1. Ephraim Bililign

Title: Measuring the temperature of granular systems
Abstract: Granular systems, or collections of athermal mesoscale particles, are immune to temperature in the conventional sense. Thus, to describe the behavior of an jammed assortment of grains, we turn to a modified thermodynamics built on forces and volumes. I will discuss the experimental measurements we made on a packing of photoelastic discs, which enabled us to identify forces and contacts throughout a granular system. Then, using the tools of statistical mechanics, I will present our analysis of one candidate for an analog to temperature: angoricity. (Faculty mentor: Dr. K. Daniels)

2. Taylor Garnowski
Title: Black Hole or Explosion? Using the Numerical Algorithm PUSH to Determine the Outcome of Collapsing Stars.

Abstract: When the cores of certain massive stars begin to contract at the end of their lives, their immediate future is uncertain. There are only two options for a star in this state: collapse all the way into a black hole or violently explode in whats called a core-collapse supernovae (CCSNe). Currently, theres no obvious indicator that tells us which option a star will take before it collapses. As a result, physicists and applied mathematicians have turned to the power of numerical modeling to help resolve this issue. In this talk, Ill present on a numerical algorithm, PUSH, that allows us to tie the fate of a collapsing star to one parameter. Ill provide a brief overview of the theory of CCSNe and the differential equations one needs to solve to model a collapsing star. Finally, Ill give some results when the PUSH algorithm is applied to a few different star types. (Faculty Mentor: Dr. C. Frohlich)

3. William Reese
Title: Statistical Properties of Solutions to Elliptic Partial Differential Equations with Random Coefficient Functions

Abstract: Elliptic partial differential equations (PDEs) are used for modeling various physical processes. Examples include vibrations of plates, fluid flow, and heat diffusion on a plate. In practice, the parameters in the PDE such as coefficient functions or boundary data are not known exactly. Hence, to model ones lack of precise knowledge about such parameters, it is natural to consider probabilistic approaches. This enables for example, characterization of statistical properties of system response, given a statistical model for the model parameters. We consider elliptic PDEs with random coefficient functinos. This can for example model heat diffusion through a heterogeneous medium with a conductivity function modeled as a random field. The purpose of this research is to study the solution behavior of elliptic PDEs with random coefficients. The equations are solved numerically, using the finite element method. The random coefficient field is represented through its Karhunen Loeve (KL) expansion. Our main objective is to study the solution of the PDE. We find that even for coefficient functions with small correlation lengths, the statistics of the solution of the PDE can be represented accurately with a few KL modes. (Faculty mentor: Dr. A. Alexanderian )

4. Brandon Summers

Title: Certifying Isolated Singularities of Plain Curves
Abstract: Consider a function f : R^{2} R where f is from Z[X,Y]. We will focus on certifying the existence of the isolated singularities of the curve f^{-1}(0). A point (x,y) is a singularity of f^{-1}, if the function f and all the partial derivatives of f vanish at (x,y). This paper analyzes two different algorithms. The first is presented in the paper, Complete Subdivision Algorithms, II: Isotopic Meshing of Singular Algebraic Curves by Burr et al. The second is presented in the paper Certifying Solutions to Overdetermined and Singular Polynomial Systems over Q by A. Szanto et al. This paper provides an overview of both algorithms and a comparative analysis which considers the accuracy of the algorithms for different types of functions. (Faculty mentor: Dr. A. Szanto)