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April 26, 2007
This Week’s Finds in Mathematical Physics (Week 250)
Posted by John Baez
In week250 of This Week’s Finds, start with a little puzzle about a game of flipping coins. Then learn about Popescu-Rohrlich game, which involves flipping coins and quantum entanglement!
Then, continue reading the Tale of Groupoidification — in which we start by recalling the history of special relativity, and use an example from relativity to ponder "atomic invariant relations". We’ll see these are just what mathematicians normally call "double cosets" — but we’ll see they’re also spans of groupoids equipped with extra stuff.
April 25, 2007
n-Curvature
Posted by Urs Schreiber
We have learned that parallel -transport in an -bundle with connection over a base space is an -functor from the -path -groupoid of to some -category of fibers.
With every notion of connection we expect to obtain notions of
1) curvature;
2) Bianchi identity;
3) parallel sections;
4) covariant derivative.
Here we describe the functorial incarnation of these concepts. We find
1) To every transport -functor is canonically associated a curvature -functor The functor is flat precisely if is trivial on all -morphisms.
2)
The curvature -functor, regarded as an -transport itself,
is always flat.
3) Parallel sections of the -bundle with connection associated with are equivalent to morphisms from the trivial -transport into :
4) General sections together with their covariant derivative are equivalent to morphisms from the trivial curvature -transport into the curvature -transport
Learning from Our Ancestors
Posted by David Corfield
Back in this post I argued against Bernard Williams’ view of science:
The pursuit of science does not give any great part to its own history, and that it is a significant feature of its practice… Of course, scientific concepts have a history: but on the standard view, though the history of physics may be interesting, it has no effect on the understanding of physics itself. It is merely part of the history of discovery.
Taking mathematics as a science, I took Robert Langlands to be on my side against Williams:
Despite strictures about the flaws of Whig history, the principal purpose for which a mathematician pursues the history of his subject is inevitably to acquire a fresh perception of the basic themes, as direct and immediate as possible, freed of the overlay of succeeding elaborations, of the original insights as well as an understanding of the source of the original difficulties. His notion of basic will certainly reflect his own, and therefore contemporary, concerns.
Now, from the interview I mentioned in the last post it appears that Connes has read Galois’ papers with profit. Meanwhile, John has been encouraging us to better ourselves by reading Felix Klein’s Erlanger program. Something I’d like to hear about are instances where people feel they have gained something by reading works from the nineteenth century or earlier, or histories on those works, especially instances where there has been some element of surprise at how not all that was good about a certain way of thinking has survived to the present day.
April 23, 2007
Who’s on the Right Track?
Posted by David Corfield
Our post at the Café.
In this interview, Alain Connes mentions work he has carried out with Matilde Marcolli on a book which treats physics for three hundred pages, and number theory for the second three hundred. Regarding an analogy they have pursued, concerning spontaneous symmetry breaking in the two fields, he writes
We know that the universe has cooled down, well, it suggests that when the universe was hotter than, say, at the Planck’s temperature, there was no geometry at all, and that only after the phase transition was there a spontaneous symmetry breaking which selected a particular geometry and therefore the particular universe in which we are. (p. 8)
This also suggests that
…the people who are trying to develop quantum gravity in a fixed space are on the wrong track.
If this latter claim were true, which quantum gravity theorists would not be ruled out?
April 22, 2007
Report-Back on BMC
Posted by Urs Schreiber
– guest post by Bruce Bartlett –
I was born in a large Welsh industrial town at the beginning of the Great War: an ugly, lovely town (or so it was, and is, to me), crawling, sprawling, slummed,unplanned, jerry-villa’d, and smug-suburbed by the side of a long and splendid-curving shore…
Thus described Dylan Thomas his childhood home of Swansea, Wales - the venue of the British Mathematics Colloquium this year :
Inspired by John’s blurb about the higher categories workshop at Fields earlier this year, I thought I’d send Urs a report-back of the (admittedly less glamorous) “BMC” , and mention a few things possibly of interest to -café patrons.
April 20, 2007
Cohomology and Computation (Week 21)
Posted by John Baez
This time in our course on Cohomology and Quantization we explained why mathematicians like to turn algebraic gadgets and topological spaces into simplicial sets — and how this actually works, in the case of topological spaces:
- Week 21 (Apr. 19) - Simplicial sets and cohomology. Two sources of simplicial sets: topology and algebra. The topologist’s category of simplices, . How a topological space gives a simplicial set called its ‘singular simplicial set’ . How this gives a functor .
Quantization and Cohomology (Week 21)
Posted by John Baez
This week in our course on Quantization and Cohomology we used Chen’s ‘smooth space’ technology to implement a new approach to Lagrangian mechanics, based on a smooth category equipped with an ‘action’ functor:
-
Week 21 (Apr. 17) - Any quotient of a smooth space becomes a
smooth space. The category of smooth spaces has pushouts.
The category of smooth spaces is cartesian closed. The path groupoid of a smooth space . The path groupoid is a smooth category. Smooth functors.
Theorem: a smooth functor is the same as a 1-form
on X.
Supplementary reading:
-
John Baez and Urs Schreiber,
Higher gauge theory II: 2-connections, draft version.
Section 6.1: proof that for any Lie group , smooth functors are the same as -valued 1-forms on
-
John Baez and Urs Schreiber,
Higher gauge theory II: 2-connections, draft version.
Cohomology and Computation (Week 20)
Posted by John Baez
This week in our seminar on Cohomology and Computation, we began to see what’s so great about simplices:
- Week 20 (Apr. 12) - Cohomology and the category of simplices. Simplices as special categories: finite totally ordered sets, which are isomorphic to "ordinals". The algebraist’s category of simplices, . Face and degeneracy maps. The functor from to Top sending the ordinal to the standard -simplex. Simplicial sets. Preview of the cohomology of spaces.
April 19, 2007
Some Conferences
Posted by Urs Schreiber
Busy with reducing the write-up lag. No time to blog.
I am trying not to try to go to too many events, but here are two more I might not be able to resist trying to attend (which may still fail even if I try).
This beautiful one
Principal Bundles, Gerbes and Stacks
17-23 June, 2007 - Bad Honnef, Germany
is right around the corner for me. At least Konrad Waldorf will go and talk about our stuff.
Not sure if John has mentioned this one anywhere yet except on his lectures site, but even if so it’s well worth mentioning it twice:
August 5 - 10 2007, Oslo, Norway
April 17, 2007
The Field With One Element
Posted by David Corfield
For some grand theory building and an answer to the question ‘What is the field with one element?’, see Nikolai Durov’s New Approach to Arakelov Geometry.
There’s something extremely intriguing about a mathematical entity which has known effects, but which has not been defined. It generates a sense of independent reality. As I mentioned in the Tuesday 8 November entry on my old blog, a vector space over the ‘field with one element’ is a pointed set. Thinking in such terms makes sense of many combinatorial facts, see TWF 187.
Here’s Durov’s answer:
The ‘field with one element’ is the free algebraic monad generated by one constant (p. 26) or the universal generalized ring with zero (p. 33).
This will need some unpacking.
The Two Cultures of Mathematics
Posted by David Corfield
Part of what intrigues me about reading Terence Tao’s blog is that he displays there a different aesthetic to the one largely admired here. The best effort to capture this difference is, I believe, Timothy Gowers’ essay The Two Cultures of Mathematics, in which the distinction is made between ‘theory-builders’ and ‘problem-solvers’. I think we have to be very careful with these labels, as Gowers himself is.
…when I say that mathematicians can be classified into theory-builders and problem-solvers, I am talking about their priorities, rather than making the ridiculous claim that they are exclusively devoted to only one sort of mathematical activity. (p. 2)
To avoid misunderstanding, then, perhaps it is best to give straight away paradigmatic examples of work from each culture.
Theory-builders: Grothendieck’s algebraic geometry, Langlands Program, mirror symmetry, elliptic cohomology.
Problem-solvers: Combinatorial graph theory, e.g, Ramsey’s theorem, Szemerédi’s theorem, arithmetic progressions among the primes.
Gowers mentions Sir Michael Atiyah as a prime example of a theory builder, and recommends his informal essays, the ‘General papers’ of Volume 1 of his Collected Works. Indeed, they convey an aesthetic which I came to admire enormously as a PhD student in philosophy. On the other hand, Paul Erdös was a consummate problem-solver. What then of the corresponding aesthetic?
One of the attractions of problem-solving subjects, which Gowers collects under the loose mantle ‘combinatorics’, is the easy accessibility of the problems.
One of the great satisfactions of mathematics is that, by standing on giants’ shoulders, as the saying goes, we can reach heights undreamt of by earlier generations. However, most papers in combinatorics are self-contained, or demand at most a small amount of background knowledge on the part of the reader. Contrast that with a theorem in algebraic number theory, which might take years to understand if one begins with the knowledge of a typical undergraduate syllabus. (p. 12)
For someone who had recently won a Fields’ Medal, it would seem strange to feel the need to defend one’s interests, but after describing a problem involving the Ramsey numbers, Gowers writes:
I consider this to be one of the major problems in combinatorics and have devoted many months of my life unsuccessfully trying to solve it. And yet I feel almost embarrassed to write this, conscious as I am that many mathematicians would regard the question as more of a puzzle than a serious mathematical problem. (p. 11)
April 14, 2007
Incandescence
Posted by John Baez
My favorite science fiction writer is coming out with a new novel!
- Greg Egan, Incandescence, Orion/Gollancz, United Kingdom, to be published May 1st, 2008.
That’s a long time to wait. Luckily, we can already read a story set in the same universe:
- Greg Egan, Riding the Crocodile.
Topos Theory in the New Scientist
Posted by John Baez
Our favorite science magazine has decided to take on Chris Isham and Andreas Döring’s work on topos theory and physics:
- Robert Matthews, Impossible things for breakfast, at the Logic Café, New Scientist, April 14, 2007.
At the n-Category Café we serve only possible things for breakfast. But, many things are possible…
April 12, 2007
Schur Functors
Posted by John Baez
As part of the Tale of Groupoidification, I’ll need to talk about Schur functors. As usually defined, these are simply functors
where is the category of finite-dimensional complex vector spaces.
An example of a Schur functor is ‘take the antisymmetrized 3rd tensor power’. In the category of Schur functors, , every object can be expressed as a direct sum of certain ‘irreducible’ objects, which correspond to Young diagrams. The example I just mentioned corresponds to this Young diagram: