# A Week Of Mathematical Outreach, The Good, The Ba(r)d And The Ugly

## A week of mathematical outreach, the good, the ba(r)d and the ugly

I like to talk about my research. I really do, even with people, who are not model theorists or not even mathematicians. It is fun to make up more or less unhinged ways of describing what you do and trying to find metaphors for very abstract thinking. So I reached out and asked whether I could be a curator for the German version of RealScientists. I followed them for a while and liked their approach of just letting scientists who are not all specialized into scicomm talk about what they do. About a week before my turn was due, I realized:

I would be the first mathematician on that account.

And I am not primarily a scicomm person. I did a lot of teaching before I got a fancy PhD position that has no teaching obligations, but I have no actual formal training. So I decided to do what I can do best: Improvise and Imitate. It is hard to actually ascertain how well that worked out, but let me tell you first what I actually did.

### The Facts

Personally, I am sometimes irritated by the way mathematical outreach is often practiced. While e.g. a biologist tells you what they do, but will not go into the technical details and all the pesky stuff, my impression was that mathematicians tend to rather talk about basics that might have nothing to do with their research at all. And I get, why that is the case. Mathematics uses its own language with little to no actual representation in the real world. Our whol shtick is abstraction and while every abstraction had a practical idea somewhere in the history of its creation, this might be $\aleph_0$ iterations ago and not useful anymore at all. I also did that, but I tried to also talk about things related to my research. And while there are things that are worse, I assume that model-theoretic motivic integration is very high on the scale of iterations of abstractions needed to understand why the hell we would be interested.

I wanted to start with things that might be known to people, who are into popsciency math. The first thing I did was a brief Rundumschlag (just to confer with the comment of a friend that Germans will just use words in their mother tongue, if there is nothing as cool in English), tell people that I do “ping-pong” between semantics and formalism. Drop a few terms that might become relevant. I told them they could ask any questions and asked them what they were interested in in particular. I gave a threaded proof of the n=4 case of Fermats Last Theorem - because someone mentioned Fermat and I figured that would not be too bad.

In my freetime, I feel mostly like a storyteller and tried to this week, so I created a couple of threads that had a narrative

• The betrayal of the integers, in which I told about the structures we learn about in roughly the order of their inclusions $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$ and where most students would find them increasingly scary. Everyone gets the natural numbers, but what the fuck is that i business and that stuff. And after I got to $\mathbb{C}$ I rolled it back up and told them that from the point of view of a model theorist, its the other way around. The complex numbers are tame and the further you go down the more unhinged nonsense you get. Of course that leads to rich mathematical structure, but combinatorical it gets worse and worse - apart from maybe the step from $\mathbb{Z}$ to $\mathbb{N}$ which are arguably equally bad. Of course everything in the context of a language, where we can talk about addition and multiplication.
• Why Gödels Incompleteness theorems are less astonishing than what is sometimes communicated, but still very great.
• Why I like the completeness theorem and compactness a lot more. duh
• What is a valued field and what are their geometrical properties.
• What did I do in my bachelor thesis and how did I find that topic. A bit about how $(a+b)²=a²+b²$ is not as stupid as you were told in school. That there are other characteristics than $0$ and why inseparable polynomials worry us.
• What did I do in my master thesis and how did I get there. What did I do in Oxford and why are local-global principles interesting?
• A very rough sketch on what interests me in my own research now.

Apart from that I did do a few interactive posts like “Do you also like to use chalkboards in your work?” or “What kind of software do you use to write?” and ended up with a few things about my personal life like that I try to not work too much on weekends and play a lot of tabletop rpgs. I also did a post, which was very important to me, how math is hard and not hard at the same time, elaborating that you do not need to be a (white cis-male) genius to be a mathematician, but that its also valid to struggle and most of the people calling everything trivial struggled themselves. I also gave them a little semimathematical riddle.

### The Good

I had the impression that people were generally interested. There were a lot of questions. People really liked my story about how the integers came to betray us in the end (or maybe more the beginning) of our mathematical journey. Generally the more narrative elements were well received, though of course I do not know whether it led to deeper understanding or not (that might very well not be the goal at all). People also liked the more sociological comments about math, they also liked to try and riddle with me.

### The Ba(r)d

I feel with the more complex math I lost the narrative structure a bit. It would have been better if I made up as elaborate stories for why we would be interested in valued fields as to why the integers betrayed us. This is harder, I think. To me it still seems like a good idea. It was a mistakte to try and explain the proof for Fermats Last Theorem for $n=4$. The proof does of course not need any sophisticated math, but I underestimated the length of the argument before I started typing it. I don’t think there was much gained from explaining this. I thought it would lead to a good narrative, because I wanted to explain how weird the integers were, but in the end this could have also been done by just mentioning how incredibly difficult the proof for the general case was. I also don’t think I was able to convey in what niche I am actually invested. I talked a bit about motivic integration, but I still struggle to explain to mathematicians what this is good for - it was not easier to do this for people without that training.

### The Ugly

My usual twitter account has about 1800 followers by now. That is also not a small number, but they are filtered by an extensive blocklist, by the occasional weird surreal joke and by a giant banner that pretty much should scare of any random libertarian. That account had 16000 and no such filters. The highlights of weird comments included:

• Someone calling me backwards for not wanting to abolish all theoretical computer science in German schools (not that I have expertise on that and I was careful with my wording, but… what?)
• Someone calling me backwards for using chalkboards (and promoting smartboards with handwriting recognition, because humans might not be able to see the difference between an i, a j and so on … but the algorithm definitely can do that)
• A few typical complaints that all funding for math should just go into cancer research

### Conclusion

Well, I don’t know! I think I did not too badly, but I do not have any data on how well I was actually able to communicate what I do. I think narrative structures are a good thing to help people navigate through our abstract nonsense. I also gained an enormous amount of respect for people who actually do mathematical scicomm.

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