«Phil Harvey wants us to partition {1,…,16} into two sets of equal size so each subset has the same sum, sum of squares and sum of cubes.» — posted by @MathsJam and retweeted @haggismaths.

Sounds implausible was my first though. My second thought was that there aren’t actually ALL that many of those: we can actually test this.

So, here’s a short collection of python lines to _do_ that testing.

import itertools
allIdx = range(1,17)
sets = itertools.combinations(allIdx,8)
setpairs = [(list(s), [i for i in allIdx if i not in s]) for s in sets]
def f((s1,s2)): return (sum(s1)-sum(s2), sum(map(lambda n: n**2, s1))-sum(map(lambda n: n**2, s2)), sum(map(lambda n: n**3, s1))-sum(map(lambda n: n**3, s2)))

goodsets = [ss for ss in setpairs if f(ss) == (0,0,0)]

And back comes one single response (actually, two, because the comparison is symmetric).

[([1, 4, 6, 7, 10, 11, 13, 16], [2, 3, 5, 8, 9, 12, 14, 15]),
([2, 3, 5, 8, 9, 12, 14, 15], [1, 4, 6, 7, 10, 11, 13, 16])]


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

(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.

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?