# MaPS – MaPSS Seminar Series

Welcome to the webpage for the Mathematical Postgraduate Seminar Series (MaPSS). We’re committed to fostering a friendly atmosphere in the school of Mathematics and Statistics. All maths postgraduate students are encouraged to present. It’s an excellent opportunity to hone presentation skills and talk about fun new topics. Most of all, it’s a great way of getting to know your fellow students. So come along and meet some friends over free pizza!

If you or anyone you know is interested in presenting, or for any other enquiries, please contact the MaPSS organisers: Eric Hester, Alexander Kerschl, Nathan Duignan, and Giulian Wiggins.

MaPSS also ran in 2015, 2016, 2017, and 2018.

# Seminars in 2019, Semester 1

All seminars will be held at 5:00 pm on Mondays in Carslaw Room 535A, with free pizza and soft drink after the talk.

## Monday, March 4th

Eric Hester (Sydney University) — Understanding multi-scale Partial Differential Equations on arbitrary domains using the straightforward differential geometry of the signed distance function

Partial Differential Equations (PDEs) are a core part of mathematical modelling in science, industry and engineering; the applications are endless! And often the most interesting problems involve complex geometries. We normally model them using PDEs which live on separate domains, with boundary conditions applied at the infinitesimal interfaces.

But that’s math, not reality. Reality is smooth! Things get fuzzy at the micro and nanoscale. And it can also be useful to do simulations this way – don’t try to simulate your PDEs on complicated domains. Instead, perturb your PDEs to implicitly model your boundary conditions.

Having smooth (but small) transitions between domains means we are considering inherently multi-scale singular perturbations of PDEs. So these approximations are only true asymptotically. What we need to know is how these smooth approximations behave in the limit. Do they tend to the correct answer? How fast? And do they work in arbitrary geometries?

This talk will examine a really useful coordinate system for analysing such multi-scale PDEs. It all comes from the straightforward differential geometry of the signed distance function. I’ll be focussing on examples from my research on modelling moving objects in fluid dynamics. No background in differential geometry or fluid dynamics is required.

## Monday, March 11th

Giuliam Wiggins (Sydney University) — Abstract Voting Theory

We look at some cute applications of representation theory to the study of election procedures and voting paradoxes.

## Monday, March 18th

Nathan Duignan (Sydney University) — Simultaneous Binary Collisions and the Mysterious 8/3

Of central importance in the N-body problem is the fact that isolated binary collisions can be regularized: that a singular change of space and time variables (first written down by Levi-Civita) allows trajectories to pass analytically through binary collisions unscathed. The resulting flow is smooth with respect to initial conditions. Curiously, we are not so lucky with simultaneous binary collisions. In a landmark paper, Martinez and Simo gave strong evidence that the best one can hope is 8/3 differentiability of the flow in a neighborhood of simultaneous binary collisions in the 4 body problem. In this talk we follow Easton by linking regularizability to the behaviour of the flow in Conley isolating blocks around the collisions. We show the 8/3 is produced from the first resonant monomial with nonzero coefficient near a degenerate singularity formed when the two binaries are separately Levi-Civita regularized. To show this, we blow-up the singularity and study the flow near the resulting two, 3:1 resonant, normally hyperbolic manifolds connected by heteroclinics. A lengthy normal form computation confirms the conjecture.