Undergraduate Poster Abstracts
Washington Experimental Mathematics Lab (WXML)
Pair Correlations in Uniform Countable Sets
Sanjay Raman/Carl Schildkraut, mentor Jayadev Athreya
We determine the pair correlations of countable subsets of n-dimensional Euclidean space satisfying natural equidistribution conditions. The pair correlations are computed as the volume of a certain regions in \(2n\)-dimensional Euclidean space which can be expressed in terms of the incomplete Beta function. For \(n=2\) and \(n=3\) we give closed form expressions, and we obtain an expression in the \(n\) goes to infinity limit. We also verify that sets of lattice points and primitive lattice points satisfy the required distribution criteria.
On Collisions of Jammed Billiard Ball Configurations
August Chen, mentor Jayadev Athreya, in collaboration with MIT-PRIMES
We consider the limiting case of multiple ball systems in two dimensions, or jammed billiard ball configurations, where all the balls have equal mass and are externally tangent to adjacent balls. Suppose for simplicity that the collisions are perfectly elastic and so no energy is lost in collisions. We wish to describe the end state of such configurations. First, we discuss a motivating random walk called Folding that models such billiard ball configurations. We characterize the orbit and transition matrix of Folding in two dimensions and find formulas for its stationary measure in special cases, while analyzing potential approaches in more dimensions. Next, we explicitly characterize the end state of “lattice-like” jammed billiard ball configurations and consider how long it takes to reach this end state. We analyze some special configurations and discuss potential more general approaches.
Graphlopedia: A Knowledge Engine for Contextual Information in Mathematics
Team: Kimberly Bautista, Aaron Bode, Riley Casper, Dien Dang, Nicholas Farn,
Graham Kelley, Stanley Lai, Adharsh Ranganathan, Michael Trinh, Alex Tsun, Katrina
Warner, mentor Sara Billey
Graphlopedia is an online encyclopedia of graphs. Our aim is to utilize graph theory and machine learning to aggregate a universal database of graphs for which research papers and references may be indexed by the graphs they refer to.
Primality Testing in Polynomial Time
Co-presenters: Rohan Hiatt, Daria Micovic, Bryan Quah, mentor: Amos Turchet
Prime numbers are fascinating objects in mathematics, fundamental to number theory and cryptography in particular. A primality test takes an integer as input and outputs whether that number is prime or composite. Most practical applications require primality tests to be efficient. Today, the largest known prime number has over twenty million digits. Proving that a number of this size is prime can be computationally expensive. Most tests are probabilistic and do not guarantee that any given prime number is in fact prime. Other tests fail to filter pseudo-primes, like Carmichael Numbers, that are in fact composite integers. In 2002, Agrawal, Kayal, and Saxena published the first deterministic primality test that also runs in polynomial time relative to the binary representation of the input. Although their algorithm represents an important breakthrough in the field of computational number theory, it is seldom used in practice. Our objective was to determine why this idealized algorithm is not practical enough compared to other primality tests. We have re-created the algorithm and optimized our initial naïve implementation by studying each step to achieve optimal complexity using Fermat’s Little Theorem, modular arithmetic, and the Fast Fourier Transform for multiplicative efficiency. To confirm that probabilistic methods are still preferred, we compared execution times and accuracy of other tests to our own results.
Gravitational Billiards
N’vida Yotcho, Stephanie Anderson, Kush Gupta, Dia Taha, mentor: Jayadev Athreya
Gravitational billiards is a theoretical vertical vacuum setting, which boundaries could be a rectangle, parabola, circle, paraboloid, sphere or any other curve of interest. In this project, our efforts were focused on studying the long term behavior of a ball moving under the influence of gravity in such peculiar conservative system. we observed that that once released the ball followed a two-dimensional projectile trajectory before colliding with the boundary. After collision, the ball bounced off the boundary following the same type of trajectory. Subsequently, we considered a trajectory between two points of ball-boundary collision to be an isolated two dimensional gravitational motion with different initial conditions. So far, our observations corroborated the physics behind our study. We hope that our project incite an interest in exploring mathematical notions ( laws, theorems and axiom) through real life event.
Experimental Algebra and Geometry Lab (EAGL)
EAGL Presentations — Changing High School Students’ Attitudes towards Mathematics
Authors: Dante Cardenas, Oscar Campuzano, and Dr. Aaron Wilson, the University of Texas Rio Grande Valley
In 2017, students and professors in the Experimental Algebra and Geometry Lab (EAGL) gave presentations to regional high school math students and also collected pre- and post-surveys. A modified version of the Attitudes Towards Mathematics Inventory (ATMI) was used as the research instrument. The results of these surveys were transferred to spreadsheets for the analysis which sought to find out if these presentations changed students’ attitudes towards mathematics. The survey responses were based on a Likert Scale, with the answers ranging from 1 to 5, where 1 represents Strongly Disagree and 5 represents Strongly Agree. The factors making up the math attitudes inventory for this instrument included Self-Confidence, Value, Enjoyment, and Motivation. The missing data we’ve encountered were inputted using the person-mean substitution method. The overall results of the analysis, which included a sample of \(N=306\) respondents, were satisfactory. There was evidence that the one-day presentations did have a slight but statistically significant positive effect on students’ attitudes towards Mathematics. These findings may have broader implications in terms of influencing students to enter degree plans in Mathematics at the college level.
Mason Experimental Geometry Lab (MEGL)
Longest Orbits over Varieties of Generalized Markoff Equations over Finite Fields
Co-presenters: Marvin Castellon, Kira Wolpert, and Seth Lee
Capturing Hyperbolic Space In Virtual Reality
Presenter: Joseph Frias
Abstract: Using the half-space model for hyperbolic geometry in dimension 3, the VR team at MEGL is working to visualize geodesics and provide a way to render hyperbolic manifolds in a virtual reality setting. Currently, geodesics and basic polyhedra have been rendered in the Unity game engine. The current work is focusing on building lattices out of these basic polyhedra.
Illinois Geometry Lab
Employing the Graph Laplacian for Understanding Electronic Structures of Matter
Co-presenters: Michael Toriyama, Ivan Contreras
The use of graph theory as a mathematical tool for understanding phenomena (such as diffusion of information in network theory) has cultivated much interest for a wide variety of practical applications. Recent efforts by our group have been focused on developing a discrete analogue of the Laplacian operator using a matrix formalism known as the graph Laplacian, as well as establishing a gluing formulae for decomposing systems into more tractable subsystems. This tool finds various applications in describes the evolution of states in quantum mechanical systems and developing a novel, foundational understanding of a discrete formulation of quantum mechanics. Consequently, this tool represents tremendous potential for more efficient simulation design in the realm of quantum chemistry from a computing standpoint, and an industrial method for rational design of novel advanced materials. Although the modern approach to computing electronic properties of matter is dominated by Density Functional Theory (DFT), we propose a simpler, graph Laplacian method for solving Schrodinger’s equation that does not require immense computing power and simultaneously runs effective ab-initio simulations. Not only can this method accelerate precise calculations of electronic properties of matter, but this may also pave way to a broader understanding of quantum interactions from a discretized grid prospect.
Statistical regularity in Apollonian Gasket
Co-presenters: Weiru Chen, Mo Jiao, Calvin Kessler, Amita Malik and Xin Zhang
Apollonian gaskets are formed by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We experimentally study the nearest neighbor spacing, pair correlation, and electrostatic energy of centers of circles from Apollonian gaskets. Even though the centers of these circles are not uniformly distributed in any ‘ambient’ space, after proper normalization, all these statistics seem to exhibit some interesting limiting behaviors.
Laboratory of Geometry at Michigan
Visualizing the Birman-Series set on the once-punctured Torus
Co-presenters: Luke Kiernan and Connor Davis
On a finite-area hyperbolic surface, the Birman-Series set is the union of all simple closed geodesics. Birman and Series proved that this set has Hausdorff dimension 1 implying it is nowhere dense and has measure zero. This is in stark contrast to the union of closed geodesics, which is dense in the surface. Using a criterion given by Buser to list all simple closed geodesics, we used SageMath to generate new pictures of the closed geodesics and the simple closed geodesics of the punctured torus up to word length 6 to demonstrate this phenomena.