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November 6, 2009
Fraïssé Limits
Posted by David Corfield
The Rado graph, or random graph, seems to be an extraordinary entity. Take countably many nodes, then for each pair of nodes flip a coin and if it shows heads, draw an edge between them. Almost surely you will have generated the Rado graph, .
Any finite graph (and indeed any countable graph) is contained in , not just in the sense of being embeddedable, but in the sense of being an induced subgraph, that is, it is the full subgraph on a subset of nodes. Along with this universality, is also homogenous in the sense that any isomorphism between finite induced subgraphs extends to an automorphism of all of .
is very robust, you can delete finitely many vertices, add or remove finitely many edges, or interchange edges and non-edges, and you still end up with a graph isomorphic to . Furthermore, the odd thing about the construction of this graph is that I didn’t have to tell you the probability of the coin showing heads. So long as is in and the tosses are independent, the Rado graph almost surely emerges.
Not only this, there are many ways to generate it without using random devices. Rado himself took the nodes to be the natural numbers, and an edge between and whenever either the th bit of the binary representation of is nonzero, or the th bit of the binary representation of is nonzero. Yet another method has us take as nodes the prime numbers equal to 1 mod 4. Then we join and if they are quadratic residues of each other.
November 4, 2009
Who Discovered the Icosahedron?
Posted by John Baez
This weekend we’re having a meeting of the American Mathematical Society here at Riverside. Julie Bergner and I are running a special session on Homotopy Theory and Higher Algebraic Structures, and there will also be two special sessions on knot theory, one run by Alissa Crans and Sam Nelson. It should be fun! And it’s starting already: Khovanov will be giving a colloquium talk today.
Notions of Space
Posted by Urs Schreiber
Today is my turn in our Seminar on A Survey of Elliptic Cohomology.
I attempted to write a survey of some central ideas in Jacob Lurie’s Structured Spaces.
You can find it here: Notions of Space.
An Adventure in Analysis
Posted by Tom Leinster
One reason I went into category theory is that I wanted a subject that would take me to different parts of the mathematical world. These last few weeks I’ve been getting my wish in spades. My hard drive contains 53 analysis papers and books that three weeks ago it didn’t. My desk is piled with library books. My floor is a mess of handwritten notes covered in integral signs.
What prompted this adventure in analysis was a problem about magnitude of metric spaces. Thanks to the contributions of a whole crowd of people, the problem has now been solved. The problem-solving process, here and at Math Overflow, took all sorts of twists and turns, many of them unnecessary. But I think I can now present a fairly straight path to a solution. To thank those who contributed — and to entertain those who were half-interested but didn’t have the energy to keep up — I give you an overview of the problem and its solution.
November 3, 2009
The Arrow of Time in Cat
Posted by David Corfield
Back here we were talking about the symmetry-breaking that takes place in mathematics by the choice of working in , which John attributed to nothing less than the ‘arrow of time’.
Why do many-to-one but not one-to-many relations get singled out for single treatment and dubbed ‘functions’? Because functions are supposed to be ‘deterministic’: the cause must determine the effect. We don’t care if the effect fails to determine the cause.
Now what is there to be said about the 2-category Cat and its three duals: , and ?
November 2, 2009
Interview with Manin
Posted by John Baez
Try this interview with Yuri Manin:
- Mikhail Gelfand, We do not choose our profession, it chooses us: interview with Yuri Manin, translated by Mark Saul, Notices of the AMS 56 (November 2009), 1268–1274.
Manin is most famous for his work on number theory and algebraic geometry. But he’s also famous for his work on noncommutative geometry, the self-dual solutions of the Yang–Mills equations, and much more. He was wide interests and erudition to spare, so it’s interesting to read his view on the history of mathematics, and its future. Check out his remarks on Feynman integrals as an ‘Eiffel tower hanging in the air with no foundation’, the role of Edward Witten in modern mathematics, and the trend towards the ‘homotopification’ of mathematics.
October 30, 2009
Generalized Operads in Classical Algebraic Topology
Posted by Mike Shulman
There are lots of ways of categorically presenting “algebraic theories;” three of the most well-known are operads, Lawvere theories, and monads. In fact, operads and monads lie near opposite ends of a continuum of such notions, ranging from “less expressive and more controlled” (operads) to “more expressive and less controlled” (monads). One uniform framework for such “notions of theory” and their corresponding “functorial semantics” is the theory of generalized operads and multicategories.
My goal in this post is to explain how a couple of fairly obscure-seeming kinds of generalized operad are actually implicit in some very classical algebraic topology. In particular, they provide a way to “make good categorical sense” out of two constructions on topological operads that have always confused me: the “category of operators” associated to an operad, and the two different monads on based and unbased spaces associated to a “reduced” operad.
October 29, 2009
This Week’s Finds in Mathematical Physics (Week 282)
Posted by John Baez
In week282 of This Week’s Finds, visit Mercury:
Learn how this planet’s powerful magnetic field interacts with the solar wind to produce flux transfer events and plasmoids. Then read about the web of connections between associative, commutative, Lie and Poisson algebras, and how this relates to quantization. In the process, you’ll meet linear operads, their generating functions, and Stirling numbers of the first kind!
October 26, 2009
Math: Folk Wisdom in an Electronic Age
Posted by John Baez
With the continued development of the nLab and the Polymath Projects, and the rise of Math Overflow as a competitor to sci.math.research, mathematicians are busy talking about new ways to take advantage of technology. The arXiv is great for papers. Blogs are great for conversations. But how to more effectively gather, store, and make accessible the folk wisdom that traditionally spread through informal person-to-person conversations?
There surely won’t be just one answer to this question. And surely the answers won’t be found just by discussion. We’ll need to grope our way there by trying many different things and seeing what works.
But humans being human, it’s irresistible and probably necessary to talk about this question. Other disciplines must be having similar discussions — does anyone know where? It would be good to see them.
Here on the -Café, conversations keep drifting towards this subject…
October 24, 2009
Mathematical Foundations of Quantum Field and Perturbative String Theory
Posted by Urs Schreiber
Following a suggestion by some publishing company, there is the idea of creating a book that collects contributions from various authors on the topic Mathematical Foundations of Quantum Field and Perturbative String Theory .
We have an idea for a proposed “Call for Papers”. But we would like to get some comments on this, from people who have experience with such issues.
October 22, 2009
Cheaper Online Textbooks?
Posted by John Baez
College textbooks are really expensive these days. In California, a law was passed to tackle this problem. But it seems to lack teeth.
One way to tackle this problem is to develop free online textbooks. I think a wiki-based approach could be good. People are trying it. Will it ever catch on?
It might also make sense for the NSF, or other funding agencies, to pay for scholars to write free online textbooks — or improve existing ones.
Now this has finally happened.
Cobordism and Topological Field Theories Week 2
Posted by Alexander Hoffnung
Welcome to week 2 of the cobordism and TFT seminar at UCR. This is the first of a series of lectures given by Julie Bergner (with occasional stand-ins by others). The previous lecture was an introductory lecture given by John Baez found here. This week’s lecture introduces cobordisms and oriented manifolds.
October 21, 2009
Aesthetics of Commutative Diagrams
Posted by Mike Shulman
I’ve recently run into the question of how best to lay out a fairly large commutative diagram. Some diagrams have a “natural” shape such as a cube or a simplex, but as far as I can tell that is not the case for the diagrams in question. They aren’t complicated, mostly just a bunch of naturality squares stuck together. But different people seem to have different aesthetic viewpoints on what makes the layout of a diagram “look good.” So I thought I’d share my data so far, and see whether anyone here has additional insights.
October 20, 2009
This Week’s Finds in Mathematical Physics (Week 281)
Posted by John Baez
In week281 of This Week’s Finds, learn about the newly discovered ring of Saturn — the Phoebe ring — and how it explains the mystery of Iapetus.
See Egan’s new applet that produces ever-expanding tilings with 10-fold quasisymmetry. Delve deeper into the history of these tilings, which date back to the Timurid dynasty. Go back all the way to the Topkapi Scroll… then go modern and check out tiling patterns in spherical and hyperbolic geometry. Finally, hear about strings in 4d BF theory, and spin foam models based on the representation theory of 2-groups.
October 19, 2009
Syntax, Semantics, and Structuralism, I
Posted by Mike Shulman
Sparked by Arnold Neumaier’s work on “a computer-aided system for real mathematics,” we had a very stimulating discussion, mostly about various different foundations for mathematics. But that thread started getting rather long, and John pointed out that people without some experience in formal logic might be feeling a bit lost. In this post I’ll pontificate a bit about logic, type theory, and foundations, which have also come up recently in another thread. Hopefully this too-brief summary of background will make the discussion in the next post about the meaning of structuralism more readable. (Some of this post is intended as a response to comments on the previous discussions, but I probably won’t carefully seek them out and make links.)
Halmos on Writing Mathematics
Posted by Simon Willerton
Over in a discussion at Math Overflow I was reminded about Halmos’ great article on writing mathematics, which I highly recommend to all graduate students (or anyone else, for that matter).
- P. R. Halmos, How to write mathematics, L’Enseignement Mathématique, Vol.16 (1970) [31 pages, 3.3MB]. Individual pages are available for those with slow connections.
Math Overflow
Posted by David Corfield
The math-blogosphere is abuzz with interest in the new Math Overflow, a mathematics questions and answers site. Already we at the Café have been helped with the answer to a query on the Fourier transform of a certain kernel, and there are some juicy questions for us to answer there too, including
- What is the size of the category of finite dimensional vector spaces?, where size is in the sense of the Leinster-Euler characteristic.
- “Wick rotation” of tropical geometry, very much a matrix mechanics kind of question.
You can read a discussion on Math Overflow, and a debate concerning its advantages relative to Lab.