SCU Mathematics Colloquium Series Schedule
Title: Solving Problems in Scientific Computing Using Maple and Matlab
Abstract: Modern computing tools like Maple and Matlab make it possible to use the techniques of scientific computing to solve realistic nontrivial problems in a classroom setting. Traditionally such problems have been avoided, since the amount of work required to obtain a solution exceeded the classroom time available and the capabilities of the students. Therefore, simplified and linearized models are often used. This situation has changed, and students can be taught with real-life problems which can be solved by the powerful software tools available. In this talk we will present some examples of Equations, Differential Equations, Symbolic Computations and Least Squares Problems and demonstrate new ways of solving problems in scientific computing.
Title: Growth Series for Regular Cubical Complexes
Abstract: A cubical complex X is a geometric object constructed by gluing together unit cubes of various dimensions along their faces. Given two vertices in X, we define the distance between them to be the minimum length of an edge-path that connects them. The growth series for X, relative to a fixed vertex V, is the power series whose nth coefficient is the number of vertices whose distance to v is n. For example, if X is a trivalent tree (an example of a 1-dimensional cubical complex), then the growth series is G(t) = 1+3t+6t^2+12t^3+... In this talk, we will compute the growth series for a certain class of cubical complexes. Namely, we consider those that are "regular" (any two vertices have small neighborhoods that look the same) and CAT(0) (a technical condition that ensures that X is contractible -- like a tree).
Title: Applications of the Jones Polynomial to Quantum Computation
Abstract: The Jones polynomial is an invariant of knots and links which, almost immediately after its invention by Vaughan Jones in the 1980's, found applications in physics through statistical mechanics and topological quantum field theories. In this talk, I will discuss its relevance to quantum computation by describing new proofs that obtaining a certain approximation of the Jones polynomial of a class of links obtained from braids is a BQP-complete problem, i.e., the hardest task which can be performed on a quantum computer. I will conclude with a quantum computing-based proof that computing the Jones polynomial is a #P-hard problem. This talk is based on joint work quant-phy/0603069 with Pawel Wocjan.
Title: Mathematics of the Rich
Abstract: Let's say you have money (yay!) and you can invest part of your money in a bank account and the other part in the stock market. How should you choose to divide your money between the two investments? And what's the best way to rebalance your investment as time evolves? We will answer these questions for a basic probabilistic model of stock market movement. Our model (like most stock movement models) relies on Brownian motion, so to answer our questions, we will need to develop some mathematical techniques from stochastic calculus that explain both Brownian motion and integration over Brownian motion. The results of these techniques, which are often surprising, form the basis on which many, many other questions in financial mathematics are answered. Background: The talk will assume knowledge of undergraduate calculus and some undergraduate probability (expected value, variance, normal distributions, and the central limit theorem). People who don't know these things will never be rich.
Unless noted otherwise, talks will be at 4:00pm Tuesdays. Room O'Connor 204. There will be refreshments before all talks in O'Connor 31 starting around 3:45pm.