Spring 2022

Spring 2022 Projects

Folding paper from planar tilings

Tiling to folding

Project Supervisor: Marcus Michelen

There are many ways to tile the plane using regular polygons.  Tilings for which there are only k different types of vertices are called k-uniform tilings.  The goal of this project is to implement an algorithm that takes in a k-uniform tiling and outputs a crease pattern for folding paper into a pattern inspired by this tiling.  Once the crease pattern is made, they can either be etched onto paper to be folded or fed into an origami simulator to create a 3D model.
Prerequisites: Programming experience, preferably in Python

"Impossible" Behavior: a Mathematical Investigation into Quantum Tunneling

Quantum Tunneling

Project Supervisor: Evelyn Richman

In classical mechanics, objects obey rigid and intuitive laws of motion; a ball left to roll down a hill will not have enough energy to roll over a second, taller hill. The same behavior might be expected of charged particles (like electrons) in “hills” made of electric potentials. However, the Schrodinger equation, which governs the dynamics of quantum particles, suggests that particles can cross potentials which have more energy than the particles themselves. This phenomenon is known as quantum tunneling. In this project, we will attempt to prove the existence of quantum tunneling by using tools from mathematical analysis and partial differential equations.
Prerequisites: Math 220 and preferably Math 310 or 320 (no programming experience necessary)

Exploring Lattice Polytopes through Problems Inspired by Combinatorial Commutative Algebra

Rhombicosidodecahedron

Project Supervisor: Kevin Tucker

Polytopes are higher-dimensional generalizations of two dimensional polygons and three-dimensional polyhedra, and are geometric objects with interesting combinatorial properties. They have applications to diverse areas of both pure and applied mathematics, and in particular are used to study toric varieties in algebraic geometry and affine semi-group rings in commutative algebra.  The geometry of polytopes is particularly well-suited to experimentation, and there is indeed much to explore as a number of easy to formulate and long-studied questions about polytopes remain unanswered. In this project, we will leverage the connection between algebra and geometry to explore certain newly defined invariants for affine toric varieties and semi-group rings over finite fields using polyhedral geometry.

Prerequisites: Math 310 or 320 and some programming experience preferred.

Arboreal dynamics and symmetries in the roots of polynomials

Rooted tree of roots

Project Supervisor: Wouter Van Limbeek

Let f be a polynomial, fix some scalar a, and consider the roots of f^n(x) = a as n tends to infinity. It turns out that all the solutions together form a graph called a “rooted tree” and f induces a map on this graph that pushes vertices towards the root (see picture). For fixed n, there are more symmetries coming from the so-called “Galois group” of f^n and these permute the numbers at a fixed distance from the root. It is conjectured that (aside from some sporadic exceptions) all possible permutations arise, i.e. the roots are “maximally symmetric”. Likewise there is a conjecture characterizing the supposedly very special polynomials that give rise to “minimally symmetric” roots. There is a lot of evidence for these conjectures, but nevertheless they remain open for most polynomials. In this project we will make computer-assisted computations to investigate unknown cases of these conjectures.

Prerequisites: Programming experience.

Numerical adventures with prime numbers

Escher

Project Supervisor: Evangelos Kobotis

The purpose of this project is to acquaint the student with the mysteries of prime numbers and explore the numerical techniques that allow us to explore them. We will look at numerical questions concerning the distribution of prime numbers, primality testing, sequences of prime numbers and other topics. Our main research goal will be to provide individual prime numbers or sets of prime numbers with interesting properties as well as relevant visualizations For our programs we will mostly use the Python programming language. There will be in person meetings but the project team will meet mostly online

 

Prerequisites: An interest in prime numbers. All background material and programming techniques will be covered from scratch.