Mathematical Modeling in Ecology: White-nose Syndrome in North American Bats
Professor Alex Capaldi
White-nose syndrome (WNS) is an infectious disease that affects hibernating bats in North America. First discovered in New York in 2006, it has spread across much of the eastern United States and has also been found in Washington. In a mere ten years, it has wiped out approximately 80% of bats in North America. This summer, we will use mathematical models incorporating difference and/or differential equations to study the spread of WNS, the population dynamics of bats, and to evaluate potential control measures.
Prerequisites: at least one of the following: differential equations, statistics, or programming experience
Set-Valued Young Tableaux and Lattice Paths
Professor Paul Drube
Standard Young tableaux are two dimensional arrays of positive integers arranged so that the numbers used increase from left to right across each row and increase from top to bottom down each column. Standard Young tableaux of a fixed shape are quickly enumerated via the famous Hook Length Formula, and there are numerous well-studied bijections between standard Young tableaux and other combinatorial objects such as integer lattice paths. In this project, we study a generalization of standard Young tableaux known as standard set-valued Young tableaux, whereby the entries are disjoint sets of integers as opposed to a single integer. The primary focus of our investigation will be the enumeration of these tableaux for a variety of basic shapes, as well as the development of new maps between standard set-valued Young tableaux and integer lattice paths satisfying specific properties.
Prerequisites: linear algebra or another introductory proof-based course
Noise-Induced Stabilization of Hamiltonian Systems
Professor 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. Due to their structure, Hamiltonian systems can never be stabilized by noise that is constant in space. This project will investigate how to deterministically perturb a certain class of unstable Hamiltonian systems in order to retain the instability and essential behavior of the systems, but allow for noise-induced stabilization to occur with constant noise.
Prerequisites: ordinary differential equations and basic probability/statistics
Classification and Characterization of Networks
Professor Karl Schmitt
Networks and graphs have become an ubiquitous way of describing systems in science, ranging from protein-protein interaction networks to power grids to the internet. The networks generated from different disciplines often exhibit attributes that are very similar within a discipline (e.g. biological:protein networks) but different between disciplines. Many graph models have been proposed to capture important traits and it can often be difficult for researchers to pick the correct model a priori.
In this summer project we will work with data mining techniques on network data (such as the www.NetworkRepository.com) and generate graphs to develop, validate, and deploy a learner/predictor for graph classification and most likely theoretical model. This will build on previous student work which created graph ‘fingerprints’ for most of the repository.
An additional direction of research is to investigate comparing graphs (and theoretical models) by developing and comparing regression models that predict the rate of graph discovery or inference.
Prerequisites: at least one of: programming experience (preferably in Python or R), statistics, or graph theory