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Mathematical Sciences

Thesis Presentations

Date: 3:30pm PDT April 24, 2012 Location: JR Howard Hall, Room 114.

JR Howard Hall, Room 114.

Mathematical Sciences Colloquium:

Jeff Cruttenden, Black-Scholes Implied Volatility: The Implied Smile and Local Volatility Model

After studying the ramifications of the constant volatility assumption in the Black-Scholes model for European options, we search for a diffusion process under risk-neutrality consistent with the distribution implied by observed option prices and market smiles.

Heather Kitada, Catalan Interpretations of Commutative Ideals, Partially Ordered Sets and Path Statistics

In this paper, we will offer several interpretations and bijections of sets of object enumerated by the Catalan numbers.  Many of these objects are characterized by binary characteristics; however, we will also show more complex structures with applications in abstract algebra.  In addition, we will demonstrate a counting argument for the number of ideals in the ring of upper triangular matrices,which is known to be the nth Catalan number.  Furthermore, we will focus on the subset of ideals that are commutative  with the hopes of illuminating partially ordered set organizations of other Catalan objects, including Dyck paths, rooted plane trees, binary trees, nonintersecting arcs, Temperley-Lieb Diagrams, and 312-avoiding permutations. 
Finally, we will explore single and bivariate Catalan generating functions that utilize statistics on Dyck paths, namely bounce and area.  It is known that this q,t-Catalan generating function is symmetric through the use of product notation; however, a combinatorial argument doesn’t currently exist. We will look into other open questions that should lead to progress in this argument.  Additionally, we will examine the role that commutative objects play in these generating functions.

Grace Zalenski,  Mathematical exploration of fluid flow through sea ice

I will describe the physical processes underlying mathematical models of sea ice and the methods used to create such models. I will address the derivation of the major equations used in modeling fluid flow in sea ice (and similar applications). I will present a numerical simulation of the evolution of a sea ice melt pond. I will also discuss the motivation for studying sea ice as well as some inherent challenges.