Mathematics Department
Undergraduate Research Symposium
Spring 2019

Tuesday, April 30 9:45-11:30 in LCB 121


9:45-10:00 Wei Yao
Mentor: Karl Schwede
Polynomial evaluation over finite and rational fields through matrices in Macaulay2 

10:00 - 10:15 Thomas White
Mentor: Christopher Janjigian 
Simulating the inhomogeneous corner growth model

10:15-10:30 Camille Humphries
Mentor: Yekaterina Epshteyn, Qing Xia
Fast Numerical Algorithms for Models with Nonlinear Diffusion 

10:30-10:45 Charlotte Blake
Mentor: Yekaterina Epshteyn
Efficient Numerical Algorithms for Automatically Processing Data with Application to Materials Science 

10:45-11:00 Gabrielle Legaspi
Mentor: Yekaterina Epshteyn
Coarsening Models 

11:00-11:15 Cassie Buhler
Mentor:Fred Adler 
Mathematical Modeling of Adaptive Therapy in Prostate Cancer 

11:15-11:30 Justin Baker
Mentor: Elena Cherkaev
Optimal Transportation Networks


Abstracts


Wei Yao
Mentor: Karl Schwede
Polynomial evaluation over finite and rational fields through matrices in Macaulay2 
Questions in algebraic geometry such as the birationality of a map between varieties and the codimension of the singular locus of a variety can be answered through relatively straightforward computations of ranks of certain matrices. Macaulay2 is a software widely used by algebraic geometers and algebraists to perform such computations among many others. However, over a polynomial ring with multiple variables and undetermined degree, these computations turn out to be slow. We implement a particular step in this computation via selecting minors in order to speed up the process.

Thomas White
Mentor: Christopher Janjigian 
Simulating the inhomogeneous corner growth model
The goal of this research was to simulate the inhomogeneous corner growth model with exponential weights. This is an important model in the Kardar Parisi Zhang universality class from physics. In order to do this, I simulated geodesics to map interactions in the discrete plane and normalized passage times, which have known behavior. As a next step, I am investigating Tracy-widom fluctuations at the interface between the concave and linear parts of the shape function. The fluctuations require further investigation, and other variations to the corner growth model are still open to study.

Camille Humphries
Mentor: Yekaterina Epshteyn, Qing Xia
Fast Numerical Algorithms for Models with Nonlinear Diffusion 
Diffusion equations, and, in particular, nonlinear diffusion models play an important role in many areas of science and engineering. In general, solutions to such models cannot be obtained analytically. Hence, there is a need for accurate and efficient algorithms that can deliver approximate solutions to these models. We consider Difference Potentials Method for numerical solution of various diffusion models. We will investigate the accuracy, stability, and efficiency of the developed algorithms, and discuss related future work.

Charlotte Blake
Mentor: Yekaterina Epshteyn
Efficient Numerical Algorithms for Automatically Processing Data with Application to Materials Science 
Our research focused on developing robust numerical algorithms that take images of crystal grains as the input and automatically output relevant data, including information about grain area, perimeter, and number of neighbors. In this presentation, we will review the work accomplished in the Fall, as well as discuss the corrections and extensions made to the algorithms this semester. We will also present the obstacles that appeared as a part of the design of such algorithms and how they were resolved. Special focus will be given to the aspects of the algorithms related to the computational geometry questions of corner identification and polygon approximation of boundaries. Finally, we will examine the efficiency of algorithms and possible improvements.

Gabrielle Legaspi
Mentor: Yekaterina Epshteyn
Coarsening Models 
Cellular networks exhibit behavior on many different length and time scales and are generally metastable. Among numerous examples of cellular networks are polycrystalline materials/microstructures. Most technologically useful materials arise as polycrystalline mi- crostructures, composed of a myriad of small crystallites or grains, the cells, separated by interfaces or grain boundaries. Coarsening results from the growth and rearrangement of the crystallites, which may be viewed as the anisotropic evolution of a large metastable system. Our project will investigate and compare different coarsening models. We will employ numerical simulations, mathematical analysis and data analytics to study and improve these models.

Cassie Buhler
Mentor:Fred Adler 

Mathematical Modeling of Adaptive Therapy in Prostate Cancer 
Prostate cancer is a hormonally driven cancer. These cancer cells need androgen, a class of male sex hormone, to survive and grow. Standard treatments for prostate cancer target androgens. This type of therapy is denoted as hormone therapy. Yet, for patients with recurring cancer, hormone therapy is not effective because cancer cells become testosterone independent over time, and consequently, the cells gain resistance and do not respond to therapy. There are studies that suggest therapy administered in intervals, as opposed to continuous treatments, could prevent this occurrence. We have analyzed mathematical models of dynamic biological systems for prostate cancer progression in order to explore the effect of treatment timing to find the most effective therapy in delaying the inevitable emergence of testosterone resistant cells.

Justin Baker
Mentor: Elena Cherkaev

Optimal Transportation Networks
Models of swarming behavior aid in disaster planning, direct the actions of warehouse robots, and can map the foraging characteristics of insects. The Monge-Kantorovich formulation of the optimal transportation problem models the best direct route for individual agents in a swarm. The presented work investigates optimal transportation and duality in a Monge-Kantorovich formulation. We reduce the Monge-Kantorovich formulation to the linear programming problem which can be efficiently solved numerically. However this formulation is not applicable in the case where the domain of travel is restricted, so that agents must travel along a particular network of paths. We extend the Monge-Kantorovich formulation to a network of paths which differs from transfer over a direct route. We show that this extended formulation can also be reduced to the linear programming problem. Finally, we investigate various applications of both the direct and network transfer using numerical simulations