# Title

Below are projects offered by IC Faculty.  Contact the faculty member to discuss working with them.

Note: funding is competitive and not all students may be supported for the H&S Summer Scholars program.

### Projects for Summer 2020

Project mentor:  David Brown
Project title:  Pruned Space-Filling Fractal Trees
Project Description:  This project explores a variant of iterated function systems (IFS) that define attractors for fractal trees.  This project considers what happens when specific compositions of functions are forbidden to be used in building the tree and its associated fractal attractor.  This effectively removes particular branches (and the branches that grow off from them) from the tree structure and hence we call these pruned trees.   Students will explore how the pruning affects the topology of the associated fractal attractor.  In many cases, pruning causes the tree to lose connectivity; while in others, this does not occur.  Even in cases where connectivity is lost, a rescaling of the pruned IFS sometimes re-establishes connectivity.   Some (non-pruned) trees exhibit fractal attractors that are space-filling and pruning sometimes results in complete loss of this space-filling property and sometimes it does not.  Students will explore the relationships among the pruning sequence, the scaling ratio, and the branching angle as they investigate the topology of the fractal attractors.  Students will use visualizations that they create in Python or Sage to provide visual clues to conjecture about the topology of the attractors and then use analysis to move toward proving their conjectures.  Background:  Students who have completed Calculus II and Math Experimentation (or COMP 17100).

Project mentors:  Ryan Moruzzi and Emilie Wiesner
Project title:  Inserting and pruning trees via matrices
Project Description: This project is focused on developing computational tools for “inserting” and “pruning” operators on graph-theoretic trees. Understanding these operators is an important ingredient for several research questions we are currently exploring in representation theory. Trees can be encoded in (0,1)-matrices, which provides a possible avenue for using linear algebra to explore the inserting and pruning operators via computer-generated examples. Students will help develop code to model the inserting and pruning operators (using Octave or other suitable computer language), as well as explore combinatorial and algebraic concepts related to trees.

Project mentor: Matt Thomas
Project title:  Developing a distributed open educational network
Project Description:  This project is part of a collaboration between a few different institutions to develop software to allow teachers to run their own educational experiments. In this project, you will work with me and other members of the team in the development of pilot experiments for calculus and linear algebra classes. This will involve developing interactive applets to be posted to the web, developing the language they are written in, developing instructional interventions using the software, and/or planning analysis of the results. There is a lot of flexibility in the direction of this project. Background will depend on the particular aspects of the project that are of interest, but could include: knowledge of calculus and/or linear algebra, interest in thinking about the development of knowledge of calculus and linear algebra, familiarity with markup languages like LaTeX or html, statistical analysis, programming in R, javascript, and/or python.