Renormalization and Computation 3
This is the third in a series of posts on Yuri Manin’s recent pair of papers applying Hopf algebra renormalization to the Halting problem. Last time I talked about the way people usually do renormalization; this time I’ll talk about the recent work by Kreimer, Connes, and others in exposing the underlying Hopf algebra in this process.
A Hopf algebra is
- An
-module for a commutative rig
which means you can add vectors and multiply them by a scalar.
- An algebra, which means you can take two vectors and multiply them. This operation is associative; there’s also a unit vector that satisfies left- and right-unit laws.
- A bialgebra, which means there’s also a coassociative comultiplication and counit, and the structures all work together. When the tensor product is the cartesian product, the comultiplication duplicates the vector and the counit is the constant map to 1 in the base field. Even when the tensor product isn’t the cartesian product, it can still be useful to think of it that way.
- A bialgebra with an involution, called the antipode.
A group is very like a Hopf algebra; in fact, a group object in the category of vector spaces and linear maps is a cocommutative Hopf algebra. You can multiply group elements and there’s a multiplicative unit; you can duplicate and delete them in equations; and you can invert them.
It turns out that Feynman diagrams form a Hopf algebra if you poke yourself in one eye and squint. First, a cut 
- given an oriented wheel in the graph, its vertices either all belong to the upper set or all belong to the lower set, and
- any edge connecting a vertex
in the upper set to a vertex
in the lower set must be directed from
Now, given a set of Feynman diagrams, consider all formal linear combinations of graph cuts. This is a vector space because you can add these things pointwise and multiply them by a scalar. We can make it into a bialgebra by defining multiplication to be a linear map
with unit
and comultiplication to be a linear map
where 
It’s graded: just count the number of vertices. And we can turn it into a Hopf algebra by defining the antipode 
Each algebra homomorphism (not necessarily preserving the Hopf algebra structure) from 
Next: given a complex group 
- a holomorphic function
on the standard disk inside the circle
- a holomorphic function
on the complement of this disk in the projective complex plane
- such that on the unit circle the original loop is reproduced as
where the product and the inverse on the right are taken in the group
Notice that
is a well defined element of
Take 
where 
As I understand this, 
Transplants without rejection
Cured diabetes in mice via pancreas transplant, but probably works on every other organ, too.
The Guinea Pigs’ Club
Stories of experimental facial reconstruction on burn victims in WWII. Also see “Billy Bishop’s Flying School“.
Colorblindness cured
Following up on my previous comments here, scientists have cured color blindness in monkeys:
Neitz’s team injected their monkeys’ eyes with viruses carrying a gene that makes L-opsin, one of three proteins released when color-detecting cone cells are hit by different wavelengths of light. Male squirrel monkeys naturally lack the L-opsin gene; like people who share their condition, they’re unable to distinguish between red and green.
At first, the two monkeys behaved no differently than before. Though quick to earn a grape juice reward by picking out blue and yellow dots from a background of gray dots on a computer screen, they banged the screen randomly when presented with green or red dots.
But after five months, something clicked. The monkeys picked out red and green, again and again. At the biological level, Neitz can’t say precisely what happened — the monkeys, named Sam and Dalton, are alive and healthy, their brains unscanned and undissected — but their actions left no doubt.
They think it will work identically in humans. If so, this means that we could do the same thing for the mutant version of L-opsin that tetrachromat women have, and make anyone (even a man) into a tetrachromat. Or, even more excitingly, a gene for infrared or ultraviolet light.
Monad for weakly monoidal categories
We’ve got free and forgetful functors 
- binary trees with
labeled leaves as objects and
- binary trees with
labeled leaves together with the natural isomorphisms from the definition of a weakly monoidal category as its morphisms.
The multiplication 
An algebra of the monad is a category 
Then the associator should be a morphism
However, it isn’t immediately evident that the associator that comes from 
for the source instead of
which we get by replacing 
so we can define 
Therefore, the isomorphism 
A similar derivation works for the unitors and the triangle equation.
A morphism of algebras is a functor 
and
so we have the coherence laws for a strict monoidal functor.
Also,
so it preserves the associator as well. The unitors follow in the same way, so morphisms of these algebras are strict monoidal functors that preserve the associator and unitors.
Where the Wild Things Are
If I were to do a movie of where the wild things are, it would be dark. When the forest grows in his room, it’s a creepy scene: the paint bubbles up, discolors, and starts to peel; the strands of shag carpet turns into centipedes that burrow into the floor, which rots away into the detrius of the forest floor. The forest itself is dark and malicious. Max’s wolf suit becomes real wolf skins, with a bare wolf skull for a helmet. He’s both exhilarated and scared of his new power; he runs, and finds that he’s supernaturally fast, like a wolf. His fingernails are sharp and hard, like claws, and he leaves gashes in the trees as he runs by.
When he reaches the water, he summons a storm to drive him across the ocean, but as he gets nearer, it blows out of control, culminating in the water-thing:
Once Max gets past the water thing, he lands on the beach–perhaps barely surviving the storm and avoiding rocks. Bedraggled, soaked, and exhausted, he moves from the wind-lashed shore for what seems to be shelter in the forest. Then he hears the wild things and gets a glimpse of the terrible yellow eyes. The dog-like thing with a horn on its nose would look something like this:
http://www.crawley-creatures.com/conceptual/hound.htm
He’s chased back to the beach when he remembers his powers, turns, and his eyes burst into a bright yellow flame; he flashes his eyes quickly at each of the attacking Wild Things, and a shock wave knocks each one back. Then returning to the dog-like one, he stares it down and the Thing writhes in pain, yelping; the others begin to comprehend the extent of his power. He finally quenches his eye magic, and the dog lies panting, trying to recover its strength.
He leads the wild things to war and conquers the entire forest. He’s crowned with even more power, then has a night-on-bald-mountain kind of rumpus, until he chases down and kills (or just nearly?) a dog in the forest and realizes it’s his own. Horrified, he gives the command to stop; the wild things do, but not willingly, and by morning his power has ebbed to the point that he can’t hold them at bay any more. He flees to the boat and has just enough magic to launch himself back to the opposite shore, and the dream fades. He finds himself awaking from a fever, with a washcloth on his forehead and some soup waiting for him.
Two separate organisms on their way to becoming symbiotic
The single-celled Hatena and the algae Nephrosolmis live independent lives: Hatena has a “mouth” with which it eats smaller creatures and organic material; Nephrosolmis gets its food from sunlight. But when Hatena eats Nephrosolmis, the algae grows inside it, discards its organelles, changes its mouth into an eyespot, and swims toward light. The algae makes enough food to keep them both alive. When Hatena reproduces, one daughter keeps all the algae, and the other goes hunting again.
See also solar-powered sea slugs.
My talk at Perimeter Institute
I spent last week at the Perimeter Institute, a Canadian institute founded by Mike Lazaridis (CEO of RIM, maker of the BlackBerry) that sponsors research in cosmology, particle physics, quantum foundations, quantum gravity, quantum information theory, and superstring theory. The conference, Categories, Quanta, Concepts, was organized by Bob Coecke and Andreas Döring. There were lots of great talks, all of which can be found online, and lots of good discussion and presentations, which unfortunately can’t. (But see Jeff Morton’s comments.) My talk was on the Rosetta Stone paper I co-authored with Dr. Baez.
‘t Hooft, Baez on becoming a good theoretical physicist
Nobel Laureate Gerard ‘t Hooft: Stuff you need to know in order to do theoretical physics, with links to sites & papers that teach it
My advisor, John Baez: How to teach stuff, Lists of good books for learning the necessary math & physics
Good object oriented coding practices = object-capability security
| OOP idea | Corresponding attribute of the Principle of least authority |
| Separation of duties | Separation of Authority |
| Information hiding | Integrity |
| Message passing | Authorization |
| Dependency injection | Authority injection |
An analogy from Mike Samuel, the Caja Tech Lead.
Great flash tutorial
http://www.kongregate.com/labs
Also http://www.kongregate.com/forums/11/topics/23746 for doing it all with free tools.
One-liner webserver
mkfifo backpipe; while true; do head -1 backpipe | (echo -n .; cut -f2 -d\ ) | xargs -L1 cat | nc -l 8080 > backpipe; done;
This is for a mac; for netcat on linux you’d say nc -l -p 8080. Note that this is in no way secure! A directory traversal attack is the most obvious one; there’s probably an injection attack, too.
Syntactic string diagrams
I hit on the idea of making lambda a node in a string diagram, where its inputs are an antivariable and a term in which the variable is free, and its output is the same term, but in which the variable is bound. This allows a string diagram notation for lambda calculus that is much closer to the syntactical description than the stuff in our Rosetta Stone paper. Doing it this way makes it easy to also do pi calculus and blue calculus.
There are two types, V (for variable) and T (for term). I’ve done untyped lambda calculus, but it’s straightforward to add subscripts to the types V and T to do typed lambda calculus.
There are six function symbols:
Lambda takes an antivariable and a term that may use the corresponding variable.
This turns an antivariable “x” introduced by lambda into the term “x”.
(Application) This takes
and
and produces
.
These two mean we can duplicate and delete terms.
The 
I label the upwards arrows out of lambdas with a variable name in parentheses; this is just to assist in matching up the syntactical representation with the string diagram.
In the example, I surround part of the diagram with a dashed line; this is the part to which the
When I do blue calculus this way, there are a few more function symbols and the relations aren’t confluent, but the flavor is very much the same.
String diagrams for untyped lambda calculus
An example calculation
How to misread Lord of the Rings
This was a great piece by Andrew Rilstone, that, except for on archive.org, seems to have disappeared from the internet. So here it is again. (more…)
Wonderful talk on applying game design to app design
I had an idea like this for training users on a capability-based UI.
http://lostgarden.com/2008/10/princess-rescuing-application-slides.html
William’s witticisms
I was running an I jumped over a plate an I dint crack my head! I’n't that amazing? I went weeeooo squlksh an I falled in some applesauce.
I have a swim head. My head pops off an I have a swim body.
(William rubs Bruce’s feet.)
Bruce: That feels good!
William: You say funny words!
Wm: Ba’guys punch people.
MJ: Are you a bad guy? You kicked Aidan.
Wm: Ba’guys don’t kick.
Marty (pretending to sword fight): ching ching! Ching!
Wm: (sings) Ching, ching, ching! I love to ching!
Wm (running downstairs to us): I’m NAKED!
MJ: We noticed.
Wm: Yeah. It’s funny. HA HA HA HA!
Mike (noticing Martin’s blue lip): Did you drink the gatorade we told you not to drink?
Marty: William gave it to me and said I should drink it and I drank it.
PhD
I’ll be categorifying the lambda calculus. The nicest way of doing this would be to develop a good notion of a 2-theory, of which “the theory of a cartesian closed category” would be one example. It’s already known how to do this for cartesian categories: the kind of 2-theory has products itself, and has one type, 
Closed categories, however, need some notion of “op,” which is contravariant. Figuring out the nicest way of doing this will be some fun research. Paul Mellies has some ideas.
Hopefully, this kind of 2-theory will also let me talk about things like the blue calculus or Tyler Close’s web calculus (where objects are web services and POST is method invocation; lambda terms are stateless objects responding to the unique “call” method) that he uses in waterken.
PhD
“I’m pleased to let you know that the Board of Graduate Studies has approved your registration as a candidate of the degree of Doctor of Philosophy…”
I Got Another Journal Publication
“Bit Commitment Blues” has been published in Journal of Craptology!
(Re-)Constructed languages
Viginia Algonquin, for the movie Jamestown:
http://www.nytimes.com/2006/03/07/science/07lang.html?8hpib
