I do quite a bit of collaboration. In fact, since after my PhD research, I have written exactly one preprint that does NOT spring from a collaboration. And there is quite a bit of technological support that flows into a good collaboration of mine. Here are some of the tools I uses and some of the thoughts I have on them.

Version control

Since I work in mathematics (and, arguably, in the fringes of Theory CS), everything I write is written in LaTeX. This is for one thing very helpful, since it means that the actual texts we collaborate on are plain text with markup. Eminently suitable for the toolkit provided by the software community.

This is a typed up copy of my lecture notes from the seminar at Linköping, 2010-08-25. This is not a perfect copy of what was said at the seminar, rather a starting point from which the talk grew.

In my workgroup at Stanford, we focus on topological data analysis — trying to use topological tools to understand, classify and predict data.

Topology gets appropriate for qualitative rather than quantitative properties; since it deals with closeness and not distance; also makes such approaches appropriate where distances exist, but are ill-motivated.

These approaches have already been used successfully, for analyzing

• physiological properties in Diabetes patients
• neural firing patterns in the visual cortex of Macaques
• dense regions in of 3×3 pixel patches from natural (b/w) images
• screening for CO₂ adsorbative materials

This is a typed up copy of my lecture notes from the combinatorics seminar at KTH, 2010-09-01. This is not a perfect copy of what was said at the seminar, rather a starting point from which the talk grew.

In some points, I’ve tried to fill in the most sketchy and un-articulated points with some simile of what I ended up actually saying.

Combinatorial species started out as a theory to deal with enumerative combinatorics, by providing a toolset & calculus for formal power series. (see Bergeron-Labelle-Leroux and Joyal)

As it turns out, not only is species useful for manipulating generating functions, btu it provides this with a categorical approach that may be transplanted into other areas.

For the benefit of the entire audience, I shall introduce some definitions.

Definition: A category C is a collection of objects and arrows with each arrow assigned a source and target object, such that

Inspired by this post over at Making Light, here, have a chart:

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!

These are notes from a talk given at the Stanford applied topology seminar by Gunnar Carlsson from 9 Oct 2009. The main function of this blog post is to get me an easily accessible point of access for the ideas in that talk.

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?

This post is to give you all a very swift and breakneck introduction to Gröbner bases; not trying to be a nice and soft introduction, but much more leading up to announcing the latest research output from me.

Recall how you would run the Gaussian algorithm on a matrix. You’d take the leftmost upmost non-zero entry, divide its row by its value, and then use that row to eliminate anything in the corresponding column.

Once we have the matrix on echelon form, we can then do lots of things with it. Importantly, we can use substitution using the leading terms to do equation system solving.

The starting point for the theory of Gröbner bases was that the same method could be used – with some modification – to produce something from a bunch of polynomials that ends up being as useful as a row-reduced echelon form.

http://arxiv.org/abs/0707.1637

Just got accepted for publication in the Journal of Homotopy and Related Structures.

Damn, this feels good!

I just received my first ever referee’s report. Yikes!

Suffice to say, the report did not, as some I’ve seen blogged about, tear me a new one. Far from it – it was civil, kind, and pointed out several areas where my article text overlapped known arguments from other people and was generally superfluous as well as several areas where my article was too curt and didn’t actually spell out the new ideas sticking in it.

Also, making the relation of my results and those I rely on to the results of the Grand Old Man in applying -techniques in group cohomology explicit and discuss these in more detail was requested.

I know I couldn’t expect to write The Perfect Article as my first submission ever. And it’s not a flat out denial. And it brings constructive comments about how to make this a better article. Still, I think my ego needs a little bit of training to learn to cope with this part of the review process.

dynkin:~/magma> magma
Magma V2.14-D250907   Wed Sep 26 2007 13:19:51 on dynkin   [Seed = 1]
Type ? for help.  Type -D to quit.

Loading startup file "/home/mik/.magmarc"

> Attach("homotopy.m");
> Attach("assoc.m");
> Aoo := ConstructAooRecord(DihedralGroup(4),10);
> S := CohomologyRingQuotient(Aoo`R);
> CalculateHighProduct(Aoo,[x,y,x,y]);
z
> exit;
Total time: 203.039 seconds, Total memory usage: 146.18MB

And this is one major reason for the lack of updates recently.

In about 23 hours, I’ll step on to the train in Jena, heading for T’bilisi, Georgia.

On Monday, I’ll give a talk on my research into -structures in group cohomology. If you’re curious, I already put the slides up on the web.

I’ll try to blog from T’bilisi, but I don’t know what connectivity I’ll have at all.

Inspired by other bloggers on Planet Haskell, I thought I’d just sit down and write a retrospection post, reviewing the past year – primarily from angles such as mathematics, computers and my generic life situation.

It divides neatly into two different sections: the months as a commercial programmer and the months as PhD student and academic careerist.

The year began still working for Teleca Systems, and with security consulting for Stockholm-based firms and frequent trips back home.

Then as the year went on and my PhD applications grew more and more, I started getting results. I got invited to Bonn for an interview with the Homology and Homotopy graduate school program – which was in the end turned down because I was more of a homological algebraist than a topologist. And the week after that, I was invited to Jena for an interview for a position doing PhD work on computational homological algebra. The interview went well, the potential advisor was nice (and a once-roleplaying gamer to sweeten the deal more) and I got the position just a few days later.

In which the author, after a long session sweating blood with his advisor, manages to calculate the A_∞-structures on the cohomology algebras and .

We will find the A_∞-structures on the group cohomology ring by establishing an A_∞-quasi-isomorphism to the endomorphism dg-algebra of a resolution of the base field. We’ll write m_i for operations on the group cohomology, and μ_i for operations on the endomorphism dg-algebra. The endomorphism dg-algebra has μ₁=d and μ₂=composition of maps, and all higher operations vanishing, in all our cases.

Elementary abelian 2-group

Let’s start with the easy case. Following to a certain the notation used in Dag Madsen’s PhD thesis appendix (the Canonical Source of the A_∞-structures of cyclic group cohomology algebras), and the recipe given in A-infinity algebras in representation theory, we may start by stating what we know as we start:

In a recent post, pozorvlak reminded me of one of the reason it is important to have a good, obvious, and quick-to-write programming language around.

He, as I, is a mathematician, spending his time thinking, finding patterns, and trying to formulate (more or less) absolute proof that his patterns hold all the time, alternatively ways to demonstrate that they may not be universal.

In the post linked above, he starts by a neat little exercise, gets interested, and goes out to look at more examples. These show a very clear pattern, and after following this pattern quite some way out, he finally believes the pattern enough to start searching for a proof: which he also finds.

I just submitted a paper to a journal.

Based on research I have done during my time as a PhD student.

Wish me luck.

Update:If you want to read the paper, I suggest you go look at arXiv:math.GR/0610374.

So, we’re back at the point where I’m hesitating whether what I tried to work out even made sense or not. So I’ll do a write up of all the things I feel certain about asserting, and ask my loyal readership to hunt my errors for me.

Don’t laugh. This is less embarrassing for me than asking my advisor point blank.

We say that a (graded) commutative ring R has depth k if we can find a sequence of elements with not a zero-divisor, each not a zero-divisor in the quotient and a ring without non-zero divisors. This definition, of course, being the first obvious point where I may have screwed up.

Now, we know (from looking it up in Atiyah-MacDonald), that for SR the localisation of R in a multiplicatively closed subset S, S(R/I)=SR/SI, that injections carry over to injections, and that the annihilator over SR of an element is the localisation of the annihilator of the element.

Accomplished:

• I am done with the coursework for the past semester. Sent off the TeXed up solution sheets to the webmaster today.
• My pattern observation seems to hold up surprisingly well – there seems to be a theorem to fetch out there somewhere.
• I have done most of the dishes. Go me!

Todo:

• FOCUS! I need to learn triangulated categories. Preferably now. I need to stop playing with Haskell and reading up on group cohomology calculations. Preferably now.
• After triangulated categories, there’s a wealth of things to look into. High priority are spectral sequences, further group cohomology, diamond lemma, path algebra quotients, A_∞, Massey products, return to model categories. Which order these are done might influence the contents of my PhD thesis significantly.
• I really need to talk to the university sysadmins about the WLan network.

Progress may be found. Just around the corner. I need a donut.

My advisor told me to go hit and as my next two cohomology calculation projects; try to do them with resolutions by hand so that I get a feeling for what’s going on. After failing spectacularily both at getting a resolution of with , he walked me through his Shiny! Gr�bner base method to get resolutions with free modules over finite p-group algebras. Armed with the minimal resolution, I sat down and started hunting products; and finally found the cohomology ring.

Or … to be exact, I found and then peeked into Carlson, et.al. for the Big List of 2-group cohomologies to see that all interesting stuff happens in anyway.

So for the benefit of any and all readers who want to see what it looks like, I’m going to walk through it again here. Nonono, you don’t need to flee all of you – just skip this entry if it’s that scary!

A paper recently up on arXiv details the errors committed by an author of a paper in Non-Linear Analysis, who, by ignoring basic conditions of theorems manages to prove most of mathematics and substantial parts of physics inconsistent.

This is the second insufficiently reviewed paper at that Journal causing some sort of waves spreading as far as to me so far. The blogospheric and medial storm around the infamous “proof” by Elin Oxenhielm of the 16th Hilbertian problem a few years ago was, at the core, sparked from her getting the paper accepted at … right, Non-Linear Analysis … and taking this publication as a token that her results were in fact true and anyone critizising here were out to steal her credit.

Needless to say, with the density displayed thus far of crackpotism and sloppy publishing, I don’t think I’ll trust NLA for anything at all in the future.

The role of almonds in Swedish marital trends

Abstract

In this paper, we investigate the Swedish folkloristic belief that almonds in the Christmas rice porridge will lead to marriage. We offer a falsification of this hypothesis.
Keywords: Christmas traditions, almonds, porridge, marriage

Introduction

One old folk tradition in Sweden is to eat rice porridge on Christmas Eve. Normally, the porridge is served with a single shelled almond stirred into the porridge. Whoever bites into the almond is supposed to make up a rhyme immediately, and also is believed to marry soon – sometimes said to be before the year ends, or during the coming year.
In [1], an information page from Skansen, a renowned open air museum focusing on Swedish culture and traditions, the following is said about this belief:
“Förr i tiden la man ofta en mandel i gröten och rörde om så att den blev gömd i den fina gröten. Den som fick mandeln sa man skulle bli gift under året.”
”In earlier times, an almond was often put in the porridge and stirred in so that it would be hidden in the nice porridge. Whoever got the almond was said to be married during the year” (translation by the authors)