# Project: Abel’s Theorem Animated

**Project supervisor:** Alex Furman

**Description:** The formula for solutions of the quadratic equation z^{2} + a_{1}z + a_{0} = 0 is familiar from high school algebra; they are given by

_{1}± √(a

_{1}

^{2}– 4 a

_{0}))

There exist more complicated, yet similar, formulae for the solution of the cubic and the quartic equations.

However, in 1824 Abel proved that there cannot be a formula, using radicals and elementary functions, that would express the solution of the quintic equation

z^{5} + a_{4}z^{4} + a_{3}z^{3} + a_{2}z^{2} + a_{1}z + a_{0} = 0 in terms of the coefficients a_{0}, …, a_{4}. In 1963 Arnold gave a geometric explanation of Abel’s theorem that revolves around motion of points in the complex plane and the images of these motions under functions and radicals. The goal of the project is to demonstrate these ideas visually through an interactive web application.

**Prerequisites:** Knowledge of javascript programming and familiarity with complex numbers and functions.

# Project: Interactive configuration spaces

**Project supervisor:** Jānis Lazovskis

**Description:** Given points in space, we can connect ones that are close to each other and build simple shapes. As the points move around (that is, as the initial configuration changes to another configuration), how do these simple shapes change? The precise ways to define these shapes and the general structure of these “configuration spaces” give insight into how complex data can be approximated in an understandable way. This project will be about exploring the foundations of topological data analysis and creating an interface through which users can interact with the data.

**Prerequisites:** Linear algebra (Math 310 or Math 320) and some programming experience.

# Project: Solution curves of families of polynomial systems

**Project supervisor:** Jan Verschelde

**Description:** A numerical solver of a polynomial system deforms a systems with known solutions into the given system. The solution curves originate at the known solutions and end at the solutions of the given system. How difficult a system is to solve depends on the difficulty of the solution curves. The goal of the project is to visualize the solution curves and the nearest singular points to the curves.

**Prerequisites:** Python programming, in particular Tkinter.