Dynamic Models of Geologic Data from Ridge-Transform Intersections

The rock that makes up the Earth is constantly fracturing and deforming. Understanding these processes is important for assessment of earthquake hazards, exploration of natural resources, reconstruction of tectonic history, and other applications. Mathematically, rock deformation is often modeled as fluid flow, which is described by partial differential equations such as Navier-Stokes, which are solved using numerical methods such as finite element models (FEMs).

This comps project can go in several directions — theoretical, computational, statistical — depending on the interests of the students, but here is a tentative plan. We first study the Navier-Stokes equations and finite difference methods. We then briefly study FEMs. We use FEM software to model a particular geologic situation (the intersection of a spreading ridge and a transform fault) where there is currently significant disagreement between theoretical models and observed data.

The project will run during the fall and spring terms. The prerequisites are a strong knowledge of multivariable calculus and linear algebra, and a willingness to learn computational tools. Prior experience with differential equations (Math 241 and 341) is not required.

Alternating Sign Matrices, Pattern Avoidance, and Domino Tilings of Aztec Diamonds

You can find the original description of this project here, but the final paper focused on a particular problem related to permutations and tilings of an Aztec diamond.  In particular, in 1992, Elkies, Kuperberg, Larsen, and Propp introduce a bijection between domino tilings of Aztec diamonds and pairs of alternating sign matrices whose sizes differ by one.  In this project we first studied those smaller permutations (viewed as matrices) which are paired with doubly alternating Baxter permutations.  We call these permutations snow leopard permutations, and we used a recursive decomposition to show they are counted by the Catalan numbers.  We then gave a simple, elegant bijection between the set of snow leopard permutations of odd length and the set of Catalan paths.  Next, we described a specific set of transpositions which generates these permutations.  Lastly, we investigated the general case, where we conjecture that the number of larger permutations paired with a given smaller permutation is a product of Fibonacci numbers.  You can read the details of our results in our paper, for which there’s a link below, and you can also use some of the code we wrote.

Boolean Numbers and Graph-Induced Sequences

Description can be found here.

Surfing on Wavelets

Description can be found here.

Strange Worlds in Number Theory

Description can be found here.

A Tour of Lie Theory

Mathematics of Games

Description can be found here.

Consequences of Alcohol Use in a Social Network of College Students

Researchers from many fields are more and more interested in how different entities–be they neurons, genes, or people–interact with each other.  Network analysis methods are increasingly utilized to understand these interactions and connections.  The analysis of network data presents many opportunities as well as challenges.  For example, the presence of interrelationships that form the connections in a network render inference based on assumptions of independence invalid.

This project will investigate how the consequences of alcohol use (such as increased risk-taking, blacking out, and hangovers) in a network of college students are associated with individual-level characteristics, while taking into account the complex dependencies inherent in network data.  Questions that this project may address include: 1.) Are consequences of alcohol use clustered within the network? 2.) Are friends more likely to have similar consequences of alcohol use than strangers?  3.) Are individuals who perceive their friends to be supportive of them less likely to have negative consequences of alcohol use after controlling for alcohol consumption?

This project will apply network analytic methods that draw from statistics, graph theory, and linear algebra to data collected from college students living in the same residence hall.  Students will explore measures of network attributes, network models, methods for visualizing networks, and statistical methods for detecting associations with network data. Added insight into these topics will be gained by applying these concepts to both the Residence Hall network data as well as networks simulated in R.

Measuring the Mississippi

Scientists at the USGS’s Upper Midwest Environmental Sciences Center (UMESC) research and monitor the ecosystems in and around the Upper Mississippi River.  Students in this comps project will have a chance to collaborate with researchers at the UMESC and analyze real data collected by these scientists. You might also have the opportunity to work with ENTS seniors who will be conducting their comps research on the Mississippi River.  The final research topic for this project has yet to be determined (it might be influenced by the ENTS research projects) but one possibility is the following:

• Submersed vegetation are important to river health, impacting both water quality and fish health.  The UMESC has collected vegetation data across many years and river locations using a variety of sampling designs and measurement methods.  Raking is a simple measurement method in which vegetation is raked up to the surface of the water by researchers in a boat. The amount of vegetation on the teeth of the rake is categorized into scores of 0 (no vegetation retrieved) to 5 (91-100% of teeth filled). There is debate about how this measurement should be used: either quantitatively using the numerical scores 0-5 or qualitatively as an ordered categorical variable (e.g. none, low density, …, high density).  We will research this question using both the data collected by the UMESC and simulations studies.

Abstract for Knots