTC | Permalink | Followups (15)