The Lazy Science

The end is near

2010-04-15T03:22:00.000+02:00

Well, as promised, the end is near.

I have begun moving thelazyscience. The future location will be at peter.krautzberger.info. It'll probably take me a few more days to (manually...) move the old posts over to the new site. However, the next post is almost finished. I still have to see if I can find a mechanism to automagically put a short post up here for each new one there, but we'll see.

It was good to have started the blog at blogspot but hopefully the new workflow (and more time) will lead to more frequent posts.

Research Notes -- Welcome to Ann Arbor

2010-03-18T22:29:00.000+01:00

After too many weeks of silence and almost two weeks in the States it is definitely time for an update.

The first 10 days at the University of Michigan were taken up mostly by bureaucracy but some old and some new ideas have evolved out of my first two meetings with Andreas Blass.

On the one hand he has renewed his interest in the forcing construction of a non-Ramsey ultrafilter that Claude Laflamme did under his supervision back in the 80s. In fact (spoiler alert) Andreas will talk about such things at the ASL Meeting in Washington this week.

On my related part I have resumed work on constructing an unstable union ultrafilter which Andreas and I once had hoped to relate to the Laflamme forcings but later turned out not to. Mostly I spent time recovering my old strategies and had lots of them shot down by Andreas. Nevertheless I see progress there and look forward to spending more time on the issue (and hopefully blog about it, too).

Blog related the great change is finally coming up. I will switch to blogofile, a blog compiler for static blogging, after I find a good place to host the new blog. I plan to switch to MathJax for better MathML support and use Disqus for comments. This will simplify my publishing process and I hope it will lead to more frequent posts. After all I have not done enough writing so that it comes naturally to me.

Other lazy people

2010-02-15T20:11:00.000+01:00

I frequently wind up with a list of interesting blog posts -- so why not include this here, too? As long as I cannot convince myself to use google reader or buzz, I'll do this the classical way (especially since I just read a complaint that less and less blogers use backlinks).

An old but very nice post on Graeme Taylor's Modulo Errors -- with beautiful linkage. I just spent 10 minutes cheering the algorithm for vehicles to succeed...

http://maths.straylight.co.uk/archives/125

Gil Kallai continues his series of posts on the concept of probability with a wonderful video by Itamar Pitowski

http://gilkalai.wordpress.com/2010/02/15/itamar-pitowski-probability-in-physics-where-does-it-come-from/
For the German speaking people -- via Christian Reinboth a nice (I'm guessing Austrian science department) video promoting mathematics.

Enjoy.

Tools for your online collaboration

2010-02-14T18:31:00.004+01:00

So the winter school in Hejnice ~~ended two weeks ago~~ is long past -- and despite my intentions I did not find the time to blog. This is primarily a sign of the quality of the winter school, both scientifically and socially. I do admit I spent the lunch breaks walking in the beautiful surrounding mountains instead of blogging...

Anyway, on the last evening of the winter school a couple of people gathered together to exchange tools for collaborating via the intertubes. I volunteered -- also with the upcoming third Young Set Theorists meeting in mind -- to make the discussion available online. Of course, the title refers to this wonderful paper by Goldstern and Judah which taught me the little bit of iterated forcing that I know.

For now I will restrict myself to freemium services. Of course, this is an open list -- drop me a comment to add to this list (hm, a google wave would be better, right?).

Phones

A much better tool than a phone is? A videophone! (especially for handwaving arguments). Namely, skype comes to mind, but there are alternatives like tokbox or google talk which are web based. With possibly lower video quality they offer other useful things like actual video conferences (whereas skype restricts you afaik to 1-1 video calls) and invitation by link. There are also numerous true VoIP/SIP clients like Ekiga. But they may have the need for some firewall configuring. For more general information, check out wikipedia.

Whiteboards

But what good is a (video)phone if you cannot write on a blackboard together? In any serious mathematical discussion, notation will become an issue sooner or later. A simple, but bandwidth friendly and flash based whiteboard is scriblink -- just go to the site and give your partner the invitation link. An alternative is dabbleboard which offers some shape recognition and also allows multiple pages in the free version and -- most importantly -- PDFs as background images. However, it is a little heavy on the bandwidth, especially latency which often annoys my voip connection.

Of course, if you want to use an online whiteboard efficiently you need some kind of tablet to write with. I personally have been very happy with a graphics tablet, a Wacom Bamboo to be exact. You can get tablets for 40€ and lower in Germany, but prices will differ regionally. Of course, I also use my Gigabyte M1028T tablet pc -- although its tablet functionality is basic (no pressure sensitivity, only moving by clicking) making writing with it less suitable for real note taking -- see the PDF section below.

~~Eierlegende Wollmilchsau~~ Swiss Army Knives

There are of course those services which offer all of the above at once. A prime example would be dimdim which offers a nice, unified service including video conferencing, instant messaging, whiteboard, pdf viewing and collaborative websurfing -- all of this available with a free account which is limited only in the number of participants (and there are premium services available, of course). Additionally dimdim's server technology is mostly open source, so you can set up your own server if you have the means. Unfortunately, I never got the video conference system to work correctly under linux. Although not quite with collaboration in mind there is also the awesome TeamViewer. It is a great remote assistance tool designed for efficient access to another computer screen. In that sense you could use it to access your home or office machine from anywhere -- if your department allows that. But in the latest version (although windows only) Teamviewer also offers Video chat and a whiteboard to communicate. For further tools look here.

Instant Chat, Online Docs and Google Wave

Personally, I have not used instant messaging for mathematics so far -- video phones seem better. However, Pidgin has a LaTeX plugin to display basic TeX code. This is of course a useful feature. I'll come back to the general problem of displaying mathematics on the web later.

I feel I must also mention Google Wave and its competitors . These are powerful tool mixing mail, chat, wikis and collaborative document editing. I have not tried any of these yet but if there's someone to collaborate with it's worth a try.

PDFs I -- what you can do with them

PDFs is the somewhat dominant standard for (compiled) TeX documents (sorry, dvi and ps fans). Besides the next section there is another aspect which makes them worthwhile -- PDF annotation. If you are like me and like to take your notes with you (for all those typos and indices that drive you mad in some papers) there is nothing better than annotating a PDF directly -- especially if you invested in a (graphics) tablet.
My favourite is the open source Xournal with excellent tablet support on both linux and windows. Alternatives are Jarnal (which also works as real time whiteboard) and (for Mac users) Skim.

Although it does not quite fit in here (or anywhere): if you feel that PDFs are inadequate to present mathematics, why don't you take a look at prezi? It offers a different angle on presentations altogether. I sometimes dream of having a prezi like ability to zoom into papers or rather proofs giving me details where I want them and letting me quickly browse through the main ideas dynamically whenever I choose to...

PDFs II -- Personal online libraries

It is convenient to store papers and other materials online. If you cannot set up a decent sftp or a version control system on your university's server, you might want to try dropbox or teamdrive. If you frequently use public computers you might want to use something more web based like google documents or the very pretty isssu that I use from time to time on this blog.

Community Sites

Of course, all science is community driven but I think (pure) mathematics could profit more from an online community than any other science or (liberal) art. The biggest player is certainly facebook -- which already has a group for, of course, the winterschool itself. Facebook attracts academia (as opposed to myspace), hence it is the more obvious place to connect -- this does not mean that you shouldn't worry about its privacy settings or rather the partial lack thereof.

On the other hand, there are a couple of science focused community sites, among them researchgate which offer science specific tools like (p)reprint lists, online references, database searches etc. This might be better for purely professional intent but I have no experience using it.

A young and incredibly successful new site is mathoverflow -- a mathematical version of the great stackoverflow. You can ask and answer questions of all sorts in a very efficient manner -- just don't get lost in all the fun.

Databases

Of course the mother of all things is the arXiv -- do I need to explain it? And then there are Google's products scholar and book search. A somewhat different database is gigapedia where you can easily search for books and find free ones. In all things beware of legal issues though.

LaTeX or displaying mathematics on the web

Of course mathematicians are used to LaTeX. On the web the best way for displaying mathematics is (from a web standards point of view) mathml. The problem is that mathml is a) too difficult to write as code directly, b) difficult to view since not all browsers view them correctly and from a visually impaired point of view it seems to be a disaster, too (see the discussion on Terry Tao's blog) and c) it is difficult to convert back to LaTeX.

There are numerous workarounds. On the one hand you can (as I do) use tex4ht to convert LaTeX to mathml. Of course, as my blog shows this is a rather tedious thing if you do not have (or want to have) control over the webserver. Alternatives are jsMath which might be superseded by mathjax. If you have a wordpress blog you can (even on your free account on wordpress.com) use this plugin -- which converts basic LaTeX commands into (rather ugly) PNGs.

The winner for best practices with mathml, I think, is the n-Category Cafe. Besides being a very active group blog they have developed impressive technologies such as mathml inclusion, the LaTeX dialect itex, the itex capable instiki with itex2mml to convert tex to mathml on the fly and all of this available in the comments, too.

Blogs, blogs, blogs

Almost last but in no way least, there are blogs. This would be worth an independent post and there are plenty of examples for this, but here we go.

They come in all colours, for an impressive list go here. Also, go to any of those blogs and check their blogroll to find many more mathematics blogs. If you don't understand what blogs are good for you might read John Baez's article. To name a few contenders for 'most influential mathematical blogs': What's new with Terence Tao, the most active single user blog I know, Timothy Gowers's Weblog and Gil Kallai's Combinatorics and more.

Of course, they are the ones that got me started with reading math blogs, but it's the small blogs that got me hooked. The diversity is a challenge (I don't understand half of what I read) but blogs form the best mathematics newspaper out there.

Polymath

At the moment the most hardcore project when it comes to online collaboration is clearly Polymath. With one paper on the arxiv, two projects finished and three projects going it is the perfect show case. Driven by the "big three" -- Tao, Gowers, Kallai -- one may argue that their power makes sure that it works (and is protected from theft). Polymath is an exemplary web project. It follows Jeff Jarvis's rule and shows the synergetic behaviour of web projects -- using multiple technologies at once: there's the blog for the main discussion, but also the authors individual blogs used partly to organize. Finally there's the wiki for fixing proper definitions and notational issues and finally they frequenly use mathoverflow to recruit new people by e.g. singling out distinct partial or dervitative questions.

But I believe it shows a glimpse of the future of mathematics. On the one hand, many problems have become too complex to be tackled by a single person or research group. On the other hand, although the techology might change considerably in the future, the idea of having researchers on all levels collaborate -- with every contribution being valued -- could be a prototype that values many soft skills, be it good writing, accessible presentation, social skills for bringing conversations to converge productively, taking a bird's view of the process to assist or acquiring empirical experimentation and implementation. It is also a very flexible approach where people can help as much or as little as they find the time for while (with proper support like Gower's current EDP posts) still being able to follow the flow and ideally being able to change their level of involvement as they please.

That's all for now. Let me know what I forgot.

Addenda

2010-02-15

Unicode characters

There was also a question regarding unicode characters and the like (instead of mathml). I just found this chart via mathoverflow -- maybe it helps.

2010-02-17

Feeds and feed readers

Feeds in either Real simple syndication (RSS) or Atom from are worth mentioning on its own. As a tool for 1-to-infinity communication it's an important technology for collaboration. You'll find feeds for all kinds of newssites and blogs, but also for each section of the arxiv. To read feeds you can use lots of different programs and web based services.

Video sites

Videos of research level mathematics are pretty rare. There is the archive of the MSRI and singular popular mathematics gems like Gowers's talk on multiplication. Also, you should check out MIT's impressive youtube channel.

To put up a video you don't need much these days, so it's strange that there's not more around -- especially since (pure) mathematics seems easier to share than, say, complicated science experiments. There are too many free video sites out there. Next to the already mentioned youtube I would point out the science video site SciVee (with its strong, yet somewhat expensive premium service) and Vimeo with its focus on original content.

Reference management

Thanks to David for reminding me that I forgot one aspect of pdf management -- reference management (see the list on wikipedia). Now there are many programs out there to get your citations, i.e., your BibTeX files organized. But there are also programs that connect the citations with the pdf, offer online database searches, tags, pdf annotation and social networking ideas.

A big list can (once again) be found on wikipedia. To present a few. I personally use referencer but David also mentioned Mendeley in his comment which has an impressive list of features including online access and social network aspects and I'll probably try it out. To give credit where it is due a few of these programs name Papers as inspiration which unfortunately is Mac only. With a different flavour there are the web-only Zotero, a powerful Firefox addon, and I, Librarian, a groupware tool.

"What is...?" Seminar -- new videos

2010-01-28T10:27:00.000+01:00

One of the most valuable experiences during my time as a PhD student lay in helping to establish a "What is...?" seminar at the Freie Universität Berlin and later/now at the Berlin Mathematical School.

I originally came into contact with the concept while visiting the University of Michigan in the winter 2007/2008. However, back in Berlin I wanted to use the theme for a different purpose. In conversations with a couple of friends we developed the idea to create a seminar by PhD students for PhD students.

This idea became central since the regular colloquium never attracted PhD students nor did the PhD students ever gather together (which thankfully now changes with the BMS). In particular, we were looking for something with a more open atmosphere.

Looking at Harvard University's experience with (from what I have been told) first having a "Basic Notions" seminar the non-trivial nature of which lead the students to compensate with a "Trivial Notions" seminar, we decided to exclude professors at first. This in fact got us some really negative responses when we sent out emails looking for all the PhD students hidden in workgroups outside our own fields (one professor in particular simply could not fathom that the presence of your "boss" might hinder a free discussion). It was rather shocking that even professors actively popularizing mathematics simply reacted with "these things only last as long as a single person is behind them" (and this was before we even started -- talk about support...).

Nevertheless, the seminar got on its way. The first semester was tough, with lots of, shall we say, "experiments", trying to find our way (and above all speakers from other fields). In the second semester a PhD student from the BMS joined us with the idea of making the seminar as part of the biweekly BMS Friday. This semester has seen yet another expansion with some talks taking place at the BMS lounge at the Technische Universität Berlin.

Since I'm now leaving Berlin it has been a pleasure to see the next generation take over. However -- and this was the whole point of the post before this melancholic rambling took over -- I still am involved in making video recordings of the talks available whenever possible. I want to stress how much I am indebted to the speaker for allowing the publication of their talks. This is especially important since the videos are sometimes not very good (see my own soon to be put up and very bad talk about topological dynamics). The point is that the seminar is a platform to experiment and test oneself which is something that students of mathematics do not get to do a lot. Therefore I think we can be very happy that so many speakers are ready to put themselves out there and learn from the experience.

Anyway, yesterday I published two more videos, Carsten Schultz's "What is Morse theory?" and Inna Lukyanenko's "What is a quantum group?". The good user experience of vimeo might lead to all of the videos eventually appearing there, but so far Inna's video is the first on vimeo and the rest is on SciVee (but another one might end up on vimeo next week, we'll see...).

Good news, bad news

2010-01-21T13:51:00.000+01:00

Just so that not another week ends without me writing a post. The bad news is that my departure for Michigan gets closer and the technicalities take up more and more time. Therefore I'm not sure I'll have much time to post in the next couple of weeks. Additionally, I'll be attending a winter school in Hejnice in the first week of February so I also need to ~~prepare~~ finish preparing my talk.

So what's the good news? Well, I have been busy on the blog but nothing has come of it yet. On the one hand I have been studying the Google App Engine so as to move the blog there -- which should make the work flow much more efficient (and the code better). On the other hand that there are three blog posts I have not finished -- so there's a chance the dry spell will be over sooner than I think. Finally, I hope to write posts during the winter school reflecting on the (possibly daily) experience.

Well, let me at least throw in some nice links worth a read. Gil Kallai turned a mathoverflow question into the kind of blog posts I really like. Over at the n-Category Cafe David Corfield ~~explains~~ muses over the "sacred" and the "profane" in mathematics (or rather for mathematicians) which made me ponder what my own "bottom line" is.

Matrices vs. idempotent ultrafilters, part 2.5

2010-01-11T14:22:00.007+01:00

Note: there seems to be some problematic interaction between the javascripts I use and blogspot's javascripts which prevents longer posts from being displayed correctly. As long as I don't understand how to fix this, I will simply split the posts.

We can also describe size and the algebraic structure.

$A$ with $F_{1}$ ( $F_{2}$ ) generates a right (left) zero semigroup (hence of size $2$ , except for $x = 0$ ).
$A$ with $F_{3}$ or $F_{4}$ generates a semigroup with $A B$ nilpotent (of size $4$ , except for $x = 0$ , where we have the null semigroup of size $3$ ).
$A$ with $G_{i}$ generate (isomorphic) semigroups of size $8$ . These contain two disjoint right ideals, two disjoint left ideals generated by $A$ and $B$ respectively.

Luckily enough, we get something very similar from our alternative for $A$ .

Proposition 0.0.4 In case $A = (\begin{matrix} 1 & 1 \\ 0 & 0 \end{matrix})$ the solutions for $B$ being of rank one consist of five one-dimensional families namely (for $x \in ℚ$ )

H_{1} (x) = (\begin{matrix} 1 & x \\ 0 & 0 \end{matrix}), H_{2} (x) = (\begin{matrix} x + 1 & x \\ - x - 1 & - x \end{matrix}),

H_{3} (x) = (\begin{matrix} 0 & x \\ 0 & 1 \end{matrix}), H_{4} (x) = (\begin{matrix} - x + 1 & - x + 1 \\ x & x \end{matrix}),

H_{5} (x) = (\begin{matrix} - x + 1 & - x - 1 - \frac{2}{x - 2} \\ x - 2 & x \end{matrix}), x \neq 2 .

As before we can describe size and structure.

$A$ with $H_{1}$ ( $H_{2}$ ) generates a right (left) zero semigroup (as before).
$A$ with $H_{3}$ or $H_{4}$ generates a semigroup with $A B$ nilpotent (as before).
$A$ with $H_{5}$ generates the same $8$ element semigroup (as before).

Finally, it might be worthwhile to mention that the seemingly missing copies of the $8$ element semigroup are alsodealt with; e.g. $- G_{i}$ generatesthe same semigroup as $G_{i}$ etc.

At first sight it seems strange that we cannot find other semigroups with two generators like this. As another friend commented, there’s just not enough space in the plane. I would love to get some geometric idea of what’s happening since my intuition is very poor. But that's all for today.

Matrices vs idempotent ultrafilters, part 2

2010-01-07T11:34:00.023+01:00

In an earlier post I gave a short introduction to an interesting finite semigroup. This semigroup could befound in the

2 \times 2

matrices over

ℚ

When I met with said friend, one natural question came up: what other semigroups canwe find this way?

The first few simple observations we made were

Remark

If either $A$ or $B$ is the identity matrix $I_{2}$ or the zero matrix $0_{2}$ the resulting semigroup will contain two elements with an identity or a zero element respectively.
In general, we can always add $I_{2}$ or $0_{2}$ to the semigroup generated by $A$ and $B$ and obtain a possibly larger one.
$A, B$ generate a finite semigroup iff $A B$ is of finite order (in the sense that the set of its powers is finite).
$A B$ has finite order iff its (nonvanishing) eigenvalue is $+- 1$ .
For $A$ of rank $1$ we may assume (by base change) that $A$ is one of the two matrices $(\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}), (\begin{matrix} 1 & 1 \\ 0 & 0 \end{matrix}) .$

So, as a first approach we thought about the following question.

Question If we take $A$ to be one of the above, what kind of options do we have for $B$ ,i.e., if $B$ is idempotent and $A, B$ to generate a finite semigroup.

Thinking about the problem a little and experimenting with Macaulay 2 we ended up withthe following classification

Proposition For $A = (\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix})$ thesolutions for $B$ being of rank one consist of four one-dimensional families, namely (for $x \in ℚ$ )

\begin{array}{l} F_{1} (x) = (\begin{matrix} 1 & x \\ 0 & 0 \end{matrix}), F_{2} (x) = (\begin{matrix} 1 & 0 \\ x & 0 \end{matrix}), \end{array}

\begin{array}{l} F_{3} (x) = (\begin{matrix} 0 & x \\ 0 & 1 \end{matrix}), F_{4} (x) = (\begin{matrix} 0 & 0 \\ x & 1 \end{matrix}) . \end{array}

Additionally, we have four special solutions

\begin{array}{l} G_{1} = (\begin{matrix} - 1 & 1 \\ - 2 & 2 \end{matrix}), G_{2} = (\begin{matrix} - 1 & - 1 \\ 2 & 2 \end{matrix}), \end{array}

\begin{array}{l} G_{3} = (\begin{matrix} - 1 & 2 \\ - 1 & 2 \end{matrix}), G_{4} = (\begin{matrix} - 1 & - 2 \\ 1 & 2 \end{matrix}) . \end{array}

Note: due to technical problems, this post continues here.

We can also describe size and the algebraic structure.

$A$ with $F_{1}$ ( $F_{2}$ ) generates a right (left) zero semigroup (hence of size $2$ , except for $x = 0$ ).
$A$ with $F_{3}$ or $F_{4}$ generates a semigroup with $A B$ nilpotent (of size $4$ , except for $x = 0$ , where we have the null semigroup of size $3$ ).
$A$ with $G_{i}$ generate (isomorphic) semigroups of size $8$ . These contain two disjoint right ideals, two disjoint left ideals generated by $A$ and $B$ respectively.

Luckily enough, we get something very similar from our alternative for $A$ .

PropositionIn case $A = (\begin{matrix} 1 & 1 \\ 0 & 0 \end{matrix})$ thesolutions for $B$ being of rank one consist of five one-dimensional families namely (for $x \in ℚ$ )

\begin{array}{l} H_{1} (x) = (\begin{matrix} 1 & x \\ 0 & 0 \end{matrix}), H_{2} (x) = (\begin{matrix} x + 1 & x \\ - x - 1 & - x \end{matrix}), \\ H_{3} (x) = (\begin{matrix} 0 & x \\ 0 & 1 \end{matrix}), H_{4} (x) = (\begin{matrix} - x + 1 & - x + 1 \\ x & x \end{matrix}), \\ H_{5} (x) = (\begin{matrix} - x + 1 & - x - 1 - \frac{2}{x - 2} \\ x - 2 & x \end{matrix}), x \neq 2 . \end{array}

As before we can describe size and structure.

$A$ with $H_{1}$ ( $H_{2}$ ) generates a right (left) zero semigroup (as before).
$A$ with $H_{3}$ or $H_{4}$ generates a semigroup with $A B$ nilpotent (as before).
$A$ with $H_{5}$ generates the same $8$ element semigroup (as before).

Finally, it might be worthwhile to mention that the seemingly missing copies of the $8$ element semigroup are alsodealt with; e.g. $- G_{i}$ generatesthe same semigroup as $G_{i}$ etc.

-->

Workin on the work flow

2009-12-22T18:21:00.000+01:00

Since I do want to try to post something once a week, I'll recap my work towards a reasonable workflow. As I hinted at before, I'm currently aiming at a workflow of LaTeX via tex4ht to mathml to blogspot. I have encountered a couple of problems, some of which I could overcome, some of which I could not and some of them turned out to be bugs or missing features.

The main problem is that my LaTeX style is too complicated for tex4ht. Now obviously this is somewhat my fault and I should (and will) simplify and update my current style of typesetting. But I intend to experiment here and it would be much easier (and possible in a more consistent fashion) if I could "just" convert my LaTeX experiments to blog experiments. In any case, it bugs me whenever I cannot find the reason why some things fail but not others, so let me recount what I could figure out.

One problem that I could find an explanation for is the SVG generation of TikZ code via tex4ht. In particular this combination cannot handle more complex text (mathematics or not) inside an svg, e.g., minipages within tikZ pictures are simply ignored entirely. Admittedly, this is documented in a short paragraph in the TikZ manual but it took me a while to find that...

Speaking of manuals. I finally stumbled upon this part of the tex4ht manual where some of the common problems and restrictions of tex4ht are described. This helped a lot with a couple of small errors like the missing \bigcap in the Matrices vs Idempotents post. Most importantly, I now know where to look these things up, so maybe there's hope for me after all.

To make up for this and as a generally easy part of the workflow I added the first PDF via the very nice issuu. This is really no extra effort since issuu offers the code for embedding in blogspot -- and I really like the viewer (as opposed to scribd).

For the future I like the idea to include audio and video as well, especially after the accessibility discussion on What's new. It seems simple enough these days to upload some flash based or html5 based audio and video based on a post and I hope to find time to experiment with such media. Up until now I used SciVee a bit for a seminar. But since they went commercial a while ago I have to see if scivee's plans are reasonable given the competition -- they seem to focus more on large entities like universities and buying bulk accounts (open access like).

All in all, it's going alright, even though the flow is far from what I want it to be. Let's see what the holidays will lead to.

Matrices vs. idempotent ultrafilters

2009-12-15T16:48:00.052+01:00

Note: as you can see I am not yet in control of how to convert LaTeX to mathml -- bear with me, but I thought I should kick myself and start posting...

The other day I was looking for someone to chat about an interesting example of ﬁnite semigroups. So ~~yesterday~~ last week I finally met up with a friend who offered to do just that. The ‘results’ of the morning we spent chatting are perfect blogging material: quite simple, mostly elementary, easily open for discussion and still carry some interest. However, it is much too long, so I'll split it into a series of posts.

So to start: what’s the example?

Example

The matrices

A = (\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}), B = (\begin{matrix} - 1 & - 2 \\ 1 & 2 \end{matrix})

generate an 8-element(multiplicative) subsemigroup. Its elements are A,B,AB,BA,ABA=-A,BAB=-B,ABAB=-AB,BABA=-BA.

So what? Well, what is interesting is that although both A and B are idempotent (i.e. A ⋅ A = A,B ⋅ B = B), their product is not, since

A B = (\begin{matrix} - 1 & - 2 \\ 0 & 0 \end{matrix}), A B \cdot A B = (\begin{matrix} 1 & 2 \\ 0 & 0 \end{matrix}) = - A B

Still, why is it interesting? Well, this example is of interest for people working with ultraﬁlters on semigroups,in particular on ℕ– one reason following from the following lemma.

Lemma Every ﬁnite (discrete) semigroup is the image of the closed subsemigroup

ℍ : = ⋂_{n \in ℕ} \bar{2^{n} ℕ}

under a continuous homomorphism

This result can be found as Corollary 6.5 in the book ‘Algebra in the Stone-Čech compactiﬁcation’ by Neil Hindman and Dona Strauss. What can we do with this?

Corollary There are idempotent elements in βℕ whose sum is not idempotent.

Proof.
Step 1: Consider the (discrete) ﬁnite semigroup generated by A and B.
Step 2: By the previous lemma, it is a continuous, homomorphic image of ℍ.
Step 3: The preimage of both A and B is a closed (by continuity) semigroup (by homomorphy) of βℕ.
Step 4: Conversely, the preimage of AB cannot contain an idempotent (or else the image of that idempotent, AB,would be idempotent by homomorphy).
Step 5: In particular, by the Ellis-Numakura Lemma, both preimages contain idempotents, say a,b ∈ βℕ.
Step 6: But ab is in the preimage of AB, hence not idempotent.

One can easily see more, i.e., a,b can even be minimal idempotents and their product is not even in the closure of idempotents,but let’s leave it at that.

Now of course one can look at other ﬁnite semigroups abstractly. But the advantageof matrix representations is that it puts some ﬂesh to the bones of abstraction. In the next post I will write about possible generalizations of this example of matrices.

Update: since fighting with mathml is tough for the time being, here is something to make up for that.

Gedankenfetzen: Nach einer Diskussion zum Lehramtsstudium

2009-12-14T23:53:00.002+01:00

I had originally planned to open this blog with a scientific post -- but what can you do. Since this post is about universities in Germany this'll be in German.

Ein paar Gedankenfetzen nach einer Diskussionsveranstaltung zum Lehramtsstudium Mathematik an der FU Berlin. Die einladende Email:

Liebe Studierende mit dem Ziel "Schule",

haben Sie manchmal das Gefühl, dass zu viel in Ihren Studiengang gepackt ist, dass Sie nicht unbedingt die richtigen Sachen für Ihren späteren Beruf lernen, dass Sie sich das Studium überhaupt irgendwie anders gedacht haben?

Wirklich ist auch unter Hochschullehrern die Meinung verbreitet, dass in der Lehrerausbildung manches verbesserungsbedürftig ist. Um einen Meinungsaustausch in Gang zu bringen und - vielleicht - Änderungen zum Positiven zu erreichen, soll es

am 14. 12. 2009 (Montag)
ab 16.15 Uhr
im großen Hörsaal der Informatik
eine Diskussionsveranstaltung
"Mathematikausbildung für angehende Lehrerinnen und Lehrer"

geben.

Als Fachleute werden dabei sein: Dr. Deiser, Prof. Lutz-Westphal, Prof. Schulz.

Alle Fachbereichsangehörigen sind herzlich eingeladen.

Mit freundlichen Grüßen

E. Behrends

Anwesend waren, na vielleicht 40 Studenten und eine handvoll Dozenten. Vermutlich litt die Veranstaltung also daran, dass nur die Studenten mit den größten Problemen sowie die Dozenten, die zufällig gerade Anfängervorlesungen halten, anwesend waren, aber genauer erschloss es sich nicht. Aber zu meinen Eindrücken.

Die anwesenden Dozenten/Professoren konnten bei den Studenten mit Einzelinitiativen punkten; seien es ergänzende Verstanstaltungen, um Lehramststudenten zusätzlich zu helfen, seien es didaktische Experimente. Trotzdem klang das nach dem sprichwörtlichen Tropfen auf den heißen Stein. Gerade die Unterschiede zum Engagement viele anderer Dozenten und die Willkür der vermeintlichen Freiheit der Forschung führen in der Lehre wohl eher dazu, dass solche engagierten Dozenten viel zu viel Kraft verschwenden, dieselben Widerstände jedesmal aufs Neue zu überwinden.

Dabei stieß eine Bemerkung des gastgebenden Prof. Behrends kaum aufReaktionen. Auf die scheinbar erfolgreichen Versuch seiner KolleginLutz-Westphal, mehr didaktische Inhalte für Lehramtsstudenten in dieVeranstaltungen einzubringen, reagierte er mit der Feststellung, dassdies ja fachlich für andere Dozenten nicht machbar wäre. Warum eigentlich? Ist es zuviel verlangt, dass sich Dozenten eine hohschuldidaktische Bildung aneignen? Es scheint jedenfalls so; Professionalisierung ist ein Fremdwort an deutschen Hochschulen. Sogar eine Vereinbarung unter den Dozenten, "best practices" o.ä. auszutauschen und gar durch persönliche Absprache innerhalb des Fachbereichs (Kollaboration!) verpflichtend zu machen, scheint undenkbar -- vor allem wohl, weil ein solches Engagement in der deutschen Wissenschaftslandschaft in keiner Weise honoriert wird; nur die Publikationsliste bringt Geld ein, sonst nichts.

Zur Diskussion um die Vorlesungsgestaltung stellte sich mir nachträglich eine grundlegende Frage. Es wurde breit diskutiert, ob und wie die "normalen" Veranstaltungen für überlastete Lehramtsstudenten ergänzt werden können, durch z.B. didaktische Übungsaufgaben oder einfach leichtere Prüfungsbedingungen. Dabei stellt sich doch eigentlich die umgekehrte Frage: Warum richtet man das Niveau der Veranstaltungen nicht an den Lehramtsstudenten aus und bietet Ergänzungen für Mono-bachelor an? Dies könnten zusätzliche Veranstaltungen mit zusätzlichem Stoff sein, dies könnte Projektarbeit bedeuten, oder es könnten mehr und schwerere Aufgaben in extra Schwerpunktvorlesungen sein. Ich vermute, dass die meisten deutschen Hochschuldozenten viel eher dafür zu begeistern wären, schwereren Stoff gesondert zu vermitteln als leichteren (oder gar "Nachhilfe" zu geben).

Der Großteil der Diskussionsbeiträge durch Studenten (die viel sagten und sogar widersprachen) wirkte jedoch gefangen im Netz der schlechten Umsetzung des Bolognaprozesses. Verzweifelt wehren sie sich, versuchen, hier und da kleine Verbesserungen vorzuschlagen, drehen sich um sich selbst, ohne zu bemerken, dass es kein Herauswinden gibt. Die Probleme bilden eher den Gordischen Knoten, der durchschlagen werden muss. In fast jedem Beitrag zur Sinnhaftigkeit des Studienstoffs wurde klar, was eigentlich jeder weiß: das eigentliche Ziel der Bachelor- und Masterumstellung wurde regelrecht boykottiert -- die Einrichtung eines originär neuen Studiengang der den rein äußeren Bedingungen des Bolognaprozess mit inhaltlicher Erneuerung begegnet, der etwas neues, etwas qualitativ anderes, aber vielleicht sogar besseres schafft.

Im Grunde bestand die Einführung des Bachelor/Master darin, dass man Vordiplom und Zwischenprüfung einfach neu deklariert hat (plus Bachelorarbeit und ein Seminarlein). Es scheint, als wollte sich bei der Einführung niemand darüber Gedanken machen, dass ein 6-semestriger Studiengang inhaltlich grundlegend anders strukturiert sein muss als ein Vordiplom. Es braucht andere Vorlesungsformen, andere Seminarformen, andere Medien und andere Curricula -- es braucht neue Ideen. Stattdessen haben wir ein umdeklariertes Vordiplom. Aber das Vordiplom war zu unstrukturiert, um den organisatorischen Ansprüchen des Bolognaprozesses zu entsprechen. Wiederum liegt ein wesentlicher Grund der mangelhaften Umsetzung sicherlich darin, dass solch eine Arbeit keinen Wert für die Dozenten bzw. die Professoren hat, das sie nicht vergolten wird. Zudem wurde durch Professor Schulz darauf hingewiesen, dass gerade beim Lehramt in den zuständigen Kommissionen Lehrer saßen und damit für das Lehramtsstudium wohl auch eine Schuld an dem heutigen Elend haben. Die Entwicklung neuer Vorlesungen wäre vielleicht ein mittelfristiges Ziel, dass durch einige wenige, bemühte Dozenten erarbeitet werden könnte, aber wie gesagt, es zählt nichts -- wer sollte sich da auch engagieren.

Ein gänzlich verkorkster Diskussionspunkt lag in der Frage nach "Anwendungen", insbesondere in der Schule. Wie passend, dass kein angewandter Mathematikdozent anwesend war. Von Seiten der Studenten schien es vor allem ein verzweifelter Ruf nach motivierenderen Vorlesungen zu sein. Jedoch trifft es für mich ein tieferes Problem: Der Lehrplan an deutschen Gymnasien ist langweilig und veraltet. Er besteht eigentlich nur aus Rechnen, das auch noch höchst langweilig gelehrt wird. Es ist wie Sportunterricht, bei dem man die ganze Zeit Zirkeltraining macht, aber behauptet, man würde Fussballspielen. Kurz gesagt, alles nach der Bruchrechnung ist eigentlich irrelevant, vor allem auf die Art und Weise, wie es gelehrt wird -- und irrelevant heißt hier sowohl für die Bildung der Schüler ganz allgemein als auch als wissenschaftlicher Inhalt. Damit stellt sich aber die Frage, wie man einem Lehramtsstudenten erklären soll, warum er sich mit (Hochschul)Mathematik auseinandersetzen soll. Und ehrlich gesagt, kann ich keinen Grund finden, solange die Lehrpläne an den Schulen nicht modernisiert werden.

Das Problem, dass aber bei all dem Zappeln im Bolognanetz am stärksten auffiel ist die Unsinnigkeit der deutschen Lehrerausbildung. Hochspezialisiert, fast ohne Wechselmöglichkeiten, viel zu lange, ohne Praxiserfahrungen und auch personell weder von den Fachbereichen noch den Studierenden zu meistern. Warum braucht es überhaupt auf universitärer Seite ein spezialisiertes Studium? Warum braucht es einen Master? Warum so komplizierte, inkompatible Studienpläne? So unmöglich es ist, dies praktisch zu fordern: das Lehramtsstudium gehört eigentlich abgeschafft. An dessen Stelle könnte auf der wissenschaftlichen Seite ein BSc oder BA treten und auf pädagogischer Seite eine professionelle Facharbeiterausbildung an den Schulen, die auf einem Bachelor aufbaut. Wäre das so fachlich so unsinnig?

Und dann war da noch Peter Monnerjahn, einsamer Rufer im Walde, der als Einziger wiederholt feststellte, dass es in der gesamten Lehre grundlegende Probleme gibt. Einerseits stieß er sogar bei den Studenten auf großes Unverständnis (was vielleicht zeigt, dass man von Studenten, die drei Jahre an der Uni verbringen, kaum erwarten kann, eine korrekte Analyse der Situation vorzunehmen), andererseits wurden seine Anmerkungen in den Abschlussworten auch noch elegant-arrogant als schlicht "nicht originell" weggewischt. Das war dann der traurige Höhepunkt, an dem ich die Veranstaltung verlassen musste.

Mein persönliches Fazit ist dreigeteilt. Auf der untersten Ebene stehen Lösungen der konkretenProbleme der besorgten Studenten. Auch wenn ich oben vom Zappeln im Netz sprach, so wenig hilft es, deswegen gar nichts zu tun. Die gerne (auch von Studenten) gestellteFrage, ob Lehramtsstudenten "dümmer" sind oder nicht, ist völligirrelevant. Alle Dozenten haben die Verantwortung, ihre Studenten so zuunterrichten, wie es am besten für die Student ist (lesenswert: Zeilberger). Dazu gehört aber auchdie ehrliche Wahrheit, dass Regeln nie für Härtefälle gemachtwerden dürfen (Härtefälle, wie alleinerziehende Eltern oder sich ihr Studium selbstfinanzierende Studenten) -- keine Regelkann sich daran orientieren, aber Ausnahmen müssen dem trotzdem Sorge tragen.

Auf der mittleren Ebene sehe ich die mittelfristigen Möglichkeiten. Solange die schwerwiegenden Defizite -- Mängel in der Verwaltung, Kürzungen von studentischen Tutoren, Vernachlässigung der und Mangel an Räumlichkeiten und auch der Mangel an (gut ausgebildeten, langfristig angestellten) Dozenten, vor allem im Mittelbau -- solange diese Defizite bestehen, wird es fast unmöglich, Veränderungen in der Lehre grundsätzlich anzugehen. Dies kann solange also nur durch Zusammenarbeit unter den Dozenten gelingen. Dabei haben die Professoren die größte Verantwortung und Verpflichtung, gerade weil Kooperation selten ist unter Professoren, die oft an mittelalterliche Kleinstfürsten erinnern. Anstatt Entscheidungen zu blockieren, müssen Sie dafür sorgen, dass die Erkenntnisse der engagierten Kollegen nicht sinnlose Einzelaktionen bleiben, sondern als "best practice" Kodex zumindest innerhalb der Universitäten Geltung erlangen.

Auf der obersten Ebene sehe ich, dass weiterhin eine langfristigeVision fehlt, wie sich die Lehre und damit die Universität alsInstitution entwickeln soll. So wenig eine solche Vision praktischorientiert sein kann, kann ich niemandem trauen, dem eine solche Visionfehlt -- nur mit einer klaren Vision lassen sich gut strukturierteVorschläge für Veränderungen machen, die mit Weitsicht sinnvolleKompromisse ermöglichen.

Testing MathML

2009-12-09T18:55:00.012+01:00

One of the tools I want to use in this blog will be MathML. I think MathML is so far the best solution to present mathematical content on the web even though the discussion on Terence Tao's blog shows that MathML has its own deficits, especially when it comes to accessibility.

Nevertheless, tex4ht allows me to wait for a good standard to develop while working with one "generator", namely LaTeX, to produce multiple outputs.

I chose blogger especially because I wanted to use MathML, e.g.

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

Of course, blogger does not make it easy, but thanks to David Carlisle a good friend of mine was able to hack enough for me to work on for now. Unfortunately, I will now have to add to the header that you really need firefox with javascript, but is that too much to ask these days?

By the way, this tag will hopefully lead to more techological experiments in the future.

Welcome

2009-12-07T17:02:00.000+01:00

After more and more intense blog reading over the last couple of months it felt right to try to start my own blog for the ~~first~~ second time.

For the moment everything in this blog will be experimental which seems appropriate given that I am not PhD student any more but have not officially started my PostDoc yet.

The main purpose of this blog (in the long run) will be to document my own research activities (in the broadest of senses, although usually just mathematics). Of course, due to the nature of the web this blog is bound to include random bits of other things, hopefully including experimental use of the web as a medium, in particular for presenting mathematics.

Enjoy, Peter.