# What does a mathematician do? This week’s plan.

Thought I’d give you a window on my week to come, since it is early Monday morning—so you can get an idea of what mathematicians do. Today is my big teaching day, I have four tutorials, all on first year calculus. I am handing back the tests the students wrote last week, and will try to convince them they must make more effort writing coherently and expressing their mathematical thoughts more carefully. Our first year students often don’t have a lot of practice in writing mathematics. Sometime during the day I will go for a swim at the Uni pool to make four hours of teaching go smoother.

Tomorrow I will chair a meeting of our Learning and Teaching Committee, we need to get going on mentoring new tutors, thinking about a L&T seminar for later in the term, and putting one of our colleagues forward for a teaching prize. I will also talk with one of my graduate students, Nguyen Le, on how we can construct a planar triangle over the rational numbers that has incenters in both blue (Euclidean), red and green (relativistic) geometries simultaneously. She has been working on triangle geometry, and we would also like to start writing a paper on some extensions of classical results involving Gergonne, Nagel, Incenter points etc.

I will need to schedule a list of mentors to oversee our new casual tutors (the School of Mathematics and Statistics employs outside help to also tutor some first year classes).

On Wednesday and Thursday I have three hours of my Algebraic Topology class. I need to have marked the problem sets before that—something else to do on Tuesday—hand them back, and give lectures on a review of free groups and non-commutative groups, and also on applications of winding and turning numbers (for example the ham sandwich theorem!) Nguyen will video one of those classes for me, and at some point, probably the weekend, I will edit it and post it on my YouTube channel (user njwildberger).

Friday I don’t have classes but I want to spend a good chunk of time with my Masters student Sharanjit Paddam who is working on rational views of classical tesselations of the hyperbolic plane. I am heading overseas the following week, to Novosibirsk (Russia) and Krk (Croatia) for two geometry conferences, will be gone three weeks, and want to make sure his project is on track.

I also need to spend some time preparing the talks I will give at these conferences. But what I also would like to do, in fact what I really want to do, this week is: think about the group structure on a cubic curve (one step up from the conic sections like ellipses, parabolas and hyperbolas) in terms of ternary forms. I have this reformulation of algebraic geometry kicking around in the back of my mind, which starts by reconsidering the basic nature of an algebraic curve; not as a set of points, but rather as… well something both more complicated but simpler to deal with. I`ll explain this at some later point when I understand it better, but suffice it to say for now that I know there are interesting newish algebraic objects which I call `pods’ floating around under the noses of classical algebraic geometers. I have been putting together Geometer’s Sketchpad worksheets that allow me to explore cubics and play around with group and pod structures.

The other thing that has been on my mind is the Cayley transform, something that I thought quite a lot about when I was a graduate student, and post-doc at Stanford. I reckon that we need a more algebraic theory of Lie groups, and I have known for a long time, although I haven’t written anything on it, that the key is replacing the exponential map with the Cayley transform, suitably interpreted.

Jack Hall did a Master’s project on related issues with me some years ago. He has finished a PhD at Stanford under Ravi Vakil, who curiously was also briefly a student of mine: he sat in on my first year algebra course at the University of Toronto when he was a precocious high school student. Jack will be taking up a position in Canberra next year, it will be nice to have him as part of the mathematics scene.

Anyway, the Cayley transform is a lovely thing, ought to be more widely appreciated and used. I would like to extend Rational Trigonometry to the Lie group setting, a big project of course, but something that I like to contemplate now and then.

I have also been feeling guilty about not posting any more videos in my WildLinAlg series for quite some time. If I can make time, I might put together a new video on the linear algebra of cubic splines. Not sure if I will get around to that though.

So that is my week, coming up! Lot’s of fun, and interesting.