Michi’s blog » archive for 'Research'

Building my academic persona

  • January 18th, 2008

http://arxiv.org/abs/0707.1637
Just got accepted for publication in the Journal of Homotopy and Related Structures.

Damn, this feels good!

(19 words)

Papers status

  • December 16th, 2007

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 A_\infty-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.

(169 words, 1 image)

Progress

  • September 26th, 2007 

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.

(65 words)

Going to T’bilisi

  • May 25th, 2007

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 A_\infty-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.

(59 words, 1 image)

Retrospection 2006

  • December 30th, 2006

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.

This is a preview of Retrospection 2006. Read the full post (574 words)

A∞-algebras and group cohomology

  • November 23rd, 2006

In which the author, after a long session sweating blood with his advisor, manages to calculate the A∞-structures on the cohomology algebras H^*(C_4,\mathbb F_2) and H^*(C_2\times C_2,\mathbb F_2).

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 mi for operations on the group cohomology, and μi 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:

This is a preview of A∞-algebras and group cohomology. Read the full post (1833 words, 70 images)

Prototyping thought

  • October 28th, 2006

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.

This is a preview of Prototyping thought. Read the full post (706 words)

Academic firsts

  • October 10th, 2006

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.

(39 words)

Localisation and ring depth

  • September 5th, 2006

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 x_1,\dots,x_k with x_1 not a zero-divisor, each x_i not a zero-divisor in the quotient R/\langle x_1,\dots,x_{i-1}\rangle and R/\langle x_1,\dots,x_k\rangle 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.

This is a preview of Localisation and ring depth. Read the full post (549 words, 18 images)

Todo and accomplishments

  • July 24th, 2006

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.

(149 words)

More group cohomology

  • June 8th, 2006

My advisor told me to go hit D_8 and U(\mathbb H) 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 \mathbb F_2 with \mathbb F_2D_8, he walked me through his Shiny! Gröbner 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 H^{\leq 2}(D_8,\mathbb F_2) and then peeked into Carlson, et.al. for the Big List of 2-group cohomologies to see that all interesting stuff happens in H^{\leq 2} 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!

This is a preview of More group cohomology. Read the full post (740 words, 54 images)

This is me losing all faith in Non-Linear Analysis

  • March 28th, 2006

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.

(151 words)

The role of almonds in Swedish marital trends

  • December 30th, 2005

The role of almonds in Swedish marital trends

Mikael Johansson, Susanne Vejdemo

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)

This is a preview of The role of almonds in Swedish marital trends. Read the full post (711 words)

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Michi is a recent PhD working in homological algebra and applied algebraic topology. This blog is his outlet for texts with some manner of thought put into them. Over at his LiveJournal intimate details and streams of consciousness might be found.
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