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Dr rer nat, Magna cum laude

July 17th, 2008

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 .

(84 words)

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
$Ext_R^{\leq 2}(k,k)$ .

This is a preview of Tour dates. Read the full post (286 words, 8 images)

A-infinity, Conferencing, Mathematics, PhD, Topology, Travel
(5) Comments

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)

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)

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.

This is a preview of Enumerating the Saneblidze-Umble diagonal in Haskell. Read the full post (2258 words)

A-infinity, Algebra, Combinatorics, Haskell, Mathematics, PhD, Programming
No Comments

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)

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.

This is a preview of D₈ revisited. Read the full post (272 words, 10 images)

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 $H^*(D_8,\mathbb F_2)$ .

To do this, I fix the group algebra
$\Lambda=\mathbb F_2[a,b]/(a^2,b^2,abab+baba)$
and the cohomology ring
$\Gamma=\mathbb F_2[x,y,z]/(xy)$
with $x,y\in\Gamma_1$ , $z\in\Gamma_2$

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
$\begin{pmatrix} a&0&bab&0&0&0\\ 0&b&0&aba&0&0\\ 0&0&a&0&bab&0\\ 0&0&0&b&0&aba\\ 0&0&0&0&a&b \end{pmatrix}$
for differentials starting in odd degree, and
$\begin{pmatrix} a&0&bab&0&0\\ 0&b&0&aba&0\\ 0&0&a&0&bab\\ 0&0&0&b&aba \end{pmatrix}$
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
$\partial_0=\begin{pmatrix}a&b\end{pmatrix}$
$\partial_1=\begin{pmatrix}a&0&bab\\0&b&aba\end{pmatrix}$
$\partial_2=\begin{pmatrix}a&0&bab&0\\0&b&0&aba\\0&0&a&b\end{pmatrix}$

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 $\Lambda^{2k}$ , keeps the first, third, et.c. elements and throws out the even ordered elements; so $x\cdot(a_1,a_2,a_3,a_4,a_5,a_6)=(a_1,0,a_3,0,a_5)$
For an element in $\Lambda^{2k+1}$ , the last element gets extra treatment, so
$x\cdot(a_1,a_2,a_3,a_4,a_5,a_6,a_7)=(a_1,0,a_3,0,a_5,ab\cdot a_7)$
For the lowest degrees, we also have
$x_0 = \begin{pmatrix}1&0\end{pmatrix}$
$x_1 = \begin{pmatrix}1&0&0\\0&0&ab\end{pmatrix}$
$x_2 = \begin{pmatrix}1&0&0&0\\0&0&1&0\end{pmatrix}$

This is a preview of An A_∞-structure on the cohomology of D₈. Read the full post (815 words, 61 images)

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 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:

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

A-infinity, Algebra, English, Homology and Homotopy, Mathematics, PhD, Research
(3) Comments

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:

This is a preview of A_∞ for the layman. Read the full post (1250 words, 1 image)

A-infinity, Algebra, English, Homology and Homotopy, Mathematics, PhD, Topology
(1) Comment

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.

This is a preview of A-infinity and Hochschild cocycles. Read the full post (1542 words, 68 images)

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about: 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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