Copoint graphs of convex geometries
Project mentor: Dr. Jon Beagley

Convex geometries are a discrete generalization of convex sets.  A graph can be generated using certain convex sets as the vertices and a relation between those sets induces the set of edges.  This graph is known as the copoint graph of a convex geometry.  One property of this graph, when the convex geometry comes from point sets in real space, is that the cliques of the copoint graph have a direct correspondence with sets of points in convex position.  The focus of this investigation is to investigate the other properties of the copoint graph, particularly whether certain induced subgraphs must occur in specific situations or are forbidden.

Prerequisites: introductory combinatorics, or another proof based course

Studying the optimal threshold of a latent variable in a bivariate distribution
Project mentor: Dr. Hui Gong

In some situations, a latent variable follows a bivariate distribution; for example, purporting to identify disease, detect creditworthiness or other conditions. Decreasing the false positive in one distribution will inevitably increase the false negative in the corresponding other distribution. An optimal selection will balance both false positive and false negative. In this study, we will focus on deriving this optimal selection in bi-normal distribution, which has been shown by several studies as a reasonable choice in diagnostic studies, especially given the mathematical convenience. A bi-normal distribution has five parameters: μ1, μ2, σ1, σ2, and ρ. The values of these five parameters decide whether we can find a closed form for the optimal threshold. In this project, we will derive this closed form if exists, or approximate this closed form if it cannot be expressed explicitly. Simulation may be applied if numerical approach is necessary. If time permits, the study may extend to other bivariate distributions.

Prerequisites: a course in statistics; exposure to probability and knowledge of program R is not required but helpful.

Asymmetric demographic models
Project mentor: Dr. Daniel Maxin

Demographic  mathematical models have a long history. The most challenging mathematical components of these models are the couple-formation functions. These functions link the number of pairs  with the number of available singles but  they are not detailed enough to include important aspects of social behavior such as: motivation for pairing which may be gender specific, scarcity or abundance of the opposite gender or economic factors.

This project consists in adapting a typical gender structured model to  better describe asymmetric demographic situations. Examples are:  a discrepancy between the two genders with respect to  pairing and/or fertility patterns,  cultural effects on pair-formation dynamics such as a prevalence of established marriages that causes  social pressures on the single population to marry,  marriage avoidance represented by certain groups that reject the traditional marriage and remove themselves from the dating pool,  a sudden drop in the size of one gender (as in the aftermath of major wars).  A secondary objective is to test some of these models against population data from countries and/or historical times in which these imbalances manifest themselves.

Prerequisites: a course in Ordinary Differential Equations and some familiarity with a computer algebra system (Maple is preferred since it is the software we will use).

Packing patterns in words
Project mentor: Dr. Lara Pudwell

A permutation of length n is an arrangement of the numbers {1,…,n}.  It is well-known that there are n! permutations of length n (for example, the 3! permutations of {1,2,3} are 123, 132, 213, 231, 312, and 321).

One permutation (p) contains another permutation (q) if there is a subsequence of the p whose digits are in the same relative order as q.  For example, p=162534 contains the pattern q=321 because the digits 653 appear in p in the same relative order as 321; that is, largest, then middle, then smallest.  This is just one of several copies of 321 inside of p.

Much of the literature on permutation patterns focuses on permutations that avoid a pattern (i.e. have 0 copies of the given pattern).  However, we can ask a related question: what is the largest number of copies of q that appear in a permutation of length n?  Finding the permutation with the largest number of copies of a pattern is known as pattern packing.

Pattern packing has been studied in permutations, and it has also been studied in words (lists of numbers with repeated digits, such as 13234).  In this project, we’ll consider pattern packing in a special sets of words that have additional symmetries.

Prerequisites: at least one of the following: discrete mathematics, combinatorics, programming experience

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