2014-15 Academic Year Colloquium Schedule

September 4, 2014

September 11, 2014

Probability is a branch of mathematics whose roots lie in gambling. While evidence of games of chance may be found in the artifacts of many ancient civilizations, the underlying mathematics that can be used to analyze these games is a far more recent development. In this talk, the mathematics underlying games of chance will be explored and the relative house advantages of many popular (and some obscure) casino games will be examined.

This talk is based on the book Basic Gambling Mathematics: The Numbers Behind The Neon, recently published by Taylor & Francis/CRC Press. Copies of the book will be available for purchase.

September 18, 2014

September 25, 2014

October 2, 2014

Classical credit risk and pricing models typically assume that the expected recovery at default is constant, or at the very least independent of the default probability. However, a large body of recent empirical evidence has challenged this assumption and shown that default rates are in fact negatively correlated with recovery rates \cite{ABRS}. Recently, Moody's Analytics proposed a model in the context of credit capital which incorporates this empirically observed correlation within a structural framework \cite{LH}. In this work we revisit Moody's PD-LGD correlation model and in the process complete and extend several results. We then price Bond and Credit Default Swaps with recovery risk using the PD-LGD model under both the Merton and Black-Cox default assumptions, and in addition compute associated risk metrics and Greeks. Our results are then compared with classical results which assume no recovery risk.

Talk Slides are available at http://www.math.msu.edu/~albert/CreditTalkAlbion.pdf.

October 9, 2014

October 23, 2014

October 30, 2014

November 6, 2014

Graphs provide terrific models, and some powerful mathematical machinery, to better understand many practical and theoretical problems. One important relationship between vertices of a graph is the length of the shortest path connecting them. We will explore a class of problems which seek to fix this distance between vertices while reducing the order of the graph.

Consider a simple graph G of order n. We say G is distance-preserving if, for all integers k such that 1<k<n, there exists an order k induced subgraph of G where dG(x,y)=dH(x,y) for all pairs of x,y∈H. We will explore the definitions and properties of distance-hereditary graphs, distance-preserving graphs, and distance-preserving trees. We will then continue with an extremal proof on the constructability of regular distance-preserving graphs. Additionally, we will discuss some open problems and see some practical applications.

November 13, 2014

November 20, 2014

We consider the effect of constraints on the number of nonnegative integer solutions of x+y+z=n, relating the number of solutions to linear combinations of triangular numbers. Our approach is geometric and may be viewed as an introduction to proofs without words.

We use this geometrical perspective to prove identities by counting the number of solutions in two different ways, thereby combining combinatorial proofs and proofs without words.

This will be an interactive talk where those in attendance will get to use triangular graph paper to construct proofs of some of the results.

This talk is based on a paper of the same name that is co-authored with Matt Haines and Ryan Huddy.