# The end is near

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.

# Other lazy people

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

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.

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.

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 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

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…).

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

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.

1. $A$ with $F_1$ ($F_2$) generates a right (left) zero semigroup (hence of size $2$, except for $x=0$).
2. $A$ with $F_3$ or $F_4$ generates a semigroup with $AB$ nilpotent (of size $4$, except for $x=0$, where we have the null semigroup of size $3$).
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 In case $A = \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}$ the solutions for $B$ being of rank one consist of five one – dimensional families namely (for $x\in \mathbf{Q}$)
$H_1(x) = \begin{pmatrix} 1 & x \\ 0 & 0 \end{pmatrix}, H_2(x) = \begin{pmatrix} x+1 & x \\ ( – x – 1) & – x \end{pmatrix}, H_3(x) = \begin{pmatrix} 0 & x \\ 0 & 1 \end{pmatrix}, H_4(x) = \begin{pmatrix} ( – x+1) & ( – x+1) \\ x & x \end{pmatrix},$
$H_5(x) = \begin{pmatrix} ( – x+1) & ( – x – 1 – \frac{2}{x – 2}) \\ x – 2 & x \end{pmatrix} , x \neq 2.$

As before we can describe size and structure.

1. $A$ with $H_1$ ($H_2$) generates a right (left) zero semigroup (as before).
2. $A$ with $H_3$ or $H_4$ generates a semigroup with $AB$ nilpotent (as before).
3. $A$ 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 also dealt with; e.g. $– G_i$ generates the same semigroup as $G_i$ etc.

It is striking to see that the orders of all finite semigroups generated by rational idempotent two by two matrices are either $2^k,2^k + 1$ or $2^k + 2$.

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.