## Graduate Student Seminar 2008-2009

Many thanks to David Kohler for organizing this second year of department-funded Grad Seminar.

at The University of British Columbia

Many thanks to David Kohler for organizing this second year of department-funded Grad Seminar.

**Fluid Instability Propagation with Stiff Boundary Conditions in Case of Low Reynolds Number. (a.k.a. What can golden syrup and latex do for you?)** - by Maria Khomenko.

This is the classic problem of a viscous fluid forming instabilities as it flows down an inclined plane. In this case the flow takes place under a sheet of latex. There are a number of physical parameters that can be measured experimentally. And then to some extent compared with theory. Note to Cameron: Ethical issues aside, just because you sold your body to science, that does not give you the right to perform perverse social experiments on your peers. (please refer to potential Dr. Christou's last week's abstract for details).

The experimental results will be accompanied by mathematical formulation of the problem.

**Stable Inverses of multidimensional power series** - by Cameron Christou.

We call a power series "stable" if its coefficients are absolutely summable. If the inverse of a given stable power series is also stable, then there exists an explicit relationship between the log-magnitude and phase of this power series when evaluated at points along the complex unit circle. I am messing with you. This was the topic of my Masters thesis, about which I will not be speaking. I just wanted to see how many people actually read the abstract to these talks. Call it perverse social curiosity. I'm really going to talk about how math can be used to promote sustainable fishing, and how fishing with explosives is probably a bad thing for the environment.

**Pattern formation and the Fekete problem: an unexpected connection** - by Igniacio Rozada.

Reaction-diffusion partial differential equations are widely used in a large array of biological applications, such as modeling animal coats. Instead of the traditional Turing analysis, I will discuss a full nonlinear analysis of the Schnakenberg model in the singular limit when the diffusivity of one of the components goes to zero. A differential-algebraic system can be derived to characterize the slow dynamics of a multispot solution, which has a very interesting connection with the problem of finding globally optimal Fekete point configurations. There will be pretty pictures.

**An introduction to orderable groups** - by Adam Clay.

This talk will be an introduction to the theory of orderable groups, with emphasis on examples. The main goal will be to introduce "the space of all orderings of a group", a new object of study which has proven very useful in addressing several long-standing open problems. Time permitting, I will also explain those open problems that have been resolved using the space of all orderings of a group, as well as outline directions for future research.

**Introduction to p-adic fields and a toy model for SL2(R)** - by Jerome Lefebvre.

As a general rule of thumb "number sets grow when a mathematician and an equation love each other very much". The p-adic numbers are very much part of that story. As p-adic numbers are rarely talked about at an undergraduate level, I'll try to introduce them through their topology and give some of their basic properties. This talk will be very elementary, requiring only basic ideas of metric spaces and ring theory. As a bonus mini-talk, I'll show how one of the models that are lying around the graduate student offices can be used to visualize SL(2,R), the group of two by two matrices with real entries and determinant equal to one.

**Regular polytopes and symmetry groups** - by Aurel Meyer.

This will be an informal talk about regular polytopes (think of Platonic solids in 3 dimensions) and their symmetry groups. I will show that these groups are all generated by reflections (Coxeter) and illustrate this with computer graphics. Conversely, all finite reflection groups are the symmetry groups of certain (semi-) regular polytopes. There are also connections to root systems and Weyl groups which I will speak about as time permits.

**Solving diophantine equations** - by Scott Sitar.

The resolution of Fermat's Last Theorem has seen the introduction of a powerful new tool into the study of Diophantine equations. In this talk, I will define and outline the basic objects and strategies of this new method: elliptic curves, modular forms, Galois representations, Frey curves, and level lowering. Using these tools, I will give a proof of Fermat's Last Theorem (which is easy!), as well as survey some other successes of the new method.

**Rational distances** - by Frank de Zeeuw.

A rational distance set is a subset of the real plane with all pairwise distances rational numbers. It's not too hard to construct an infinite rational distance set within a line or a circle, but if you do not allow 3 points on a line or 4 on a circle, the current record is a set of 7 points, found a few years ago with a computer. On the other hand, no one has been able to show that a rational distance set, no 3 on a line or 4 on a circle, must be finite, or even that it cannot be dense in the plane. Together with Jozsef Solymosi I showed that the only algebraic curves that contain infinite rational distance sets are lines and circles. I'll explain the main ideas behind this, most of which are very accessible. You can find the article on the arxiv here.

**Subverting the hassle of e-documents** - by Mclean Edwards.

Subversion is a popular revision control system that is installed and ready to use on some of our math servers. A revision control system is great because it keeps a history of all of the changes you make to a document in a central location, and allows you to revert to a previous version at any time from any location. Furthermore, it is designed to be used by many people concurrently, and can intelligently deal with multiple people editing something simultaneously. It does take some technical knowledge and has a small learning curve, so I'll be guiding you through getting started and using subversion on a day-to-day basis. If you use multiple computers and/or do collaborative research, you'll not want to miss this.

A laptop that can ssh to math.ubc.ca (or a close friend with such a laptop) is necessary to get the most out of this seminar (you'll need to bring the laptop, possibly with friend attached). If you plan to come, please email me beforehand (mcleane) so I can setup an account. If you forget to email, don't worry, you'll just end up with a generic account name.

Even if you use only one computer, Subversion is perfect for thesis writing. I use it myself every day and figure I save about 20 minutes weekly in time and worry because of it.

For the second half, I'll present a couple of tips and tricks I find helpful and time-saving when writing LaTeX code. Then we'll open the floor and hear from you. Do you know of any LaTeX tips yourself? Please bring some along with you and share with the rest of us. We'd all love to hear them!

**The strong law of small numbers** - by David Kohler.

Introduced by Gardner in 1980 and then by Guy in 1988, the strong law of small numbers states that "there aren't enough small numbers to meet the many demands made of them". We will have a look at some of the many examples provided by Guy to illustrate the law, everyone will be asked wether the shown pattern persists or not. This talk is intended to be amusing and light, absolutely no technical knowledge will be needed, anyone is welcomed!

**Cubic surfaces** - by Alex Duncan.

This will be a fairly informal exposition of smooth cubic hypersurfaces in space. These surfaces have many beautiful properties. They can always be described as a blow-up of six points (in "general position") on 2-dimensional projective space. Also, there are the famous 27 lines sitting inside the surface. The automorphisms of cubic surfaces are related to permutations of these lines and the Weyl group of E_6. The talk should be accessible to anyone, concepts like blow-ups and projective spaces will be introduced as needed.