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September 28, 2007
New Blog
Posted by David Corfield
I was wondering when Lamarck would be proved right and John’s (academic) children would show his acquired characteristic of wanting to broadcast over the Web. Now Jeff Morton has begun to blog at Theoretical Atlas.
Progic III
Posted by David Corfield
So we’ve seen, in this thread that, even when we work with nice basic finite sets, the probability monad doesn’t get along too well with logical structure, there being none of that pleasant adjointness between categories of predicates over sets.
However, the probability monad, , does come along with other structure of its own. In particular is a Riemannian manifold, with the Fisher information metric, which is crucially invariant under reparameterization. Recall that this space is composed of maps , i.e., probability distributions over . The Fisher information metric (see, e.g., section 4 of Guy Lebanon’s thesis) takes on a simple form for finite , (p. 21).
September 27, 2007
Detecting Higher Order Necklaces
Posted by Urs Schreiber
You know a conference is a good one when there’s no time to report from it. Categories in Geometries and Physics is of this kind.
But before I forget it, I want to record a couple of things worth mentioning and remembering.
I had very long and interesting discussions with Nils Baas. Let me share the following question he poses:
Suppose you have a lot of silver rings. You join them to form a necklace. Then, given more such silver rings, you form more such necklaces. Then, from all these necklaces, you build, by similarly joining them, yet another necklace: a necklace of necklaces. A second order necklace. And so on.
Now suppose on a table sits a huge pile of silver rings. How do you decide if they form an order necklace?
Shouldn’t there be something like a higher order knot/link invariant which detects knots of knots and links of links?
Further topics today: a comparison with Enrico Vitale’s work on weak categorical cokernels and mapping cones, as well as a speculation on weak Lie -algebras triggered by discussion with Pavol Ševera.
September 26, 2007
Rotations in the 7th Dimension
Posted by John Baez
Squark emailed me an interesting question about spin groups, which I take the liberty of reproducing here:
Dear John,
I would appreciate to hear your thoughts on the following matter.
In dimensions we have the exceptional isomorphisms. In complex form:
In dimension 8 the triality isomorphisms play the same role, in a way:
What about dimension 7? Does anything of the sort happen there?
Best regards,
Squark
September 25, 2007
An Invitation to Higher Dimensional Mathematics and Physics
Posted by Urs Schreiber
I mentioned that I was aked to give a public lecture in the context of Categories in Geometry and Physics next Friday, to high school kids and journalists. I am imagining the lecture might proceed along the following lines.
Since this is still in progress, I would enjoy receiving comments and suggestions. The following already incorporates advice by John Baez and Tom Leinster.
An Invitation to Higher Dimensional Mathematics and Physics
In which sense is summing two numbers a 2-dimensional process? Everybody who knows that is the same as will be lead in this talk to a simple but profound result in a branch of mathematics known as -category theory. This simple insight in higher dimensional mathematics alone will be sufficient to make understandable some fancy ideas in modern theoretical high energy physics.
Progic II
Posted by David Corfield
My research time at present largely consists of the time it takes me to travel to Canterbury and back. Here’s what I came up with on the subject of progic on yesterday’s trip down.
A probabilistic predicate, , I said back here, is a map from to . But another way to look at it is as a conditional probability distribution on the set given . This is a map in the Kleisli category of the probability monad. In fact, our probabilistic predicate is a map . Now we can look at the composition of a distribution over and this .
Let’s take a set of dogs, is a probability distibution over , perhaps recording my degree of belief in which dog I saw just now as it dashed past. Then if is the predicate recording my belief as to whether is a poodle, then is my degree of belief that I just glimpsed a poodle.
This is one of Mumford’s random constants, i.e., probabilistic predicates over {*}, or maps in the Kleisli category .
September 24, 2007
The Catsters Strike Again: “String Diagrams”
Posted by John Baez
- The Catsters, 5 lectures on String diagrams.
Obstructions, Tangent Categories and Lie N-tegration
Posted by Urs Schreiber
Last week I visited Yale, where I talked with Hisham Sati about supergravity, Chern-Simons 3-bundles and other higher structures in String theory, with Mikhail Kapranov about parallel surface transport, with Todd Trimble about Hecke algebras, groupoidification and the sociology of quantum gravity, and with Anton Zeitlin about BV formulation of Yang-Mills theory. With a short stop at home I went from there straight to Croatia, where I am currently attending Categories in Geometry and Physics.
The first day here was quite remarkable, with an astonishing amount of Croatian media attention for us followed by a couple of rather impressive talks. After Pavol Ševera’s talk I had the chance to chat with with him about differentiation and integration of Lie -algebras, much to my delight.
Last night at about 3 am I learned that I am supposed to give a public talk to high school kids and journalists next Friday. And sure enough, later towards the pm part of the day I found myself being interviewed by some TV team on how I plan to convey the ideas of Categories in Geometry and Physics to the broad public.
Trying to think of something which is both interesting and relevant, as well as easy to understand and expressible in terms of lots of pictures, I thought I’d give a talk revolving around the Eckman-Hilton argument, illustrated by two party-balloons joined at their tip and labeled “A” and “B”.
(The very moment that I am posting this, a journalist sends me the following two questions:
Q1. Your field of interest in mathematics is the relation between categories and quantum physics. During the Split categories conference you will be giving a popular lecture on this topic to the high school students, so can you briefly summarize that lecture for our readers?
Q2. You are also one of the admins at -category cafe blog. As it has become an interesting and influential internet based communication and discussion tool for resarchers in category theory, can you tell me a bit more about it - how did the idea start, how it developed, and what is its influence on the category maths community professional work?
)
Hence there’d be many fun things to write about here. And I should. But right now I will instead talk more about two of the issues I had mentioned recently in Obstructions for -bundle lifts and related.
The Virtues of American Scientist
Posted by John Baez
A lot of us old-timers feel sad about the decline of serious popular science magazines like Scientific American and New Scientist. Perhaps with the rise of science blogging we don’t really need such magazines anymore. But I’m not so sure…
Lately I’ve been browsing through this magazine at my local bookstore:
It makes me feel the way I used to feel when reading Scientific American. They’ve got articles on all sorts of topics, and tons of book reviews in each issue. They’re well-written, clear, and — best of all, when compared to some of the competition — moderately demanding, not dumbed-down or drawn like a moth to the candle flame of lowest-common-denominator sensationalism.
Why are they so good? Maybe it’s because they’re run by a scientific society, Sigma Xi. The way multimedia conglomerates demand their newspapers, magazines and journals keep boosting their profits may eventually kill certain writerly crafts… but magazines put out by professional organizations may do better. Let’s hope so! A lot of the best things in life are created just for the love of it.
September 20, 2007
The Catsters’ Latest Hit: “Adjunctions”
Posted by John Baez
Who knew category theory could be so popular? The Catster’s videos on Monads have taken Youtube by storm! As of this moment, the first episode has garnered 3758 viewings, prompting one disgruntled teenager to suggest that the Catsters are cheating in the battle for high ratings:
> Its really funny that your most of
> your subscribers have no videos, are
> subscribed to only you, and have only your
> videos as favorites. Do I sense a youtube
> cheater?
Of course, we Catsters fans know it ain’t so — they’re just satisfying a hitherto unnoticed craving for math lectures on YouTube!
And the Catsters are on a roll. Now they’ve tackled adjoint functors, also known as ‘adjunctions’:
- The Catsters, 7 lectures on Adjunctions.
Their edgy, low-budget production makes for gripping cinema. This is the real stuff.
September 19, 2007
Deep Beauty: Understanding the Quantum World
Posted by John Baez
There were many amazing mathematicians in the 20th century — people with world-transforming powers, like Gödel, Mac Lane and Grothendieck. But surely, no matter how short your list of greats, John von Neumann would have to be on it. From topics so abstract as the foundations of set theory and quantum mechanics, to topics so practical as game theory, the Manhattan project and the first computers, he seemed to be everywhere… right at the cutting edge.
Soon there will be a symposium honoring the 75th anniversary of von Neumann’s book The Mathematical Foundations of Quantum Mechanics:
- Deep Beauty: Mathematical Innovation and the Search for an Underlying Intelligibility of the Quantum World, Princeton University, October 3-4, 2007. Organized by Hans Halvorson.
Hans Halvorson is a philosopher at Princeton. I really like his idea that new mathematics may be needed to make quantum mechanics more intelligible, and that philosophers should get involved. I think it’s true.
September 18, 2007
Progic
Posted by David Corfield
My colleague here in Canterbury Jon Williamson is part of an international research group, progicnet, whose aim is to find a good integration of probability theory and first-order logic. For one reason or another, some technical projects get counted as philosophy, while others don’t. Where I’m sure I’d have a hard time getting funding for 2-geometry, progicnet’s goals are philosophically pukka.
Now, as Jon says,
We see then that there are a plethora of combinations of probability and logic, and that these approaches are being investigated in some detail.
But is there a ‘natural’ way of doing it? In the quest to have philosophy take notice of category theory, all we need do is use it to blend probability theory and logic powerfully. Does anyone know of any progress along these lines? Perhaps involving the probability monad?
September 16, 2007
The Catsters on YouTube: “Monads”
Posted by John Baez
You can see all sorts of silly things on YouTube these days — from guys flying in wingsuits to math professors making mistakes converting miles to kilometers. But what about serious stuff? Can you watch math lectures on YouTube?
Can you learn category theory?
Can you, say, learn about monads?
Yes! Now you can!
Just try these videos from the new sensation out of Sheffield — the group that has all the British press raving — the Catsters!
September 14, 2007
Groupoidfest 07
Posted by Urs Schreiber
I was asked to forward the announcement of a conference called
Groupoidfest 07
to the -Café. No webpage or similar official site for this seems to exist at this moment. What does exist are two email announcements circulated by Paul S. Muhly. So for the time being I simply reproduce these:
From: Paul S. Muhly
Subject: Groupoidfest 07, First Announcement
Date:September 12, 2007
Dear Friends,
I am writing to announce that Groupoidfest 07 will be held this Fall in Iowa City, at the University of Iowa. The dates are November 3 and 4. Mos