Michi’s blog

On purity and essence of mathematics

• June 15th, 2008

I seem, lately, to be so densely planned that all I can do for my blog is to react on blog posts from Ben Webster at the Secret Blogging Seminar.

He has, recently, written a post inspired by the xkcd comic on purity in the sciences. The comic is funny, and rings true, but Ben brings up a severe criticism of the premises of the comic that rings back to my own years as a hotheaded undergraduate.

You should read all of Ben’s post, but if you don’t, you should at least read the following:

Restarting high school topology

• May 21st, 2008

My two high-school kids came by today. We’ve been trying to get a new teaching session together since early February, but they had a hell of a time all through February, and all our appointments ended up canceled with little or no notice; and then I spent March and April on tour.

We pressed on with knot theory. Today, we discussed knot sums, prime knots, knot tabulation, behavior of the one invariant (n-colorability) we know so far under knot sums, Dowker codes, and we got started on Conway codes for knots. Next week, I plan for us to finish up talking about the Conway knot notation, get the connection between rational knots and continued fractions down pat, and start looking into new invariants.

Parallel and cluster computing with MPI4Py

• May 18th, 2008

First off, I’d ask your pardon for the lull in postings - this spring has been insane. It has been very much fun - traveling the world, talking about my research and meeting people I only knew electronically - and also very intense.

To break the lull, I thought I’d try to pick up what I did last summer: parallel computing on clusters. It’s been a bit of blog chatter about SAGE and how SAGE suddenly has transformed from a Really Good Idea to something that starts to corner out most other systems in usability and flexibility.

Matlab? SciPy bundled with SAGE and the Python integration seems to be at least as good, if not better.
Maple? Mathematica? Maxima? Singular? GAP? SAGE interfaces with all those that it doesn’t emulate.

Tour dates

• March 6th, 2008

Edited to add Galway

I’ll be doing a “US tour” in March / April. For the people who might be interested - here are my whereabouts, and my speaking engagements.

I’m booked at several different seminars to do the following:

Title: On the computation of A-infinity algebras and Ext-algebras
Abstract:

For a ring R, the Ext algebra carries rich information about the ring and its module category. The algebra is a finitely presented k-algebra for most nice enough rings. Computation of this ring is done by constructing a projective resolution P of k and either constructing the complex or equivalently constructing the complex . By diligent choice of computational route, the computation can be framed as essentially computing the homology of the differential graded algebra .

Being the homology of a dg-algebra, has an induced A-infinity structure. This structure, has been shown by Keller and by Lu-Palmieri-Wu-Zhang, can be used to reconstruct R from
.

Introduction to Algebraic Geometry (3 in a series)

• March 4th, 2008

I’m going to move on with the identification of geometric objects with functions from these objects down to a field soon enough, but I’d like to spend a little time nailing down the categorical language of this association. Basically, we have two functors I and V going back and forth between two categories. And the essential statement of the last post is that these two functors form an equivalence of categories.

Now, first off in this categorical language, I want to nail down exactly what the objects are. In the category the objects are solution sets of systems of polynomial equations. And in the category , the objects are finitely presented Noetherian reduced k-algebras.

The functor acts on objects by sending an algebra R to the solution set of the polynomial equations generating the ideal in a presentation of the algebra.

Introduction to Algebraic Geometry (2 in a series)

• February 21st, 2008

I want to lead this sequence to the point where I am having trouble understanding algebraic geometry. Hence, I won’t take the usual course such an introduction would take, but rather set the stage reasonably quickly to make the transit to the more abstract themes clear.

But that’s all a few posts away. For now, recall that we recognized already that any variety is defined by an ideal, and that intersections and unions of varieties are given by sums and intersections or products of ideals.

This is the first page of what is known as the Algebra-Geometry dictionary. The dictionary is made complete by a pair of reasonably famous theorems. I won’t bother proving them - the proofs are a good chunk of any decent commutative algebra course - but I’ll quote the theorems and discuss why they matter.

Introduction to Algebraic Geometry (1 in a series)

• February 21st, 2008

I’m growing embarrassed by my lack of understanding for the sheaf-theoretic approaches to algebraic (and differential) geometry. I’ve tried to deal with it several times before, and I’m currently reading up on Algebraic Geometry again to fill the void that the finished thesis, soon arriving travels and non-existent job application responses produce.

So, why not learn by teaching? It’s an approach that has been pretty darn good in the past. So I thought I’d write a sequence of posts on algebraic geometry, introducing what it’s supposed to be about and how the main viewpoints develop more or less naturally from the approaches taken.

Varieties

The basic objective of algebraic geometry is to study solution sets to systems of polynomial equations. That is, we take some set of polynomials in some polynomial ring over some field . And we write for the set of all simultaneous roots to all these polynomials:

PROPs and patches

• February 15th, 2008

Brent Yorgey wrote a post on using category theory to formalize patch theory. In the middle of it, he talks about the need to commute a patch to the end of a patch series, in order to apply a patch undoing it. He suggests a necessary condition to do this is that, given patches P and Q, we need to be able to find patches Q’ and P’ such that PQ=Q’P', and preferably such that Q’ and P’ capture some of the info in P and Q.

However, as such, this is not enough to solve the issue. For one thing, we can set Q’=P and P’=Q, and things are the way he asks for.

Now, I wonder whether we can solve this by using PROPs (or possibly di-operads or something like that). Let’s represent a document as a list of some sort of tokens. We’ll set the set of all lists of length , and we’ll set to denote operations that take a list of length n and returns a list of length m.

My topology students move into knot theory

• February 1st, 2008

So, here’s the plan for my 10th grade topology students.

Today, we’ll abandon algebraic topology completely, and instead go into knot theory. I’ll want to discuss what we mean by a knot (embedding of in ), what we mean by a knot deformation (thus introducing isotopies while we’re at it) and the Reidemeister moves. Also we’ll discuss knot invariants - and their use analogous to topological invariants.

Later on, we’ll continue with other invariants; definitely including the Jones polynomial, and possibly even covering Khovanov homology. One possible end report would be to explain a bunch of knot invariants and show using examples how these have different coarseness.

Edited to add: I got myself some damn smart students. They figured out the Reidemeister moves on their own - as well as minimal crossing number in a projection being highly relevant - with basically no prompting from me. I’m impressed.

Algebraic surface toys!

• January 25th, 2008

At the start of the German Year of Mathematics, the Oberwolfach research institute has released an exhibition and the software they used to produce it. The software, surfer, is a really nice GUI that sits on top of surf and lets you rotate and zoom your algebraic surfaces as well as pick colours very comfortably.

They have a whole bunch of Really Pretty Images at the exhibition website, and I warmly recommend a visit. If you can get hold of the exhibition, they also have produced real models - with a 3d-printer - of some of the snazzier surfaces, so that one could have a REALLY close encounter with them.

But also, I’d really like to show you some of my own minor experiments with the program.

This is the interior of a Klein Bottle, using the “standard” realization as an algebraic surface given by Mathworld. In other words, I’m using
(x^2+y^2+z^2+2*y-1)*((x^2+y^2+z^2-2*y-1)^2-8*z^2)+16*x*z*(x^2+y^2+z^2-2*y-1)=0
for the defining equation. It kinda looks a bit like a Sousaphone in my opinion.

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!

AMS-MAA JMM 2008 Liveblogging, day 1

• January 7th, 2008

I’m exhausted.

I’m completely exhausted.

And I just got through the first day.

However, I also managed to meet up with S from the university interested in me. We had a really nice chat, and I feel rather good about it.

Other things done today - listened to an interesting talk generalizing Koszul algebras based on the highest degree ring generator of the Ext algebra. Listened to bits and pieces of a talk on Koszul and Verdier duality. Saw Flatland - The Movie (with Martin Sheen playing the main character, Arthur Square).

I also chatted with Cliff Stoll - whose sales pitch for the Klein Bottles is immensely entertaining; NSA - who don’t want me; Maplesoft - who are interested in me; Mathematica - who pointed me to their website; various e-Learning companies; and many many other exhibitors.

Also, got tired, hungry and WET. It’s bloody raining here.

Checking email 4000 times a day

• November 20th, 2007

In a recent column at The Chronicle of Higher Education, the columnist writes

I’m a latecomer to it, in part because I have a very hit-or-miss interest in new technologies. (I still don’t own a cell phone, for example, though I check my e-mail 4,000 times a day.)

Now. There are 24 hours in a day. 1 440 minutes. 86 400 seconds. Thus, checking e-mail 4 000 times in a day would require you to check your inbox every 21.6 seconds. Day and night.

Either the author is innumerate or hyperactive.

High school topology restarting

• November 16th, 2007

Today, I told my two bright students about abstract and geometric simplicial complexes, about the boundary map and the chain complex over a ring R associated with a simplicial complex Δ, and assigned them reading out of Hatcher’s Algebraic Topology.

The next couple of weeks will be spent doing homology of simplicial complexes, singular homology, equivalence of the two, neat things you can do with them; and then we’ll start moving towards a Borsuk-Ulam-y topological combinatorics direction.

I might end up pulling combinatorics papers from my old “gang” in Stockholm on graph complexes, and graph property complexes, and poke around those with them.

Wreath products

• October 29th, 2007

In a conversation on IRC, I started prodding at low-order wreath products. It turned out to be quite a lot of fun doing it, so I thought I’d try to expand it into a blog post.

First off, we’ll start with a definition:

The wreath product is defined for groups G,H and a G-set X by the following data. The elements of are tuples . The trick is in the group product. We define

(or possibly with a lot of inverses sprinkled into those indices)

Consider, first, the case of with the nontrivial G-action defined by gx=1, g1=x. We get 8 elements in the wreath product . Thus, the group is one of the groups with 8 elements - . We shall try to identify the group in question using orders of elements as the primary way of recognizing things. Consider an element ((x,y),z).

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.

Coq and simple group theory

• August 5th, 2007

Trying to make the time until my flight leaves tomorrow go by, I played around a bit with the proof assistant Coq. And after wrestling a LOT with the assistant, I ended up being able to prove some pretty basic group theory results.

And this is how it goes:

This sets the stage. It defines a group as a group object in Set, but without the diagonal map. It produces a minimal definition - the left identity and inverse follow from the right ones, which we shall prove immediately.

Today seems to be a day for posting…

• July 12th, 2007

ComplexZeta asked me about the origins of my intuitions for homological algebra in my recent post. The answer got a bit lengthy, so I’ll put it in a post of its own.

I find Weibel very readable - once the interest is there. It’s a good reference, and not as opaque as, for instance, the MacLane + Hilton-Stammbach couplet can be at points.

The interest, however, is something I blame my alma mater for. Once upon a time, Jan-Erik Roos went to Paris and studied with Grothendieck. When he got back, he got a professorship at Stockholm University without having finished his PhD. He promptly made sure that nowadays (when he’s an Emeritus stalking the halls) there is not a single algebraist at Stockholm University without some sort of intuition for homological algebra.

The why and the what of homological algebra

• July 12th, 2007

I seem to have become the Goto-guy in this corner of the blogosphere for homological algebra.

Our beloved Dr. Mathochist just gave me the task of taking care of any readers prematurely interested in it while telling us all just a tad too little for satisfaction about Khovanov homology.

And I received a letter from the Haskellite crowd - more specifically from alpheccar, who keeps on reading me writing about homological algebra, but doesn’t know where to begin with it, or why.

I have already a few times written about homological algebra, algebraic topology and what it is I do, on various levels of difficulty, but I guess - especially with the carnival dry-out I’ve been having - that it never hurts writing more about it, and even trying to get it so that the non-converts understand what’s so great about it.

So here goes.

Blogging seminars - a new fad!

• July 6th, 2007

They simply do not end. Now, Cornell grads and pre-grads have started the Everything Seminar - which has absolutely brilliant discussions about the forbidden minor theorem in graph theory as well as a fascinating overview over constructing homological algebra as embedded in the theory of modules over .

Connected to this comes the observation that by constructing calculus using the tricks used in synthetic differential geometry, we end up with - again - modules over , and some very fascinating discussions are sparked as to subtle and interesting connections between these two viewpoints!

How on earth I am going to keep up with the interesting sprouting discussion group blogs I shall never know. Maybe it’s getting to the point where we’ll start an -blog?

Linking in hushed voices

• June 12th, 2007

All the cool kids are doing it, so I’ll tag in too.

There’s a secret blogging seminar going on. And the people seem to be writing interesting things - what little they managed to write before the hushed rumour mill started.

Enumerating the Saneblidze-Umble diagonal in Haskell

• June 11th, 2007

IMPORTANT: Note that the implementation herein is severely flawed. Do not use this.

One subject I spent a lot of time thinking about this spring was taking tensor products of A_∞-algebras. This turns out to actually already being solved - having a very combinatorial and pretty neat solution.

Recall that we can describe ways to associate operations and homotopy of associators by a sequence of polyhedra K_n, n=2,3,.., called the associahedra. An A_∞-algebra can be defined as being a map from the cellular chains on the Associahedra to n-ary endomorphisms of a graded vector space.

If this was incomprehensible to you, no matter for this post. The essence is that by figuring out how to deal with these polyhedra, we can figure out how to deal with A_∞-algebras.

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 -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.

Young Topology: The fundamental groupoid

• May 4th, 2007

Today, with my bright topology 9th-graders, we discussed homotopy equivalence of spaces and the fundamental groupoid. In order to get the arguments sorted out, and also in order to give my esteemed readership a chance to see what I’m doing with them, I’ll write out some of the arguments here.

I will straight off assume that continuity is something everyone’s comfortable with, and build on top of that.

Homotopies and homotopy equivalences

We say that two continuous maps, f,g:X→Y between topological spaces are homotopical, and write , if there is a continuous map such that H(x,0)=f(x) and H(x,1)=g(x). This captures the intuitive idea of step by step nudging one map into the other in formal terms.

Two spaces X,Y are homeomorphic if there are maps , such that and .

Two spaces X,Y are homotopy equivalent if there are maps , such that and .

Interview with a blogger

• April 30th, 2007

The website/forumsite Mathetreff, run by the Bezirksregierung (region government) Düsseldorf, just performed a mail interview with me.

Here it is, translated to english, for your enjoyment.

MT: Dear Mr. Johansson, you are an expert on mathematics blogs. Thus first off a double question: What is a blog, and what do you do in your algebra blog?

MJ: A blog, or weblog as the name started, is basically just a comfortable way to publish texts sequentially on the web. As such blogs aren’t much more than websites with administration aids. The interesting starts when you involve interactions - comments on the texts or easy linking between different blogs.

For this, there are both many different blog softwares, that make these aspects easy accessible and automated, and many websites that network blogs. Thus, it happens that farreaching blog debates occur, where through a networking site several blog discuss or even fight over some question.

Modular representation theory - when Maschke breaks down

• April 21st, 2007

This post is dedicated to Janine Kühn and her Proseminar-lecture.

We had, in my first representation theory post, a mention of Maschke’s theorem. This states that if the characteristic of our field doesn’t divide the group order, then simple and irreducible mean the same thing.

Now, obviously, the actual proof you normally see first deals with a construction that works for when the characteristic doesn’t divide the group order - which uses 1/|G| at one point. So, what happens when this is impossible to work with? When the conditions of Maschke simply do not hold?

The very simplest answer is that then we can get modules that are glued together by simple modules with some meshing. Such that they aren’t direct sums any more. The ways we can glue together modules are through extensions - i.e. we can glue together A,C by forming a short exact sequence
0 → C → B → A → 0
and we’ll have that B is a module such that B/C=A. Now, the typical such module is the direct sum of A and C - and if Maschke holds, this is indeed all there is.

looksay - today’s Haskell snippet

• April 18th, 2007

Conway’s Look-and-say sequence

Modular representation theory: Simple and semisimple objects

• April 2nd, 2007

Representations of categories

The basic tenet of representation theory is that we have some entity - the group representation theory takes a group, the algebra representation theory most often a quiver, and we look at ways to view the elements of the structure as endomorphisms of some vectorspace. The attentive reader remembers my last post on the subject, where was given a group action by the rotations and reflections of a polygon surrounding the origin.

There is a way, suggested to me by Miles Gould in my last post, to unify all these various ways of looking at things in one: groups - and quivers - and most other things we care to look at are in fact very small category. For groups, it’s a category with one object, and one morphism/arrow for each group element, and such that the arrow you get by composing two arrows is the one corresponding to a suitable product in the group. A quiver is just a category corresponding to the way the quiver looks - with objects for all vertices, and arrows for all edges.

Sudden responsibilities

• March 28th, 2007

I just met up with the workgroup in the Deutsche Mathematikervereinigung (German Association of Mathematicians) with interest spanning “Information and Communication” - which turns out to mean that they care about libraries, about communicative tools for mathematicians, and spend their time thinking about these things, and meeting at conferences.

Met up with is to say participated in their workshop.

Stunned them all with the relevation that I blog. “Wow, a real, live blogger! Here!”

And promptly got elected to their executive committee.

Kinda unexpected result of going to a conference.

4th Carnival of Mathematics posted

• March 23rd, 2007

The fourth Carnival of Mathematics is up at EvolutionBlog.

Featured this time around are homological algebra, representation theory, Rubik’s cube solutions, Bernoulli processes, topology, number theory, and much much more.

Representation theory - basics

• March 21st, 2007

Many interesting groups have a very geometrical definition: transformations that fix certain symmetries is one of the historical origins of group theory.

Thus, one of the most interesting classes of finite groups are the rotation and reflection symmetries of a regular polygon. These are called , for a n-sided polygon. Thus, for a triangle, we can label the corners a,b,c, reading clockwise, and enumerate the possible transformations by the positions the corners end up in. Thus we get the elements:
identity: abc -> abc
τ: abc -> bca (rotation by 120 degrees)
τ²: abc -> cab
σ: abc -> acb (reflection fixing a)
στ: abc -> bac
στ²: abc -> cba

Now, if we fix one equilateral triangle - say the one spanned by the points , and , then these transformations of the triangle can be extended to rotations and reflections of the entire 2-dimensional plane . As such, we can write down matrices for the group elements, starting with
and

The rest of the group elements we can realize as matrices by just multiplying these with each other.

Carnival of Mathematics #3

• March 9th, 2007

And it is with pride that I welcome you all to my first issue, and the third issue all in all, of the Carnival of Mathematics. I probably should apologize as well - my announcement stated March 8th, but that was before I really looked at the dates involved, so we did, alas, miss the international women’s day. We haven’t had quite the rush that Mark CC enjoyed, but we’ll make a good one even so.

First out, from the first half of our submissions, we have a grand tour of didactic topics, starting out with Michael Tang, who shows us why negative times negative is positive, with a touch of ring theory into the mix. Following that, Rebecca Newburn discusses equation solving strategies and Laurie Bluedorn takes a historical view on the age of introduction of formal arithmetic. To finish it up, jd2718 tells us about teaching complex numbers and your humble host has a manifesto of sorts about stimulating strong students.

Bright students and topology

• March 2nd, 2007

Today, I started an experiment together with the local specialised secondary school. I’ll be taking care of two of their brightest students, meeting them roughly once a week, and taking them on a charge through algebraic topology. At the far end shimmers knot theory and other funky applications; and on the way there, I hope for many interesting spinoffs and avenues.

They got, today, Armstrong’s Basic Topology, and an extract from the German topology book by Jänich, and on monday, we shall go over the formal definition of topological spaces, and of continuous functions together.

I plan to keep updates on our progress here on the blog - with the questions I send them off with each meeting as well as some sort of discussion about how this setup is working out, if at all.

For the first trip, the questions I dumped in their laps were:

• What topologies are possible on the set {0,1}?
• What topologies are possible on the set {0,1,2}?
• Which are the continuous functions between the topologies above?
• Give an example of a continuous and a discontinuous function each for the following cases
  1. with the standard topology on
  2. with the standard topology on
  3. with the discrete topology
  4. with the finite-complement topology

Carnival of mathematics, issue 2

• February 23rd, 2007

The second carnival of mathematics is up over at Good Math, Bad Math. It’s again a nice, good read. Go.

The third carnival of mathematics is to be hosted by yours truly on the International Women’s Day (maybe we can get a theme going? Women in mathematics, anyone?). Submissions to me directly or over the submission form.

First carnival of mathematics is up!

• February 9th, 2007

Over at Alon’s place, Abstract Nonsense, the first issue of the forthnightly Carnival of Mathematics is up.

Go there. Read. There’s a LOT of good blog posts there.

D₈ revisited

• February 7th, 2007

I have previously calculated the A_∞-structure for the cohomology ring of D₈. Now, while trying to figure out how to make my work continue from here, I tried working out what algebra this would have come from, assuming that I can adapt Keller’s higher multiplication theorem to group algebras.

A success here would be very good news indeed, since for one it would indicate that such an adaptation should be possible, and for another it would possibly give me a way to lend strength both to the previous calculation and to a conjecture I have in the calculation of group cohomology with A_∞ means.

So, we start. We recover, from the previous post, the structure of the cohomology ring as k[x,y,z]/(xy), with x,y in degree 1, and z in degree 2. Furthermore, we have a higher operation, m₄, with m₄(x,y,x,y)=m₄(y,x,y,x)=z.

Announcing the Carnival of Mathematics

• February 2nd, 2007

Alon Levy, over at Abstract Nonsense has just announced the first issue of a brand new Blog Carnival: the Carnival of Mathematics.

Go take a look. Submit your own blog posts. And then check it out in a week - the carnival is scheduled for the 9th.

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.

An A_∞-structure on the cohomology of D₈

• November 30th, 2006

As a first unknown (kinda, sorta, it still falls under the category of path algebra quotients treated by Keller) A_∞-calculation, I shall find the A_∞-structure of .

To do this, I fix the group algebra

and the cohomology ring

with ,

Furthermore, we pick a canonical nice resolution P, continuing the one I calculated previously. This has the i:th component Λⁱ⁺¹, and the differentials looking like

for differentials starting in odd degree, and

for differentials starting in even degree. The first few you can see on the previous calculation, or if you don’t want to bother, they are

Now, armed with this, we can get cracking. By lifting, we get canonical representating chain maps for x,y,z described, loosly, by the following:

x takes an element in , keeps the first, third, et.c. elements and throws out the even ordered elements; so
For an element in , the last element gets extra treatment, so

For the lowest degrees, we also have

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 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:

A_∞ for the layman

• November 7th, 2006

I recently had reason to describe my PhD work in complete laymans terms while writing letters to my grandparents. This being a good thing to do in order to digest your ideas properly, I thought I might try and write it up here as well.

It will, however, push through some 100-odd years of mathematical development rather swiftly. Try to keep up - I will keep it as light as I can while not losing what I want to say.

Algebra

In the 19th century, a number of different mathematical efforts ended up using more or less precisely the same structures, though not really recognizing that they were the same. This recognition came with Cayley, who first brought the first abstract definition of a group.

A group is a set G of elements, with a binary operation *, such that the following relations hold:

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.

A-infinity and Hochschild cocycles

• October 20th, 2006

This blogpost is a running log of my thoughts while reading a couple of papers by Bernhard Keller. I recommend anyone reading this and feeling interest to hit the arXiv and search for his introductions to A_∞-algebras. Especially math.RA/9910179 serves as a basis for this post.

If you do enough of a particular brand of homotopy theory, you’ll sooner or later encounter algebras that occur somewhat naturally, but which aren’t necessarily associative as such, but rather only associative up to homotopy. The first obvious example is that of a loop space, viewed as a group: associativity only holds after you impose equivalence between homotopic loops.

Computational Group Theory in Haskell (1 in a series)

• October 18th, 2006

This term, I’m listening to a lecture course on Computational Group Theory. As a good exercise, I plan to implement everything we talk about as Haskell functions as well.

The first lecture was mainly an introduction to the area, ending with a very naïve algorithm to generate a permutation group from a set of generators. Next week will bring less naïve algorithms with not quite as horrible complexity.

Before the algorithm can be brought, we’d want some undergrowth: we’d want to be able to work with permutations at all. So, we’ll start with the basic group theory and permutation implementations. A lot of this is stolen or rewritten from this permutation group code.

Our code will make use of two libraries, so if you collate code snippets while reading this, you’ll want to use

If you don’t want to bother with that, the code is available here.

Weekly Report: Settling down again

• October 8th, 2006

I get the feeling that my pledge to write the weekly reports regularily has been less than successful. So I’ll try to renew that pledge: I shall try to keep up the regularity of my weekly report.

Since last authored, I have been running a mathematics camp for 10th grade kids in mathematics-oriented schools. There are (apparently) 3 or 4 of those in Thüringen, and we had a good portion for each. As a new camp leader, I was very much the odd one out trying to get into the social circles; since kids and leaders alike tend to meet at least two-three times a year for math camps, math competitions et.c. It was fun, though, as expected, and a LOT of alien culture to get into. They do it so very much not like the swedish Unga Forskare.

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 with not a zero-divisor,