"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...).

Good news, bad news

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.

• $A$ 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 $AB$ 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 $A=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)$ the solutions for $B$ being of rank one consist of five one-dimensional families namely (for $x\in \mathbb{Q}$)

$$H_1\left(x\right)=\left(\begin{array}{cc}\hfill 1\hfill & \hfill x\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right),H_2\left(x\right)=\left(\begin{array}{cc}\hfill x+1\hfill & \hfill x\hfill \\ \hfill -x-1\hfill & \hfill -x\hfill \end{array}\right),$$

$$H_3\left(x\right)=\left(\begin{array}{cc}\hfill 0\hfill & \hfill x\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right),H_4\left(x\right)=\left(\begin{array}{cc}\hfill -x+1\hfill & \hfill -x+1\hfill \\ \hfill x\hfill & \hfill x\hfill \end{array}\right),$$

$$H_5\left(x\right)=\left(\begin{array}{cc}\hfill -x+1\hfill & \hfill -x-1-\frac{2}{x-2}\hfill \\ \hfill x-2\hfill & \hfill x\hfill \end{array}\right),x\ne 2.$$

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 $AB$ 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 also dealt with; e.g. $-G_i$ generates the 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

In an earlier post I gave a short introduction to an interesting finite semigroup. This semigroup could be found in the $2\times 2$ matrices over $\mathbb{Q}$.

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