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 $\mathbb{Z}/12$) and inversion (music jargon for negation in $\mathbb{Z}/12$). 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:
- Alissa S. Crans, Thomas M. Fiore, Ramon Satyendra, Musical actions of dihedral groups.
Yes, that’s my student Alissa Crans, of Lie 2-algebra fame!
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:
- 2nd School and Workshop on Quantum Gravity and Quantum Geometry, Corfu Summer Institute, September 13–20, organized by John Barrett, Harald Grosse, Larisa Jonke and George Zoupanos.
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.
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 $\epsilon-\delta$ 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.
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 $n$-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:
$n$Café-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.
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…
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’:
- Michael Barr and Charles Wells, Toposes, Triples and Theories.
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’.
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.
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 $n$Café 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
May 18, 2009
A Prehistory of n-Categorical Physics
Posted by John Baez
I’m valiantly struggling to finish this paper:
- John Baez and Aaron Lauda, A prehistory of n-categorical physics (draft version).
Perhaps blogging about it will help…
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 $n$-categories and $n$-stacks) and their applications to geometry, topology and physics! They’re having another workshop soon:
- Higher Structures in Topology and Geometry III, June 4th-5th, 2009, Courant Research Centre, Göttingen, organized by Giorgio Trentinaglia, Christoph Wockel, and Chenchang Zhu.
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 $(\infty,1)$-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
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?
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.
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.
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
$\Phi(X) = 1 + X + X^2.$
Given that terminal coalgebras can be said to have cardinality $i$, in which categories will I find such a thing?
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…
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 $Dist$ or $Prof$?
May 5, 2009
Gerundives
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.
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 $n$Lab page. We are still looking for volunteers for sections 5 and 6.
May 4, 2009
The Foibles of Science Publishing
Posted by John Baez
The latest news about Elsevier journals and Scientific American.