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
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
Just got accepted for publication in the Journal of Homotopy and Related Structures.
Damn, this feels good!
- December 16th, 2007
- 10:34 am
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.
- September 26th, 2007
- 4:25 am
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"
> Aoo := ConstructAooRecord(DihedralGroup(4),10);
> S := CohomologyRingQuotient(Aoo`R);
Total time: 203.039 seconds, Total memory usage: 146.18MB
And this is one major reason for the lack of updates recently.
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.
So, my MSc advisor, J
This term of teaching ends next week.
When I got back from T’bilisi, just over a month ago, I had research leads that I expect will end in three different publications.
I was slated with writing one LARP report for a swedish gaming magazine, and a series of various popular mathematics articles for the local student-run mathematics magazine here.
All in all, very many things converged this June/July for me.
It has started paying off though – the gaming article is published, and yesterday I submitted the first of the T’bilisi articles to the Journal of Homotopy and Related Structures as well as to the arXiv.
I now am listed on the arXiv with three papers, out of which one is already published, one is rejected (not unjustly so), and one is just submitted for review.
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 Kn, 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.
- December 30th, 2006
- 7:31 pm
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.
- 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 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:
- 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.
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 22nd, 2006
- 11:05 am
The term has started. In full force. No seating in the lunch cafeteria, lot’s of people all over the place, lot’s and lot’s of new students, and lectures and examples classes kicking off all over the place.
I’m leading an example class this year: linear algebra and geometry part 1 for the maths majors. One of six different examples class sessions for the same course. And apparently, my good tradition of going out drinking with my students keeps up: I went to the exchange students term-start party last friday, and while partying with the swedes and finns of the scandinavian Stammtisch on the dance floor, a girl squeezes through the crowds past us and asks me in passing if I’m not the examples class teacher. Turns out she registered for my class.
First contact with the students is on tuesday morning.
- 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.
- October 10th, 2006
- 12:22 pm
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.
- October 8th, 2006
- 11:08 am
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 Th
- September 11th, 2006
- 10:07 pm
I haven’t really been updating much here – and especially not the category Weekly Report. Slowly, it’s time to get around to it.
Now, there is a specific reason updates have been slow: I’ve been travelling. A lot. With very varying internet access and even more varying energy to spare for writing. It all started with two weeks of vacation in Sweden – spending time with my lovely fiancee, meeting old friends, and generally relaxing. She proposed (of sorts), and we’re getting married next summer. After the vacation, I went to Leeds, to the Triangulated Categories workshop, and then back home to Jena – only to go off again within just over a week, for the First Copenhagen Topology Conference, tightly followed by a master’s class in Morse theory lead by Ralph Cohen, and a simultaneous workshop on Morse theory and string topology.
- September 5th, 2006
- 1:24 pm
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.
Got treated today to a really nice workout in group cohomology; most of which is well worth sharing, since seeing it done once gave me a lot of insight.
So, if we pick and view it as the set 0,1,2,3,4,5,6,7,8,9 and with the group operation given by a*b = a+b % 10, then one standard 2-cocycle is the function
That this actually does form a cocycle would be the same as requiring
which is to say that the number of carry bits generated when adding three digits does not depend on associativity.
This cocycle classifies the group extension
with the first map taking and the second taking
Now, this is a nontrivial extension – which is equivalent to it not being a coboundary – by the following calculation:
Suppose f=dg. Then f(a,b)=g(a)+g(b)-g(a*b). So, since f(0,0)=0, we get g(0)-g(0)+g(0)=0, so g(0)=0. For any b≤8, we also get 0=f(1,b)=g(b)-g(b+1)+g(1), so g(b+1)=g(b)+g(1) and thus by induction, g(b)=bg(1) for all 0≤b≤9.
But, now, 1=f(1,9)=g(9)-g(0)+g(1)=10g(1)=0, which is a contradiction.
- 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!
- 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.
The weekly reports have been dead for a while. Reason? The blog has been dead for a while.
The old computer running this website had some problem all of a sudden about 3 weeks ago. These problems appeared as a complete lockdown of the system – no response to anything. So my brother – with me on the other side of a telephone, tried to reboot the box; but couldn’t get it back up online again. He was headed out to a LARP anyway within hours – and so couldn’t really do much more about it.
End result? I joined forces with a good friend of mine; we split hardware costs for a slick new box – an Asus barebone box with a 64bit processor and a gig of RAM. It received the harddrive and network interface from the old box, and was with that good to go – only .. processor architecture changed; and so for optimal performance, it’d be a nice idea to actually use a new system install that took advantage of the extra available bitwidth.
One predominant tendency in the algebra/category theory camp is to seek out the minimal set of conditions needed to be able to perform a certain technique, and then codifying this into a specific axiomatic system. Thus, you only need to verify the axioms later on in order to get everything else for free.
One such system is the theory of triangulated categories. This pops up in homological algebra; where you like to work with Tor and Ext – both of which turn out to be derived functors, generalizing the tensor product and the homomorphism set respectively. With the construction of the derived category, we can find a category, in which the tensor product in that category is our Tor, and the hom sets is our Ext.
It’s been a while since I managed to write one of these. The reason is simple enough – my weekends have been packed; and I don’t get around to it during the weeks.
During the last three weekends, first my parents and my brother, and then for the last two and the week inbetween my fiancée, have been visiting me in Jena. Thus, I have covered more ground in these three weeks when it comes to tourism than I probably will be able to do in several months. I have seen the Blue Man Group in Berlin (WOW!), I have seen the Dornburg, the Feengrotten and Weimar. I have eaten at the expensive luxurious restaurant at the top of the old university tower in Jena (it’s bloody scary, but quite cool – the restaurant is on the 29th floor; in a city where only one single house goes above 10 floors).
I just received in the mail a bunch of prints. Of my article “Computation of Poincaré-Betti series for monomial rings”, produced from my Master’s thesis for the “School and workshop on computational algebra for algebraic geometry and statistics” in Torino 2004. It is now being published in the Rendiconti di Istituto Matematico di Universita di Trieste, on pages 85-94 of Vol. XXXVII (2005).
Damn, it feels good. Reviewed and everything. If you’re curious, my manuscript can be found at http://math.su.se/~mik/torino.pdf or at the arXiv as math.AC/0502348.
This is the second weekend in a row spent to more or less large part in the office, working with the product structures on cohomology. Reason for this is that I’m getting my share of the department seminars now – I’m to walk us through the Yoneda cohomology product; the cohomology-as-Hom-in-a-derived-category viewpoint; their equivalence to one another as well as to the cup product; and then talk about restriction and corestriction (i.e. what happens to cohomology when we go between kG- and kH-modules for H a subgroup of G)
This is all not really very bad – I really, really, REALLY need to get a solid grip of this myself too. Only; when I started working on it, I thought I had 4 days and not 11 to prepare in – and dove right into it. Maybe a bit too deeply, so when I (monday) found out I didn’t need to get it done THAT quickly, I kinda dropped most of it for the rest of the week. And now, I need to find a decent proof that cup products = Yoneda products. And I just realized that my books don’t really cover it.
I’m back in Jena now. The last week was spent working myself to the brink of unconsciousness trying to grasp homotopical algebra, simplicial objects, model categories and any and all things Alex sent my way. With some 6-8 hours each day spent on lectures and discussions explicitly held to enable me to understand what was going on, I ended up being halfdead from the mental exercise.
In addition, since I was back in Sweden, outside lecture times was spent almost exclusively socializing in one way or another. Meeting friends. Shopping. Watching movies and spex. And then top it off with an endlessly long trip back.
I left Jena going to Stockholm on Saturday. Thus, much of the week past has been spent in preparation for the trip – reading up on homotopical algebra; getting all the paperwork together and getting my things together for the trip.
Along with “Make sure you learn homotopical algebra” and “Get back primed and ready to teach when you come”, I also was instructed by my advisor to get in touch with $MATHEMATICIAN, who currently resides at Mittag-Leffler and whom he knew from earlier. He is, I am told, very good, very knowledgeable and definitely a resource to be tapped if I should have any chance of it whatsoever.
I desperately, sincerely need to get a better cheap travel route to Jena. This trip now took me €150-170 somewhere, but had a travel time of more than 13 hours. There has GOT to be a better way to go.
Right. It’s thursday. And I had some sort of hopes to do my weekly reports on saturdays. Only, last saturday found me back in Nuremburg, in the middle of a marathon party-after-party session with the RPG crowd there.
Last week was very much characterized by getting various conditions for my being allowed to go to Sweden next week, and getting various bits and pieces of general paperwork in order. In addition to that, I held my first lesson – an examples class in Algebra. Right now, we’re doing modules: general definitions and then the structure theorem for finitely generated modules over a PID. I have already noticed for my self what has been painfully obvious from observing bloggers and friends whining about their students – it’s obvious as soon as you set foot in the classroom who knows what’s going on; and these are the only ones who will give you any sort of life indication. I already started despairing about getting reactions from the people not running up to speed – especially since these also don’t bother handing in any kind of work for me to correct either. End result: I have no idea if I’m doing any good for those who need me, and only get responses from those who don’t. The teacher’s lament.
My first week has passed. Today is saturday; and the move took place monday. So far, I’ve been running around doing bureaucracy and little else (I managed to leaf through the first 5 pages of Evens: Cohomology of Groups). Along the lines – I’ve received a summons to appear in front of the immigration authorities to explain my moving in, I’ve ran circles around the city trying to get someone to approve my swedish birth certificate et.c.
My apartment is small, neat and nice. It’s some 4×5 meters, with bed, bookcase, two tables, wardrobe, kitchenette, toilet with bath, balcony. And then all the things I brought with me – including a bookcase, three tables, computer, books-books-books, and much much more. I’ve gotten around to some interior decoration as well – putting up my swedish and my franconian flags on a wall. The endeffect is pretty – although I periodically have to remind myself that my putting up a swedish flag is no longer a sign of right-extremism but rather a sign of keeping in touch with home.
Now that my blog returns to its status of a PhDiary as I actually got a PhD position, I will introduce one flavour of regular postings. Instead of keeping in touch with people by mailing lists, livejournal, and everywhere else, there will be weekly postings here about life as a German PhD student.
So far, my entrance into German academics has had one feature above all else. Bureaucracy.
In order to even look at my contract, I needed to go, specifically therefore, to Jena, to fill out a questionnaire. This questionnaire is geared towards ascertaining that I am a good representant of the German state and its ideals. So, there are questions upon questions upon questions about my involvement with Stasi, my involvment with former DDR, whether I went to party schools, whether I’ve held party offices, et.c. et.c. Not to mention the centimeter-high stack of papers I got home to fill out on my own. With complete curriculum vitae from the age of 14. And Gods only know how many different obscure decisions to be made and forms to be filled in.