## Graduate Student Seminar 2009-2010

Thanks to David Kohler and Robert Klinzmann for the organization of this year's graduate seminar.

at The University of British Columbia

Thanks to David Kohler and Robert Klinzmann for the organization of this year's graduate seminar.

**What is mathematical finance?** - by Peter Bell.

Basics of Mathematical Finance will be presented using concepts from probability. Risk measures and option replication strategies will be introduced, which allow the introduction of fundamental theorems of asset pricing. Brief results from the author's research concerning regularity of profits under technical trading rules will be also presented.

**When is an infinitesimal isometry on a Hermitian manifold holomorphic?** - by Kael Dixon.

I will present this open problem in complex geometry via the example of a Hopf surface. This is (roughly) the only compact Hermitian surface that admits infinitesimal isometries which are not holomorphic, and it exhibits some interesting behaviour which is indicative of some known results in higher dimensions. This talk will not require very much knowledge of differential geometry, with most of the discussion on the more intuitive level of group actions by Lie groups.

**What are stochastic differential equations?** - by Nicole Jinn.

This talk will be an introduction to the theory of stochastic differential equations (SDEs), with an emphasis on the connections to statistics. In any case, I could look at SDEs from two different perspectives: applied mathematics and statistics.The differences between the two perspectives will quickly be explained. I will also briefly mention some of the results of a directed studies project that I am doing and highlight the connection to SDEs. My project is about modeling diffusion of one particle into a two-dimensional system of spatially homogeneous particles. The eventual goal is to have a working molecular dynamics simulation representative of the system in question, as well as to validate one of two theoretical predictions for the diffusion coefficient and its dependence on spacing between particles.

**What is probability theory?** - by Tyler Helmuth.

We all know some probability, if nothing more than what we help first year students with in the tutorial center. But what makes a concept probabilistic as opposed to measure-theoretic? I'll try to answer this question, which sheds some light on the strange language and notation used by probabilists.

**What is mathematical finance?** - by Tao Cheng.

Nowadays, mathematicians plays a more and more important role in the financial world. In this talk, I will briefly introduce the development of mathematical tools in finance. Then I will focus on the most famous formula in mathematical finance field, namely the Black-Scholes Formula (a SDE problem mathematically), and explain how it works in pricing financial derivatives.

**What is Hilbert's thirteenth problem?** - by Alex Duncan.

One of the most ubiquitous operations in mathematics is the solution of polynomials in one variable. Unlike the situation for quadratics, cubics and quartics, Abel's Theorem tells us that we cannot find a general formula in radicals for polynomials of degrees greater than 4. Of course, one can always find roots numerically but this is unsatisfying. There are relatively nice solutions to quintic and sextic equations using functions more general than radicals. Motivated by these constructions, Hilbert's thirteenth problem asks if there is a general solution to a seventh degree polynomial of a particular form. Arnold and Kolmogorov found that the answer was yes, and furthermore, their solution could be extended to polynomials of arbitrary degree. However, we shall see why their answer was probably not what Hilbert was seeking.

**What is a number?** - by Maxim Stykow.

In this talk I will present several ideas ranging from Euclid to Conway about how to put our intuitive and also sometimes not so intuitive ideas about what a number is on a rigorous foundation. Some questions you might have that I'm going to answer are: How to create something out of nothing? Why is a proof by induction actually a proof? What is a surreal number?

**What are p-adic numbers?** - by Robert Klinzmann.

We will construct p-adic numbers and real numbers by enlarging the rational numbers. This shows that lR is in some sense very special. Finally I will present some basic facts of p-adic analysis.

**What is a vector bundle?** - by David Steinberg.

A vector bundle is a continuously varying family of vector spaces; for example, the set of lines that are tangent to a smooth curve is a vector bundle. In this talk, we will draw pictures, give examples, state applications, and (time permitting) learn how vector bundles can mend a heart broken by Liouville's theorem. Our emphasis will be on intuition, and all technical details will be suppressed.

**What is statistical physics?** - by Dennis Timmers.

In statistical physics we study models for interacting particles. One of the main open questions is to find models which exhibit a phase transition (think about water turning into gas at 100 Celcius). First I will introduce all the big words people use in statistical physics. Then I will show a result of Lebowitz and Penrose on the existence of a phase transition for a certain class of models. If there is some time left I can talk about some extensions of the Lebowitz and Penrose result.

**What is game theory?** - by Alex Jakobsen.

This talk will be a friendly introduction to the basic concepts of game theory, starting with the Nash solution concept (more appropriately, the "Cournot-Nash" concept - you'll see why). My goal is to highlight some of the central results, illustrate some interesting examples, and to give an idea why game theory has become so important in economic analysis (and other disciplines, too). Technicalities will be kept to a minimum, so it should be easy for everyone to walk away with a good idea of what game theory is all about.

**What is an expander graph?** - by David Kohler.

This short talk aims at describing expander graphs and some of their fascinating applications to other fields of mathematics and computer science. Wether you are interested in coding theory, complexity theory, probability theory, number theory or group theory, (with or without a flavour of geometry and linear algebra on the side) there will be something for you. And if none of these really speak to you, at least you'll get a nice promenade in the mathematical landscape.

**Some facts about elliptic curves** - by Robert Klinzmann.

Elliptic curves are very old mathematical objects but they are still element of current research. In my talk I will define the notion of an elliptic curve E motivated by some beautiful connection to meromorphic functions on complex tori. We will illustrate (but not prove) that an elliptic curve E carries an abelian group structure and will define its group of rational points, which leads to the famous Mordell-Weil Theorem. Time permits I will say something about the conjecture of Birch and Swinnerton-Dyer.

**Finite reflexion groups and root systems** - by Tobias Friedel.

In geometry finite reflection groups (subgroups of the orthogonal group of an Euclidean space that are generated by reflections) appear, e.g., as groups of symmetries of certain regular polytopes. We will establish a correspondence between those groups and root systems, finite subsets of R^n that were first mentioned in Lie theory but have since then been observed in many different contexts like combinatorics or cluster algebras to name just a few. With help of those sets we will manage to give a nice description of finite reflection groups as well as a complete classification.

**Turning analysis in reaction-diffusion equations, when it works and when it doesn't** - by Ignacio Rozada.

In the 1950's, reaction-diffusion equations were proposed as a model for the symmetry-breaking process that organisms undergo as they transition from single cells to embryos. There are plenty of other applications that involve them, most notably as a mechanism responsible for pattern formation in organisms. Perhaps the main reason for the popularity of the model is that linear stability analysis provides considerable insight into the types of patterns that the stationary solutions will have. The talk will serve as an introduction to reaction-diffusion equations and Turing analysis. For the people already familiar with the subject, we will also discuss the limitations that occur when considering domains in 2 or more dimensions.

**Knots, tensors, and statistical mechanics** - by Kael Dixon.

In this short talk, I will breifly introduce knot theory and abstract tensor diagrams. Then I will show how combining these ideas gives the Yang Baxter Equation, which originated in statistical mechanics. Don't worry, there's not actually going to be any statistical mechanics in the talk.

**Non-measurable sets** - by Terry Soo.

We will discuss the nonmeasurable sets defined by Vitali, Ulam, and Shelaha-Soifer.

**Hyperbolically speaking...** - by Simon Rose.

For millennia, many misguided mathematicians moved to prove that Euclid's parallel postulate was a consequence of his other axioms. In this talk I will discuss hyperbolic geometry, and draw a picture or two proving all of these valiant efforts wrong.

I will also discuss relationships between complex analysis and the classification of all two-dimensional geometries, why hyperbolic geometry can be described (somewhat tongue-in-cheek) as "God's Geometry", and given time, a discussion of extant three dimensional geometries.

**Can you hear the shape of a drum?** - by Ramon Zarate.

This question is the title of Mark Kac's famous 1966 paper. This talk is an undergraduate level talk on some of the math behind this peculiar question and a brief outline of the techniques involved. This talk is meant to make PDE's attractive to a non PDE public.

There will be some functional analysis, some variational methods and some asymptotic analysis (all at a very basic level) along the worlds best ratio: (interest of result)/(difficulty of proof) in an amusing jigsaw-puzzle proof! All this in a context of basic geometry.