25 August, 1pm, Linköpings Universitet. The topology of Politics.

1 September, 10am, KTH. Combinatorial species, Haskell datatypes and Gröbner bases for operads.

1 September, 1pm, KTH. Topological data analysis and the topology of politics.

2, 3, 6, 7 September, 11am, KTH. Mini-course on Applied algebraic topology.

I’m happy to give out details if you’d like to come and listen.

1. I’m going on the job market in the fall. I’m looking for lectureships, tenure tracks, possibly 2 year postdocs if they are really interesting.

2. I’m very interested in adding visits, adding seminars, adding anything interesting to this itinerary. If you want to meet me, send me a note, and I’m sure we can find time. My email adress is easy to google, and listed in the site info here.
I have current research and survey talks mostly ready to go for:

Barcodes and the topological analysis of data: an overview

In the past decade, the use of topology in explicit applications has become a growing field of research. One fruitful approach has been to view a dataset as a point cloud: a finite (but large) subset of a Euclidean space, and construct filtered simplicial complexes that capture distances between the data points. Doing this, we can translate any topological functor into a functor that acts on these filtered complexes, and reinterpret the results from the functors into information about the dataset. I’ll illustrate the basic results from the field, and give an overview of where my own research fits into the emergent paradigm.

Persistent cohomology and circular coordinates

Using a new variety of the algorithms described by Zomorodian, we are able to compute cohomology persistently, use persistence to pick out topologically relevant cocycles, and convert these into circle-valued coordinates. This allows us to generate topological coordinatizations for datasets, opening up for new approaches to data analysis.

Dualities in persistence

Starting out with a point cloud, the use of the barcode of persistent Betti numbers to characterize properties of the point cloud is well known. Expanding the amount of information we extract, we are led to study persistent homology and cohomology, in both an absolute manner, studying H(X_i), and in a relative manner, studying H(X_∞; X_i). We are able to show that these four possible homology functors are related by dualization functors Hom_k(-, k) and Hom_k[x](-, k[x]), and are able to use this to translate between all four corners. This way, we can choose to compute our barcodes in whatever situation an application motivates, and translate the resulting the barcode into whatever situation we want to interpret.

Period recognition with circular coordinates

In work joint with Vin de Silva and Dmitriy Morozov on computing circular coordinates for datasets may be used in the analysis of dynamical systems and in signal processing. We have experimental results that show a high resistance to spatial noise in reconstructing the period of a periodic process while avoiding Fourier transforms and running differences. Ideally, the use of algebraic topology in time series analysis for dynamical systems and signals will complement existing methods with more noise-robust components, and I discuss to some extent how this may be achieved.

Implementing Gröbner Bases for Operads

In a paper by Dotsenko and Khoroshkin, a Buchberger algorithm is defined in an equivalent subcategory of the category of symmetric operads. For this subcategory, they prove a Diamond lemma, and demonstrate how the existence of a quadratic Gröbner basis amounts to a demonstration of Koszularity of the corresponding finitely presented operad. During a conference at CIRM in Luminy, me and Dotsenko built an implementation of their work. I’ll discuss the implementation work, and what considerations need to be taken in the implementation of Gröbner bases for operads.

Parallelizing multigraded Gröbner bases

In work together with Emil Sköldberg and Jason Dusek, we use the lattice of degrees for a multigraded polynomial ring to parallelize Gröbner basis computations in the multigraded ring. We show speedups in implementations in Haskell using Data Parallel Haskell and the vector package, as well as in Sage using MPI for parallelization and SQL for abstract data storage and transport.

The topology of politics

While the use of data analysis in political science is a mature field, the development of new data analysis methods calls for updates in the choice, use and interpretation of these methods. We set out to investigate the use of the topological data analysis methods worked out at Stanford on political datasets in an ongoing research project. I’ll illustrate initial results, partly well-known, on the geometry and topology of parliamentary rollcall datasets.

Stepwise computing of diagonals for multiplicative families of polytopes

In recent work together with Ron Umble, we demonstrate a way to use basic linear algebra to compute cellular diagonal maps for families of polytope such that each face of each polytope in the family is given by products of other polytopes in the same family. Using the algorithm we describe, we are able to recover the Alexander-Whitney diagonal on simplices, the Serre diagonal on cubes as well as the Saneblidze-Umble diagonal on associahedra.

3. This summer, I will be present at :
ATMCS in Münster, Germany, June 21-26
Topology and its Applications in Nafpaktos, Greece, June 26-30
The International Conference
“GEOMETRY, TOPOLOGY,
ALGEBRA and NUMBER THEORY, APPLICATIONS”
dedicated to the 120th anniversary of Boris Delone in Moscow, Russia, August 16-20
Symposium on Implementation and Application of Functional Languages near Amsterdam, Holland, September 1-3
International Congress on Mathematical Software in Kobe, Japan, September 13-17

First, Second, …

1st, 2nd, …

And, because this chart is kinda tricky to read, here’s the log-scaled version of the same chart:

For the log-chart, I stopped stacking the numbers.

ETA: Changed the log-chart from a line-chart to a bar-chart after feedback from the readership of bOINGbOING. Hello and welcome!

(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