The title is the most complicated part of this talk! Come hear about the construction of surreal numbers and how they can help us analyze some games involving dominos, sticks, and flag-football. I've just read John Conway's wonderful little book "On Numbers and Games". This talk will be a brief overview of the basic ideas he presents.
Any compact Riemann surface can be realised as the quotient space of the upper half plane by a group of automorphisms with certain nice properties. I will give a brief discussion of this realisation. Following this, I will discuss how we may use this realisation to obtain other information about Riemann surfaces.
We will introduce the notion of an Elliptic curve over the complex numbers, which is a genus one curve together with the choice of a point, from an analytic perspective. We will then give a survey of results on the uniforimsation of these curves, and the methods neccessary to give an algebraic description of an Elliptic curve. Once we have this algebraic description, we will free to move to more things more exotic than the complex numbers, for example p-adic fields. Unforunately, in these cases, we lose the analytic description we had over the complex numbers, however, if there is time, we will discuss briefly an analytic uniformisation that one has over p-adic fields. This talk should be viewed as introductory and as a companion (yet logically independent) talk to Aaron Wootton's talk of last week.
Do you feel like you have been tied in knots? Well your DNA has been tying and untying itself in knots since your conception. We will discuss the basic ideas behind link, twist and writhe; which are topological invariants useful in studying your DNA. We will then use this information to understand the Supercoiling effect of DNA and why biologists and mathematicians alike are interested in the formula given by Calugareanu that Lk=Tw+Wr. This talk will cover material I am investigating for my Master's thesis, but it should be thought of as introductory in nature. Only prerequisite is that you can tie your shoes!
We will look how to construct orthonormal wavelets via multiresolution analysis (MRA). That is, we will see how to construct orhonormal wavelets based upon the existence of closed subspaces of L^2(R). If time permits, we will prove for any natural number n, there exists an orthonormal wavelet f with compact support s.t all the derivatives of f up to order n exist and are bounded.
There are cannonical cohomology classes defined on the Moduli spaces of pointed curves of genus g. The Witten-Kontsevich theorem determines the structure of the subring generated by these classes. This theorem is a statement about a generating function which satisfies an integrable hiearchy of partial differential equations. A different formulation of the problem leads to a generating function whose expansion gives the enumeration of maps (graphs with additional structure) on genus g surfaces.
Fourier's solution of the heat equation created tidal waves in the mathematical waters of his time. In this talk, I will discuss the state of mathematical physics prior to Fourier, an introduction to Fourier's Method, the questions his method posed to the mathematical community and its influence in Algebra, Analysis, and Number Theory in the following century. The discussion will be a survey of the topics presented in a series of lectures by G.W. Mackey.
Morse Theory and its generalizations play an important role in many applications of differential topology. This talk will present an introduction to Morse Theory, discuss some results of Smale and Milnor, and present some applications of differential topology.
This talk is the second of two independent but completely related talks. In this talk I will introduce the notion of a Belyi function on a surface. Then I will dicuss the correspondence between surfaces on which Belyi functions exist (Belyi surfaces) and subgroups of the fundamental group of the sphere minus three points. I will also state (without proof!!) Belyi's Theorem which relates algebraic curves defined over Q and Belyi surfaces. To finish, I will describe the Grothendieck correspondence between Dessins d'Enfants and Belyi surfaces and some of the interesting results which have emerged from this.
The neat fact that partial derivatives commute leads us to transformations that take solutions of special PDE's to other solutions. This is related to transformations that take surfaces of certain types to other surfaces of the same types.
The classic problem of "the gambler's ruin" can be analysed from a Martingale perspective. Moment generating functions can be used to obtain approximations for the probability of success of a player. Such approximations combined with a few calculations yield a rule of strategy. I will briefly review moment generating functions, conditional expected values, Markov chains and martingales. Then the rule of strategy will be stated and illustrated with a few examples.
Did you know that there was a connection between the fundamental groups and differential equations? Well, Riemann and Hilbert did, and so should you, so come and discuss with me the Riemann-Hilbert correspondence.
Group cohomology depends on projective resolutions. Resolutions can be difficult to calculate even for small groups. Morita equivalence makes resolutions easier to compute even for large groups. In this talk I will explicitly calculate a couple of Ext groups and then explain how Morita equivalence is used.
In this talk, I will demonstrate a few strategies for finding rational points on a variety and give examples showing how these methods may fail or succeed.