# Fall 2018 Projects

Visualizing the unit ball of the AGY norm

Grad student mentors: Carsten Peterson and Matt Stevenson

Undergraduate researchers: Vignesh Jagathese, Jason Liu, and Jacob Shulkin

[Final poster]

Project Description: Avila-Gouezel-Yoccoz defined a norm on a vector space attached to a surface built out of triangles. This norm plays a major role in the work of the Fields medalists Avila and Yoccoz, and is also related to the work of the Fields medalists McMullen and Mirzakhani. Since its definition only recently in 2013, we still do not have a detailed understanding of this norm. For example, it is unknown if the norm contains enough information to reconstruct the precise shape of the surface, as is famously the case in a related situation. The goal of this project is to write a computer program to draw pictures of two-dimensional slices of the unit ball. Such a picture will be a convex set in the plane. It probably won’t be a round circle in this case, but it may look like some sort of distorted circle. It may or may not have corners, and it may or may not have flat edges. This project will empirically resolve these mysteries and lead directly to ongoing research. See the extended abstract for more details and for pre-requisiteshttp://www-personal.umich.edu/~alexmw/AGY.pdf.

Geodesics of polyhedral Finsler metrics in Euclidean space

Grad student mentors: Francesca Gandini and Mark Greenfield

[Final poster]

Project description: In this project we will study polyhedral Finsler metrics on Euclidean space. These are (often asymmetric) metrics whose unit ball is a convex polyhedron. A familiar example is the taxicab (L^1) metric. One interesting feature about these metrics is that some pairs of points have infinitely many distinct paths which minimize distance (geodesics). Our goal in this project is to understand, for as large a class of metrics as possible, what the collections of geodesics between different pairs of points look like. This will consist of formulating and proving (or disproving) various conjectures about how the shape of the unit ball influences the collections of geodesics. Another goal is to write a program which draws these collections for us.

Prerequisites: Math 215 and 217 (or equivalent proof experience) essential. Math 451 and/or 490 would be very helpful. Computer experience for at least half the team needed (basic proficiency in Mathematica, Sage, or Python should be sufficient).

Modeling convex projective geometries

Grad student mentors: Samantha Pinella and Robert Walker

Undergraduate researchers: Rudreshwaran Ranganathan, Steven Schaefer, and Hanissa Shamsuddin

[Final poster]

Project description: The main goal of the project is to produce 3D models of the following objects, which are examples of convex projective geometries: https://www.math.u-psud.fr/~benoist/dessin/convexe1anime.html

Along the way, we will learn about the words in the title, the geometry of these objects, and the symmetry groups of these objects. We will also study deformations of these objects which preserve the topology of the symmetry groups. Open questions we may explore include (1) How do the geometries degenerate with large perturbations? (2) Can you take a two dimensional slice of one of these objects which has exactly two disjoint open line segments in the boundary? (3) What happens to translation distance of group elements as you perturb the object? (4) Does the topological complexity of the symmetry group decrease as the geometry degenerates? (the last question would be hard)

Prerequisites are Math 217 or equivalent. Recommended background is at least two distinct elements of the set {algebra, topology, programming skills}.

A visual approach to complex analysis

Undergraduate researchers: Yuxuan Bao, Yucheng Shi, and Justin Vorhees

[Final poster]

Project description: To accurately plot the graph of function from the complex plane into itself, we would generally need to draw a four (real) dimensional picture. But there are other ways to do this. By using a counterclockwise ROYGBIV coloring of the complex plane, we can visualize the argument or phase of complex valued functions. It turns out that if the function in question is holomorphic, then the “phase plot” captures all essential information about the function, in that any two holomorphic functions with the same phase plot are necessarily positive real multiples of each other.

In this project, we will use an interactive complex function visualizer that allows the position of the mouse to be used as an input of a second complex variable into function. As we move the mouse around the screen, we are actually playing with a four (real) dimensional object, two dimensions at a time!

There are several different directions I would like to take this project. The phase plots make it easier to understand many geometric properties of holomorphic functions.

1. Monodromy and Analytic Continuation: Use the phase plot visualization to understand Riemann surfaces arising as branched covers of planar domains. Then investigate the shape of the Bergman kernel and mapping properties of the Bergman projection associated to certain Riemann surfaces with boundary.
2. Moving from qualitative to quantitative information using phase plots: Zeroes and Poles of polynomial functions are easy to detect from phase plots. The Gauss-Lucas Theorem says that the zeroes of the derivative of a polynomial in the complex plane necessarily lie inside the convex hull of the zeroes of the original polynomial. A stronger version of this theorem can be expressed in terms of curves of constant argument. Therefore, understanding these curves for various holomorphic functions is of interest.
3. Complex Dynamics: Iterating holomorphic functions often results in fractal pictures. We will use the phase plot visualizer to better understand the complex dynamics of various polynomials and transcendental functions

Pre-requisites  Experience in Math 555 (Complex Variables) or an equivalent course is very strongly suggested (but exceptions could be made).  A proof based Advanced Calculus and/or Real Analysis course is also highly recommended.  Coding skills, especially Mathematica, MatLab and/or Sage are also desirable.  Special attention will be given to applicants with experience with Xcode, Swift and using the iOS libraries and frameworks. 