Mathematics Colloquia

Fall 2003

Oct. 10 (12 pm, LA5-267): Alfonso Castro, Harvey Mudd College.

From college algebra to topology and nonlinear elliptic partial differential equations.

Building on elementary notions such as when a system of linear equations has a solution (Fredholm alternative) results on the solvability of nonlinear partial differential equations (NLPDE) will be discussed. The role of topological properties of level sets in the existence of multiple solutions to NLPDE will be presented.

Oct. 24 (2 pm, LA5-267) Scott Annin, CSU Fullerton.

Rootless matrices.

Given a square matrix over C, we ask: Does it have a square root? If the matrix is diagonalizable, you may know that the answer is always "yes". But what about in general? And what about cubed roots, fourth roots, fifth roots, and so on?

In this talk, I will describe joint work with a Cal State Fullerton student on the classification of rootless matrices. A rootless matrix is a square matrix over C that fails to have a square root, cubed root, fourth root, and so on. This work generalizes results published by B. Yood in Mathematics Magazine (June, 2002) by using the machinery of Jordan Canonical Forms. After reviewing this important tool, I will use it to tackle the classification problem described above. The results may surprise you, and they lead to a host of other intriguing questions as well.

Oct. 31 (12 pm, LA5-267) Marco Sammartino, Universita di Palermo, Italy.

Asymptotic methods in option pricing.

We consider the problem of the pricing a European option in presence of transaction costs and stochastic volatility. It is well known that when the volatility is constant and when the transaction costs are absent this problem is solved by the celebrated Black-Scholes pricing formula. The presence of the transaction costs makes the problem more difficult because a perfect replication through delta hedging is no longer possible. Following Davis, Panas and Zariphopoulou we address this problem seeking for the optimal hedging strategy through utility maximization. This leads to the problem of solving a Hamilton-Jacobi-Bellman PDE.

We solve this equation first supposing small proportional transaction costs and following the asymptotic procedure suggested by Whalley and Wilmott (1997). Then we suppose that the volatility follows an Ornstein-Uhlenbeck stochastic process which is fast mean-reverting with a normal invariant distribution. We finally find a pricing formula for a European call option which involves, at the leading order, the classical Black-Scholes price, and the correction terms that we calculate explicitly.

Nov. 14 (12pm, LA5-267) Todd Ebert, CSULB.

Solving the Hat Problem with an Infinite Number of Players.

The “Hat Problem” with n players represents a game in which each player is assigned to wear a hat whose color is unbeknownst to him and is determined by a coin toss. The players are then led to a room where each has three choices: to guess (either red or blue) the color of his hat, or to pass. During the game no communication is allowed and the choices are written in secret. The players share an $n million prize if at least one player guesses, and all guesses made are correct. They win nothing if no one guesses or if there is an incorrect guess. If the players are allowed a strategy session before the game, what strategy should they adopt in order to maximize their chances of winning? Hint: one strategy is to assign one player to guess while all others pass. Such a strategy has a 50% chance of success. The players can do much better than this!

The first half of the talk will be devoted to solving the hat problem for the case when n=2 ^k -1. On our way to the solution we will encounter linear error-correcting codes and derive some of their properties using linear algebra over the field Z₂. The optimality of these codes will then be shown using an elegant probability argument for which Kent Merryfield was among the first to propose.

For the second half of the talk we will consider the Hat Problem for the case when there are an infinite number of players P₁, P₂, …, who choose in that order. Players win money as follows. If player P_i correctly guesses, then she and the players before her share a $1 million prize. In the advent of an incorrect guess all players must relinquish their winnings up to that point and the game continues. For this game we will demonstrate a guessing strategy that, if adopted by every player, will make the following statements true with probability one:

• An infinite number of players will venture a guess.
• All but a finite number of the guesses will be correct.

• Every player wins an infinite sum.

Dec. 5 (12pm, LA5-267) Robert Mena, CSULB.

Using matrices to count the rationals.

Cantor is deservedly well-known for having shown both that the rational numbers could be counted and that the real numbers could not be so. However, his diagonal method of counting left a lot to be desired. This talk, to which any junior in mathematics is welcome, introduces a family of 2 by 2 matrices to set up a truly computable one-to-one correspondence between the positive rationals and the positive integers. In addition to the matrices, a fabulous sequence that should be better known, the Stern-Brocot sequence will be introduced and used to set up the correspondence.