Constructing 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 convexly independent sets. The focus of this project is to investigate the other properties of the copoint graph, particularly whether certain induced subgraphs must occur in specific situations or are forbidden. This is a continuation of a project from 2018.
Prerequisites: introductory combinatorics or another proof-based course.
Biomath Modeling: Pollen Competition
(Project mentor: Dr. Alex Capaldi)
Many aspects of plant sex remain a mystery to the biological community. This summer, we will use mathematical and computer simulation models to study the dynamics of pollen competition and to determine successful evolutionary strategies. In particular, we will use an agent-based modeling (ABM) approach. ABMs are models where individuals (agents) are unique and autonomous and interact with each other and their environment locally. ABMs have become a popular modeling process in the last decade due to the rise of computational power so we will be able to answer novel biological problems with new modeling techniques!
Prerequisites: At least one of the following: differential equations, statistics, programming experience.
Generalized Motzkin Paths and Standard Young Tableaux
(Project mentor: Dr. Paul Drube)
Blurb: Standard Young tableaux are two-dimensional arrays of positive integers in which numbers increase from left to right across each row and increase from top to bottom down each column. There are numerous well-studied bijections between standard Young tableaux of fixed shape and combinatorial objects such as Dyck paths, and standard Young tableaux with at most three rows have been placed in bijection with the distinct class of lattice paths known as Motzkin paths. For this project, we will focus upon a generalization of standard Young tableaux known as standard set-valued Young tableaux, in which each entry of a tableau may contain more than one integer. We will look to enumerate specific classes of standard set-valued Young tableaux, and will work to develop new bijections between such tableaux and different varieties of integer lattice paths. In the case of standard set-valued Young tableaux with at most two or three rows, this will involve a consideration of generalized (colored) Motzkin paths, prompting a more general consideration of such integer lattice paths.
Prerequisites: linear algebra or another proof-based course.
Noise-Induced Stabilization of Hamiltonian Systems
(Project mentor: Dr. Tiffany Kolba)
The phenomenon of noise-induced stabilization occurs when an unstable deterministic system of ordinary differential equations is stabilized by the addition of randomness into the system. This is quite a surprising and intriguing phenomenon because one’s first intuition is often that noise will destabilize, rather than stabilize, a system.
Hamiltonian systems are characterized by a Hamiltonian function that is constant along each solution curve. Hamiltonian systems arise in many physical applications, where the Hamiltonian function represents the total energy of the system. Due to their structure, Hamiltonian systems can never be stabilized by noise that is constant in space. This project will investigate how to stabilize a certain class of unstable Hamiltonian systems by constructing a noise term whose strength depends upon the space variables.
Prerequisites: an ordinary differential equations course and a probability/statistics course