Developing a Supplementary Secondary School Math Curriculum
The local charter school, Arcadia, which accepts students in grades 6-12, prides itself on using a variety of teaching techniques to teach students at whatever level they need when they arrive. Some students arrive ready to start Algebra, other students need extra review of elementary topics.  All students currently have a study hall during the last hour of their school days; the principal would like to set aside Tuesdays and Thursdays for students needing extra challenge or extra help on math to spend that hour working on math.  He would like a curriculum created for these Tuesday/Thursday meetings for one (or both) of these groups of students. This project will consist of learning about learning styles and developmental progress of young teens, creating a collection of lesson plans and any necessary supplements (such as worksheets or educational videos), testing those lessons on actual students, and then adjusting as necessary.

Spatial Economics through a Mathematical Lens:  City Structure and the Breakdown of General Equilibrium
Description can be found here.

Emergence of Synchrony in Pulse-Coupled Neurons
Emergence is the study of the natural phenomenon of how complex structures, behaviors, and patterns arise out of interactions among relatively simple parts. Examples include the complex behavior of ants and other communal insects, schools of fish, and the collection of neurons that comprise the brain. In our project, we will study the spontaneous synchrony that sometimes arises in a collection of weakly coupled oscillators.  In particular we will examine what kinds of conditions that do and do not give rise to this synchronicity.  We will study mathematical models of simple oscillators and different ways the individual oscillators in a large population can be connected.  Our goal will be to identify which conditions lead to synchrony or other ordered behaviors and which do not.  The tools we will bring to bear on this problem include calculus, chaotic dynamics, simulations, and elementary graph theory.

Hierarchical Modeling
Hierarchical, or multilevel, models are a natural extension of linear models (or glms) that are often used when data doesn’t meet the “independence” assumption needed for many basic statistical inference methods.  In this project, we will explore hierarchical modeling through applications in ecology that lend themselves nicely to the hierarchical framework. The applications that we will study often have two components that generate the data we observe: the ecological process and the observation process.  For example, suppose we are interested in modeling the number of moose in northern Minnesota over time.  The ecological process might suggest that we should model moose abundance as a function of habitat and weather covariates.  The observation process, which is determined by the sampling design, must also be modeled since the observed moose often undercount the actual number present in the population because detection of animals is often imperfect.

During this project you will learn about different types of hierarchical models used in ecology and how statistical inferences are made about these models using both maximum likelihood and Bayesian estimation methods.  A good understanding of probability is a must for these models, so dust off your probability notes! You will also work with real data, learn how to simulate your own data in R, and obtain MLE and Bayesian estimates in R and JAGS.

Measuring Sets in Fractional Dimensions
Description can be found here.

Matching to Produce Causal Estimates in Non-experimental Settings
Description can be found here.

Knot Theory Seminar
Description can be found here.

Topics

• Product-to-Sum Formula for Knots on a Torus
• Generators of the Muller’s Algebra of Curves
• Calculating the Jones Polynomial Using a Quantum Computer