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:

Thesis written

• February 15th, 2008

In a mean push, these last two weeks my advisor has read three different drafts of my thesis. And I’ve worked on getting the corrections in quickly. The last push started yesterday, when I got a bunch of corrections in the morning, had the last draft ready at 4pm, and then sat reading it myself until 1am.

My advisor took it home with him, spent the evening on it, and had his batch of corrections in the morning.

Hence, today at 10-ish when I got myself in to the office, I had two batches of corrections in front of me, and a printer closing at 2pm. So I worked - and now, well, it’s done.

That’s it.

It’ll get printed.

Then read.

In May, we should get all the comments back from the external examiners.

Still away

• September 9th, 2007

However, I am enjoying the Scottish countryside and just - today - turned years of age.

Politics and absences

• August 4th, 2007

First off, Alexander Borovik has been writing a couple of times about a REALLY nice-sounding mathematical village in Turkey.

And it turns out, the village got closed this summer, with the government officials citing “education without permission” as their reason to close it.

Alexander is sending a petition to the prime minister. You should sign it.

In other news, I’m currently just waiting for Monday to come along. Why Monday? Because that’s the day I’m going back to Stockholm again. Once there, I’ll spend a couple of weeks spending time with friends and family, and then I will go and vow fidelity and those other things. The 25th of August, in case you’re about to ask.

All in all, this means that posting will be sparse if existent until mid-September - when I arrive, fresh out of my honeyforthnight, to Sydney; where the Magma research group host me for some 5-odd weeks. There I expect to have office space, an internet connection and a computer.

In-between times

• July 17th, 2007

These are the times that eat my productivity. The times that ensure that entire days go by and I afterwards feel nothing have happened at all. These times that are too short for productive work - where I know from the beginning that I cannot sit down and do something - too little time for reading, for coding, for writing, for .. well .. anything. And yet, while trying to get through them, they are obviously too long. An hour here, an hour there, interspersed with lunch, then coffee, then a seminar, and all of a sudden out of an 8 hour workday, the only vaguely productive thing that got done was hearing the seminar.

Fragmentation kills my productivity. With a fragmented workday, I have the time available neatly chopped up in pieces of free time that fall in-between. That are too short, but yet cover almost the entire workday.

Still no new blogging

• June 28th, 2007

Too harried to blog.

Will miss this carnival.

Bugger.

Response to Heath Raftery

• February 9th, 2007

While I’m still on the subject of writing code with the PFP library, I may as well join in on a discussion that got pulled into the Carnival of Mathematics exposition.

Heath Raftery writes about weird probabilities in dice discussions, a problem very much reminiscent of the Monty Hall problem, both in the amount of controversy it generates when people discuss it, and also, it seems on at least half the people in the discussion so far, in how much the terms use end up confusing people.

So, in a comment to the Carnival post, the suggestion came from Alex, that a simulation be used to settle the whole question. And since I just wrote some things with PFP, I might as well write another!

The question, as posed by Heath, is

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.

On religion

• January 27th, 2007

I decided on a whim to look in at the Dilbertblog, where the top post at the moment has Scott Adams calling all atheists that discuss on the net irrational, using a rather neat strawman carbon copy of most discussions of faith between believers (i.e. mostly Christians) and atheists he has seen on the web.

At the end of it comes a question: supposing it could be established that there’s a truly infinitesimal (in the physical sense, not the mathematical) chance of a God existing - he quotes “one in a trillion” as a model probability; would it then be fair of an atheist to claim the non-existence to be proven?

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:

Why Blog?

• November 16th, 2006

The Community College Dean has written about why he blogs, and asks any and all readers to tack on to his effort.

My blog is not very anonymous. It is occasionally personal, occasionally political and throughout a venting location for thoughts, and a place where I formulate myself in higher detail - so to speak a scratchpad, but public enough for me to allow others to read it.

I write it to formulate my own thoughts further, find possible errors, start discussions, or just jot down the viewpoints that illuminated some point of some argument for me. I do it in public because I thouroughly enjoy the conversations it sparks.

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.

Weekly Report: Parties and lectures

• October 22nd, 2006

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.

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.

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.

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.

OpenGL programming in Haskell, a tutorial (Part 2)

• September 19th, 2006

As we left off the last installment, we were just about capable to open up a window, and draw some basic things in it by giving coordinate lists to the command renderPrimitive. The programs we built suffered under a couple of very infringing and ugly restraints when we wrote them - for one, they weren’t really very modularized. The code would have been much clearer had we farmed out important subtasks on other modules. For another, we never even considered the fact that some manipulations would not necessarily be good to do on the entire picture.

Some modules

To deal with the first problem, let’s break apart our program a little bit, forming several more or less independent code files linked together to form a whole.

First off, HelloWorld.hs - containing a very generic program skeleton. We will use our module Bindings to setup everything else we might need, and tie them to the callbacks.

Hello, Planet Haskell

• September 14th, 2006

Since the content rate of haskell-related posts is going up, the feed of this blog will get added to Planet Haskell. Hi, Planet!

OpenGL programming in Haskell - a tutorial (part 1)

• September 14th, 2006

After having failed following the googled tutorial in HOpenGL programming, I thought I’d write down the steps I actually can get to work in a tutorial-like fashion. It may be a good idea to read this in parallell to the tutorial linked, since Panitz actually brings a lot of good explanations, even though his syntax isn’t up to speed with the latest HOpenGL at all points.

Hello World

First of all, we’ll want to load the OpenGL libraries, throw up a window, and generally get to grips with what needs to be done to get a program running at all.

This code throws up a window, with a given title. Nothing more happens, though. This is the skeleton that we’ll be building on to. Save it to HelloWorld.hs and compile it by running

ghc -package GLUT HelloWorld.hs -o HelloWorld


Monthly Report: Conference summer draws to an end

• September 11th, 2006

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.

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, 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.

Carry bits and group cohomology

• July 27th, 2006

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
f(b,c)-f(a*b,c)+f(a,b*c)-f(a,b)=0
or regrouped
f(a*b,c)+f(a,b)=f(a,b*c)+f(b,c)
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.

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.

Weekly Report: Back up again

• July 23rd, 2006

The weekly reports have been dead for a while. Reason? The blog has been dead for a while.

Hardware woes

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.

Right.

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.

Triangulated categories

• July 17th, 2006

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.

Group cohomology revisited: Quaternionic unit group

• June 20th, 2006

In a previous installment, we calculated with some amount of success. For that post, I said that I was going to calculate the cohomologies of and of by hand - and I’ve been at it for the latter group since then. With some help from my advisor - mainly with executing the obvious algorithms far enough that I get decent material to work with - I know have it.

The group

So, for starters, we need a presentation of such that we can work well with it. We all know that . So due to ij=-k and , we can just pick any two of the i,j,k and call them x and y. Then , and iji=ik=j so xyx=y. This gives us the presentation

Admin: Wordpress 2.0

• June 19th, 2006

This blog just migrated to WP 2.0. Should anything be odd, please notify me.

The migration was to a large portion motivated by commenting problems I’ve been told about. I hope that my readers out there will be able to comment now; possibly even without logging in!

Weekly Report: Power tourism

• June 18th, 2006

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).

More group cohomology

• June 8th, 2006

My advisor told me to go hit and 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 with , 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 and then peeked into Carlson, et.al. for the Big List of 2-group cohomologies to see that all interesting stuff happens in 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!

First academic publication!

• June 6th, 2006

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.

My first group cohomology - did I screw up?

• May 21st, 2006

I thought the seminar on tuesday would possibly benefit from something not very often seen - explicit examples. So I started working through one. I wanted to calculate and give explicitly in a series of ways the product structure - as Yoneda splices, as chain map compositions and as cup products.

Now, has a very nice resolution as a -module - all cyclic (finite) groups have canonically a really cute minimal resolution - given by

with the last map taking and the other maps alternatingly being multiplication with and with .

So this gives as a nice projective (in fact: free) resolution to work with. We now can observe that since any map has to respect the group action, which is trivial on , and so any map is determined by its value on 1. Thus we get the sequence of dual modules

Weekly Report: From 0 to 100 in 6 seconds flat!

• May 20th, 2006

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.

Weekly Report: Mathematics makes my ears bleed

• May 14th, 2006

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.

Weekly Report: Preparing to leave

• May 7th, 2006

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.

Weekly Report: Forgot all about it!

• May 4th, 2006

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.

Reading Merkulov: Differential geometry for an algebraist (4 in a series)

• April 24th, 2006

Suppose we have a presheaf of abelian groups over and pick a point . On the collection of all abelian groups defined over some neighbourhood of (disjoint union) we put an equivalence relation which identifies and precisely if there is some open in the intersection where and coincide. (or more precisely, their restrictions coincide). The set of equivalence classes turns out to be an Abelian group called the stalk of the presheaf at .

So, with more fluff introduced, the stalk is all the elements in the presheaf that are defined above any neighbourhood of the point, and counted as the same if they seem to be.

For an open set and a point there is a canonical group morphism which sends an element to its equivalence class. This image is the germ of at .

Weekly report: Settling in

• April 22nd, 2006

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.

Question for the mathematical audience

• April 20th, 2006

I have now been staring at this particular sentence for way too long, and thus will start using any and all communication lines I can find to get assistance. Either I’m being way too stupid, or the author neglects to mention some salient detail.

Setup: is a group homomorphism, , . can be given the structure of a -module by pulling back through , i.e. we define for and .

So far it’s all crystal clear for me. However, it then turns out that we’re highly interested in using a morphism and I cannot for the life of me find out how such beasts are guaranteed to exist. If it where , I wouldn’t have any problems with it; but then the stuff I need/want to do with it don’t work out.

Weekly posts - a new feature at this blog

• April 15th, 2006

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.

Why I keep organizing congresses

• April 5th, 2006

Once upon a time, I wasn’t passionate about mathematics. Up to grade 6, I even disliked it quite a bit - it consisted of only mechanical plugging away of numbers, and training of multiplication tables that I had the feeling I already mastered.

Then something changed. Subtly at first - in grade 7, it started to gain texture, it got beyond the rote calculations ever so slightly. And so I started devouring the old popular mathematics texts my father kept in his bookcases. Soon, I stumbled across a new word - “integral calculus” - and of course asked my father to explain it. And thus it was that I, at the age of 13, got introduced to limits, derivatives and integrals.

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.

Report from Villars (5 in a series)

• March 12th, 2006

For the last two half-days of the conference, I managed to take a break in skiing precisely when the conditions were at their very worst; with sight down to a few meters and angry winds. Miles Gould and Arne Weiner, however, managed to sit in a chair lift that kept stopping every 5 meters - AND they managed to break a T-bar lift. Suddenly the rope broke, they told me, and they had to ski down to the warden with the T-bar in the hand.

First out in this mathematical expose, though, is André Henriques, talking about

An operad coming from representation theory

There is a way to connect to a finite Lie algebra first it’s universal enveloping algebra and quantum groups . From representations of , one path leads on over braided tensor products to braided tensor categories. Such categories are described by operads, which occur in the study of Gerstenhaber algebras and their homology.

Report from Villars (4 in a series)

• March 8th, 2006

I haven’t been able to get around to skiing since the last update - I may, or may not, go out in the slopes after this updates. The weather is growing warmer and wetter; and doesn’t really invite to skiing as it previously did.

However, we have had more talks. First out, yesterday evening, was Pascal Lambrechts

Coformality of the little ball operad and rational homotopy types of spaces of long knots

The theme of interest for this talk was long knots; i.e. embeddings of into such that outside some finite region in the middle, the embedding agrees with the trivial embedding . The space of all such is denote , and the item of study is more precisely the rational homology and rational homotopy of the fiber of the inclusion of into the space of all immersions of into .