(Data taken on February 23, from the current state of olympic medals achieved at that date, and from the Wikipedia page listing populations of the nations of the earth, taken the same date)

This, here, is me trying it out. With any luck, the appearance of a new toy will get me back to actually blogging some more – it’s been winding down a bit much here lately.

Coordinatization

First off, a few words on what we mean by coordinatization: as in algebraic geometry, we say that a coordinate function is some or possibly some , with all the niceness properties we’d expect to see in the context we’re working.

A particularly good example is Principal Component Analysis which yields a split linear automorphism on the ambient space that maximizes spread of the data points in the initial coordinates.

Topological coordinatization

The core question we’re working with right now is this:
Given a space (point cloud) X, and a (persistent) view of , can we use some map to generate a map inducing that map?

In topology there are ways around to compute spaces of maps directly from available homological information. The key phrase here is the Eilenberg-Moore spectral sequence, by which we compute using . We note that with the compact open topology.

We fix a field k of coefficients. We can find a spectral sequence that computes

Variations of this spectral sequence allow us to compute . A homotopy class of maps thus corresponds to a cohomology element from .

Monad approximation

Define
(unordered pairs)
and similarily unordered tuples

we can get a map, for pointed spaces

In the limit, we get the union .

The Dold-Thom theorem tells us:

Suppose we are really looking for maps .

We pick chain complexes . There is a hom chain complex consisting of graded maps with the homotopy/induced boundary

For this complex, we get
is the family of chain homotopy classes of chain maps .

Recall: are chain homotopic if there is a family such that .

Homotopic space maps yield homotopic chain complex maps. And thus

How, though, do we get from to ?

Cosimplicial spaces

A simplicial space is a contravariant . It has a total space or realization given by

In the dual setting of cosimplicial spaces, the total space is a subspace of .

There is a cosimplicial space

whose total space is .

Furthermore, there is a spectral sequence starting at (well understood space!)
going through

which recovers for us.

The mapping space maps to , which sends to chain maps. Differentials give us conditions on chain maps to come from actual maps on the topological level .

Suppose now that is a simplicial complex, and is known, say isomorphic to .

We pick as a model space, and work in . We pick out conditions form the spectral sequence above to pick out chain maps that might come from topological maps.

Suppose we had some chain map that sends .

We look for a chain homotopic map such that is a simplex (this will be WAY too optimistic, usually) or at least for small and close together in graph distances.

Yi Ding: there are likely many simplicial maps inducing any given . This makes optimizations on quality of maps (as measured in smallness properties of preimages of given simplices) troublesome, since we have a large search space and are likely to get stuck in local minima.

The topological approach creates very many maps, but we just want ONE – albeit with quality assurances. The big question is how to get there:

Shrinking the space? Homotopy collapses to reduce the number of generators?

Ask for more side conditions? If so, which?

Minimize the size of preimages and kernels – smoothing the resulting map?

Run with one condition until we reach a minimum, then change conditions to get out of that particular sink, repeat until it all looks stable?

Thus we need to search for strategies to improve convergence to actually good maps.

The questions we pose are similar to:

Jeff Ericksson, et al.: Given X, , how do we find a good representative ?

Finding good maps is really about finding good representatives in . Given simplicial bases we can pick out a first basis
(Kronecker ) for .

Finally: many problems in data analysis call for analyzing contractible spaces. So if we pick out the notion of a boundary, doing all of this relative to a boundary would give us increased power to use the resulting methods in data analysis.

Lecture 7, featuring the introduction of monads based in adjoints, with the connection between the monoid of endofunctors and the Haskellite specification of monads.

Lecture 8, featuring Eilenberg-Moore algebras, initial algebras for datatype specification, Lambek’s lemma and structural induction and recursion with endofunctor algebras.

During class, and in unrelated conversations afterwards, though, the question emerged:

If Formally differentiating datatypes gives us zippers? What happens if we formally integrate datatypes?

Since last week Wednesday, the second lecture is up on the Haskell wiki.

In this course, we will teach category theory from first principles with an eye towards its applications to and correspondences with Haskell and the theory of functional programming. We expect students to previously or currently be taking CS242 and to have some level of mathematical maturity. We also expect students to have had contact with linear algebra and discrete mathematics in order to follow the motivating examples behind the theory expounded.

Wednesdays at 4.15.

Online notes will appear successively on the Haskell wiki on http://haskell.org/haskellwiki/User:Michiexile/MATH198

come this fall, I shall be teaching. My first lecture course, ever.

The subject shall be on introducing Category Theory from the bottom up, in a manner digestible for Computer Science Undergraduates who have seen Haskell and been left wanting more from that contact.

And thus comes my question to you all: what would you like to see in such a course? Is there any advice you want to give me on how to make the course awesome?

The obvious bits are obvious. I shall have to discuss categories, functors, (co)products, (co)limits, monads, monoids, adjoints, natural transformations, the Curry-Howard isomorphism, the Hom-Tensor adjunction, categorical interpretation of data types. And all of it with explicit reference to how all these things influence Haskell, as well as plenty of mathematical examples.

But what ideas can you give me to make this greater than I’d make it on my own?

Here are two others. Different data source, different point in time, but what are they?

They are all linked in pairs – one coloured and one black linked together. They are not sports related. And they are taken from real world data. The colours are relevant and constitute a hint in their own right.

So, first I downloaded the zipcode database at Mapping Hacks, and unpacked the zipfile in my working directory.

Then, I loaded the data into R

So, now I have an R frame containing a lot of US cities, their geographical coordinates, and their zip codes. So we can start playing with the plot command! After rooting around a bit, I ended up settling on the smallest footprint plot dot I could make R produce, by setting the option pch=20 in the plot options. Hence, I ended up with a command basically like this:

where the +1 after the modulus is to make even 0-values plot, and the cex parameter sets the point size to something small and pretty.

We can continue this, tweaking the divisor to extract all the other digits of the zip code, and we end up getting:

and the result

and the result

and the result

and finally

and the result

And then, of course, we can zoom in on data too. So we can do things like extracting Californian zip codes

to get

or, we could extract New York zip codes:

or even extract, say, the zip codes starting with 10 or 11, covering New York City and surroundings and take a closer look