After about 5 semesters, one paper, one erratum (submitted to JHRS) and one thesis, and after taking two oral exams and delivering one 30 minute talk on my research, I am now (modulo the week or two it takes to produce my certificate) entitled to the title of *doctor rerum naturalium*.

Next stop is the topology in computer science workgroup at Stanford, where I have accepted an offer for a postdoc research position up to 3 years (conditional on my good behaviour :-).

*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

.

- January 18th, 2008
- 4:26 pm

http://arxiv.org/abs/0707.1637

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

Damn, this feels good!

- September 26th, 2007
- 4:25 am

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.

*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.

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.

- February 7th, 2007
- 6:36 pm

I have previously calculated the A_{∞}-structure for the cohomology ring of D_{8}. 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, *m*_{4}, with *m*_{4}(x,y,x,y)=m_{4}(y,x,y,x)=z.

- November 30th, 2006
- 2:28 pm

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 Λ^{i+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

- November 23rd, 2006
- 3:48 pm

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 μ_{1}=d and μ_{2}=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:

- November 7th, 2006
- 4:18 pm

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:

- October 20th, 2006
- 3:15 pm

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.