Log(M)
Log(M) - The Laboratory of Geometry and the university of Michigan
The Laboratory of Geometry is a semester long research opportunity for undergraduate students at the University of Michgian. In small groups students explore a mathematical research topic. Projects often have computational and visualization parts. Here I share the projects I have mentored in the past years.
The motion of three interacting point vertices on the surface of a dumbbell as computed by the students.
Supervised with Dan Cianci and grad student mentor: Yuxin Wang
Undergraduate researchers: Christian Capanelli, Meixuan Sun, Yupeng Zhang
Colloquially, we refer to a vortex as anything that causes nearby particles to follow a roughly circular trajectory. For instance, water flowing down a drain or hurricanes in the atmosphere are examples of these types of vortices. One way to try to capture this notion mathematically would be to look at the curl of a 3-dimensional vector field. Recall that the curl of a vector field measures the tendency of nearby particles traveling along the vector field to rotate. So if we think of the vector field as giving the velocity of a particle in some fluid at a point in space, then the curl measures how “vortex-like” the fluid is at a particular point. Indeed, in fluid mechanics the curl of a vector field is called the vorticity.
In this project we will study an idealized fluid in two dimensions where the vorticity is concentrated at certain discrete points called point vortices. These point vortices behave like particles and will follow paths determined from the configuration of other nearby point vortices (similar to how planets follow orbits depending on the configuration of nearby planets). They provide an interesting toy model for understanding fluids. The goal of this project is to computationally investigate the dynamics of these point vortices in relatively “complicated” geometries. For instance, what happens to their motion when the point vortices are obstructed by impenetrable islands or when their motion is constrained to curved surfaces (e.g. spheres and tori) instead of the plane? Initially, we will try to understand the equations that govern the point vortex dynamics in these geometries. Then we will focus on how to (efficiently) compute the trajectories (i.e. numerically solve the equations that govern the dynamics).
Robinson Schensted Knuth Correspondence
The Robinson Schensted Knuth correspondence establishes a map between randomly distributed permutations and Standard Young tableaux. The students have generated a growth process on random permutations, which then induces a growth process on the corresponding Standard Young tableaux. Fluctuations in this growth process give rise to the Tracy-Widom distribution.
Fredholm determinants vs longest increasing subsequence
Computation of the Tracy-Widom distribution via the evaluation of Fredholm determinants was also investigated which led to the right probability density function. This was contrasted with an Monte-Carlo simulations of uniform random pertubations, whos longest increasing subsequence can be rescaled to converge to the Tracy-Widom distribution, as shown in the orange histogram.
Numerical Approaches to the Tracy-Widom Distribution
Supervised with Guilherme Silva and graduate mentor Yuchen Liao
Undergraduate researchers: Yuxuan Bao, Chen Shang, Jiacheng Xu, John Dolan
Project description: You and your friend are sightseeing in New York, walking from Columbus circle to the Brooklyn bridge. Your friend has a route planned that takes them along all interesting sights. You, on the other hand, don’t want to be seen near a map and instead take arbitrary turns leading south and east, trying to see as many sights as possible. You don’t see all the sights but arrive earlier to the Brooklyn Bridge. What is the chance that you wait for your friend for a short time? The Tracy-Widom (TW) distribution will tell you.
This naive example is one of many modern problems in mathematical physics, number theory, statistics, random networks, data science, which are deeply connected to the TW distribution. However, unlike in the Gaussian case that gives rise to the bell-shaped curve, visualizing the graph of the TW distribution is a formidable task. Luckily enough, there are now many different ways to express the TW distribution in mathematical terms. Perhaps the most prominent ones are 1. expressing the TW in terms of a solution to a nonlinear differential equation, 2. calculating the TW using a large determinant expression and 3. extracting the TW from a random model.
The goal of this project is to explore these three different mathematical venues from the numerical perspective. With numerical experiments we want to explore several aspects of the TW, compare the different methods of computing a TW and visualize them to gain intuition. Time permitting, novel related distributions will also be explored under the same perspective.
Probability density function and histograms compared.
Supervised with Andrei Prokhorov and grad student mentor: Elizabeth Collins-Wildman and Benjamin Krakoff
Undergraduate researchers: Xiaoqi Peng, Wenhao Deng, and Hexin Cui
In this project we studied the zeros of solutions to Painlevé ODEs. There are six Painlevé ODEs which are second order nonlinear equations with applications in other fields of mathematics and physics. They admit families of rational solutions. As the degree of these solutions gets large, their complex valued zeros fill out certain shapes in a complex plane, for example triangles for Painlevé III and combinations of triangles and rectangles for Panlevé IV. It can happen that the solutions depend on extra parameters and it is intriguing to observe how zeros move as parameter changes. We want to focus on the less studied family of rational solutions to the Painlevé VI equation. We expect that it depends on parameters. Our first goal is to compute and visualize its fascinating zeroes pattern.
Zeros of the a certain Umemura polynomial in dependence of a parameter. These zeros can be linked to the poles and zeros of rational solutions to the Painlevé V equiation.
Conway’s Gama of Life
Winter 2021 Supervised with Katie Storey
Grad student mentor: Benjamin Riley and Carsten Sprunger
Undergraduate researchers: Faye Jackson, Erya Du, Trey Smith and Kevin Ke
To the left: Life forms the students discovered for a new set of rules for Game of life. Conway’s original rule takes into consideration of each cell’s immediate 8 neighbors. In this extension more layers (two for the left image) of cells are counted as alive and weighted by the inverse square of their distance to the cell evaluated. This gave rise to new life forms such as still lifes, oscillators and gliders.
In this LoG(M) project we studied Conway’s Game of Life, a 2D cellular automaton. The rules of Conway’s game of life are deceptively simple. A cell can be alive or dead:
1. A living cell with two or three living neighbors survives to the next time step
2. A cell with no or one living neighbor dies
3. A dead cell with exactly three neighbors will be alive in the next time step
These rules seem to balance the population and allow for interesting dynamics. We will implement our own Game of Life and study patterns and behavior of this system. After studying the many life forms that can support themselves in this rule set, we will investigate a few questions:
- Can we find alternative rules with similar properties as Conway’s rules that count cells as alive that are not immediate neighbors?
- Can we come up with a probabilistic version of Game of Life?
Modes of the infinite string
Fall 2021
Grad student mentor: Bernardo Bianco Prado
Undergraduate researchers: Christopher Tait, Florence Wu, Pengrun Huang
The oscillations of a string with variable density that is clamped on two ends is described by a differential equation that fits regular Sturm-Liouville theory. The differential operator of such a problem has real eigenvalues and eigenfunctions that are orthogonal and complete.
An infinitely long string no longer fits this theory, however, in certain cases the differential operator has complex eigenvalues with eigenfunctions “orthogonal” in a (non-positive definite) bilinear form. In the literature such eigenfunctions are known as quasi-normal modes.
In this project we try to expand solutions to the infinite string equations using these quasi-normal modes. We will study the connections between differential operators and their discretizations. The main goal is to develop and implement an algorithm that computes the solution to an infinite string with variable density on a bounded interval excited by an external force.