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November 19, 2009
Mathematical Emotion
Posted by David Corfield
Continuing the season of Collingwood on mathematics, here is an extract from The Principles of Art (1938):
A symbol is language and yet not language. A mathematical or logical or any other kind of symbol is invented to serve a purpose purely scientific; it is supposed to have no emotional expressiveness whatever. But when once a particular symbolism has been taken into use and mastered, it reacquires the emotional expressiveness of language proper. Every mathematician knows this. At the same time, the emotions which mathematicians find expressed in their symbols are not emotions in general, they are the peculiar emotions belonging to mathematical thinking. (p. 268)
[‘Symbol’ is used here to mean “something arrived at by agreement and accepted by the parties to the agreement as valid for a certain purpose” (p. 225).]
Anyone who doubts such ‘emotions belonging to mathematical thinking’ exist need only read an edition or two of This Week’s Finds. They are dripping with emotion, as the modern phrase has it.
November 18, 2009
Cobordism and Topological Field Theories Week 3
Posted by Alexander Hoffnung
Welcome to week 3 of the cobordism and TFT seminar at UCR. The previous lecture by Julie Bergner can be found here. There she talked about categories of cobordisms.
In the present lecture, Julie introduces symmetric monoidal categories and defines a notion of topological field theory.
To keep everyone apprised of the state of this seminar, I should say that in “real time” we are about to hit week 9. In “blog time” we are still not caught up, but I am sure we will get there soon. The nice thing is that I can give you a preview of the weeks to come below the fold.
November 16, 2009
Combinatorial-Game Categories
Posted by Mike Shulman
I just got back from the category theory Novemberfest at CMU, which was plenty of fun. One especially nice talk was by Geoff Cruttwell, who talked about axiomatizing the structure that exists on Joyal’s category of combinatorial (Conway) games. It turns out that the category of (finite) games is initial among such “combinatorial-game categories,” which implies a clever way to construct invariants of games. And naturally, the question of a terminal object is related to ill-founded or infinite games.
A Rose by Any Other Name
Posted by David Corfield
Pity Ola Bratteli who, when checking out how cited he is, has to take into account the “common misspellings Bratelli, Brattelli and the less common Brateli and Blatteli” of his last name. For someone who has given his name to the rather important Bratteli diagram, this is unfortunate.
Bratteli diagrams are ways of depicting approximately finite -algebras.
November 13, 2009
The 1000th Post on the n-Category Cafe
Posted by John Baez
Whoo-hoo! It’s our 1000th post! 
We’ve got four new hosts at the café, and we’d like you to meet them…
November 11, 2009
This Week’s Finds in Mathematical Physics (Week 283)
Posted by John Baez
In week283 of This Week’s Finds, see galaxies in visible, infrared, and ultraviolet light:
Read the mystery of who discovered the icosahedron: Theaetetus or the ancient Scots. Read about Lieven le Bruyn’s detective work which helped solve this case! And learn an amazing fact, proved in the last book of Euclid’s Elements, which relates regular polygons with 5, 6, and 10 sides.
November 10, 2009
Courant Algebroids From Categorified Symplectic Geometry
Posted by John Baez
guest post by Chris Rogers
‘Higher symplectic geometry’ is a topic that has come up recently in posts here in the -Café and in some ideas presented by Urs and others in the Lab. This is a subject I’ve been thinking a lot about lately, and I just finished a (rough!) draft of a paper that attempts to establish some connections between two different approaches to generalizing, extending, and/or categorifying symplectic geometry. So of course I happily accepted John’s invitation to write a guest post about some of this work. I hope by doing so I can contribute a little something to the ongoing discussion and learn more about the different ideas people have on the subject.
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!