Comments on A Neighborhood of Infinity: Hopf Algebra = Group + Monad

Comoputational monads go back to the calculation o...

2008-11-22T05:25:00.000-08:00

Comoputational monads go back to the calculation of sine tables in antiquity, and then intersected algrebra under the influence of formal logic at about the time of the Asclepios (Hermetic monadology). This is not well known, because the true origin of differential calculus lies here. This history was touched on by Delambre on route to the SI (metric system), which accordingly holds the planet together! As errors of estimate, CMs then disappear behind the Weierstrass Approximation Theorem in tandem with statisticsal mechanics, and then colonize cosmology via Stone's extension to compactifications. The fabulous superstrings now fall to earth as M-branes, familiar as rigid rotators, or the atoms of the Ancients. Of course, this is only half of what topology now does, which is what bought me to this blog.

DiedLaughing

Kevin,

If you're interested in solving larg...

2008-08-19T17:32:00.000-07:00

Kevin,

If you're interested in solving large linear systems there will be some bindings to the standard C libraries out there. But I suggest asking in haskell-cafe where I've seen people discuss these things. haskell-cafe is a pretty friendly place and there are a quite a few mathematicians 'there'.

Incidentally, the benchmark for compactly computing homology is probably this.

Sigpfe -- thanks for that!

It's pretty ama...

2008-08-19T17:16:00.000-07:00

Sigpfe -- thanks for that!

It's pretty amazing that it takes 20 lines to code to compute homology. (I'd been dreading trying to define a chain complex class in C++...)

The coolest thing about all this ribbon graph stuff is the way it links algebra and geometry (and physics!). For instance, Kontsevich has a great paper where he shows how to understand the cohomology of moduli space by thinking about homotopy-associative algebras. My project is ultimately related to this.

The only reason Haskell might not be appropriate for what I want to do is that at some stage I'll want to invert very large matrices. But I presume that Haskell can just call some fast + optimized C program to do this part, is that right?

Thanks,
Kevin

Kevin,

Chain complexes and homology?
2008-08-19T17:03:00.000-07:00
Kevin,

Chain complexes and homology? Funny you should ask :-)

Years ago I read the paper by Penner on fat/ribbon graphs, matrix models and the computation of the Euler characteristics of moduli spaces and thought it was pretty amazing. I'd love to see some related Haskell code.

I (apparently) got fellow blogger Mikael Johannson interested in Haskell a while back and I think he did a bunch of homological algebra type things with it.

Great blog!

Haskell does seem to be the p...

2008-08-19T16:51:00.000-07:00

Great blog!

Haskell does seem to be the perfect language for pure mathematics. I've been trying to do some computations involving ribbon graphs and the moduli space of curves in C++, but from reading your blog it looks like it would be far easier in Haskell.

Do you know of any implementations of things like chain complexes, homology, etc. in Haskell?

Best,
Kevin

Very interesting! I wonder what happens when you m...

2008-08-08T14:20:00.000-07:00

Very interesting! I wonder what happens when you monadify/Kleislify the axioms of other algebraic structures, say monoids, fields or vector spaces.

I.e.
class Field m a where
zero :: m a
one :: m a
add :: a -> a -> m a
mult :: a -> a -> m a
...

instance (Monad m) => Field m Float where ...

With this setup: What is V Float Float? Because of the instance given in the previous paragraph this has (additionall to vector addition) a bilinear add and a bilinear mult.

Also, what is Maybe Z? Or [Z]? All of these come, thanks to the instance (Monad m) => Group m Z, equipped with a compatible multiplication, (>>= mult).

Got it! I was looking at toListWith's data signat...

2008-08-05T16:30:00.000-07:00

Got it! I was looking at toListWith's data signature and I mixed up the (k, a) there and the (k, a) in V.

Christian,

Elements of vectors are written ...

2008-08-05T09:12:00.000-07:00

Christian,

Elements of vectors are written as lists of pairs (k,a) where k is the coefficient and a is the basis vector label. I need to collect together pairs with the same basis label. Data.Map can do this. But fromList and toList use the *first* element of the pair as key, not the second element, and they sort on the first element. So I needed to swap.

The code would have been slightly shorter if I'd used lists of (a,k). But it's more conventional in mathematics texts to write the coefficient before the basis element.

I'm pretty sure the definition of V is correct because the quickChecks work. Of course 'pretty sure' isn't a proof.

I'm not sure if I need to read on to understand it...

2008-08-05T09:03:00.000-07:00

I'm not sure if I need to read on to understand it but it seems to me that x in reduce x is already (k, a). I don't understand why there is a need to fmap swap x first. Is the definition of V a typo?

Those were just the the kind of resources I was lo...

2008-08-03T15:10:00.000-07:00

Those were just the the kind of resources I was looking for. Greatly appreciate it. Hope you ran a good marathon!

That was a terse reply. I was on my way out to run...

2008-08-03T15:04:00.000-07:00

That was a terse reply. I was on my way out to run half of the San Francisco marathon.

Anyway, I found this useful for getting the correspondence between a category theoretic notion of monads and the actual notation as used in a Haskell program. I also enjoy lots of papers by Uustaluand Vene as they come with category theoretical language and Haskell code so it's easy to see the correspondence.

In that case there so many papers I don't know whe...

2008-08-03T05:12:00.000-07:00

In that case there so many papers I don't know where to start!

Well, I do have a fair amount of background (thoug...

2008-08-03T01:29:00.000-07:00

Well, I do have a fair amount of background (though quite modest, I should say) in computer science. Thanks!

I think the intersection of Haskell programmers an...

2008-08-02T20:57:00.000-07:00

I think the intersection of Haskell programmers and mathematicians is still fairly small and there aren't too many books, if any at all, on the subject geared to someone with a categorical, but not computer science background.

I am not exactly sure if you have answered the fol...

2008-08-02T20:41:00.000-07:00

I am not exactly sure if you have answered the following question before, but pardon me if I am repeating it anyway.

Are there some good online resources (articles, books,blogs, etc.) that may help one to (re)learn Haskell from a (more) categorical perspective?

Thanks,