# Talk - Syntax and Semantics (Trans Math Day 2020)

### 16 Dec 2020 -

I’ve been busy with some assignments and grading, so it took me a while to post this. We got there eventually, though! I gave a talk at an online conference for Trans Math Day on December 5th. There were a lot of interested speakers, so the organizers gave us 5 minute, 10 minute, and 20 minute spots. I was given a 5 minute talk, which is a borderline impossible assignment – Obviously I was still exited to give it, there were just a slew of challenges to work out.

I’m not a complete stranger to this format. As an undergraduate, I was given a 15 minute slot to present my thesis work (on Abelian Autaomata Groups), and I wanted to discuss as much as I could (after all, I was proud of that work!). I ended up leaning into the manic energy and powered through, barely stopping to breathe.

I didn’t want to give that kind of talk, through. This was an online conference, and nobody present knew me. It’s fun to give a comically fast paced presentation when you know everyone knows you, but in a room of strangers, it’s better to be professional. So I had to find something which was simultaneously interesting and explicable in 5 minutes… I settled on Syntax and Semantics.

There are *lots* of fun theorems that place restrictions on the complexity
of an object (its semantics) based on how simple that object is to describe
(its syntax). I decided to give a short outline of the idea, followed by one
example from algebra and one example from analysis. In the end, I used 7 minutes.
While I shudder to think of using almost 150% of my allotted time, I think
every talk goes around 2 minutes over. So I’ll happily think additively
instead of multiplicatively in this instance :P

I asked if the talk could be recorded, but in the interest of protecting the privacy of some attendees, the organizers politely declined. I still have the slides, though, and sometime soon I’ll put up that blog post I keep promising outlining some more examples…

As a last aside, the conference was a lot of fun. It was great to be in a (zoom) room with a bunch of other trans mathematicians, and many of the talks were extremely interesting! One person gave the most lucid account of Riemann-Roch I’ve ever seen (which is made more impressive by the size of the time slots), and I even left with some fun problems to think about once I have some more free time! All in all, I hope this becomes an annual event, and I hope to continue being involved ^_^

Syntax and Semantics as an Organizing Principle

The key observation in Mathematical Logic is that the syntax of mathematics (the symbols we write on a page) is different from the semantics of mathematics (the meaning we as humans prescribe those symbols). By studying them separately, we can prove theorems of the form “any object which is easy to describe cannot be too complicated”. In this talk we will survey a collection of results of this form which the speaker has found useful in their own research and education.

The slides are here