PURE Math Interns

2013

 
2013_Intern_Photo.html

Intern Advisors: Dr. Luis García-Puente - Sam Houston State University

                            Dr. Brian Wissman - University of Hawai`i at Hilo

Graduate Assistants: Anastasia Chavez - University of California, Berkley


Interns

Sarah Baumgardner - University of Tennessee

Brittany Boribong - University of Scranton

Andy Fry - Western Oregon University

Cody Kalā - Univeristy of Hawai`i at Manoa

Armando Salinas - Arizona State University

Reina Shintaku - University of Guam

Raven Showels - University of Texas at Austin

Reuben Tate - University of Hawai`i at Hilo

Amanda Urquiza - University of Arizona

Gautam Webb - Colorado College

Kathreen Yanit - University of Guam

Andrew You - Duke University

Projects


Accessibility Polynomials of Abelian Sandpiles            

Armando Salinas, Reuben Tate, Andrew You, directed by Dr. Luis García-Puente


Abstract               Presentation


Avalanche Polynomials            

Andy Fry and Gautam Webb, directed by Dr. Luis García-Puente


Abstract               Presentation


Riemann Problem For Various Speeds of Traffic

Sarah Baumgardner and Reina Shintaku, directed by Dr. Brian Wissman


Abstract               Presentation


Nonlinear System of Conservation Laws - Shallow Water Equations            

Brittany Boribong and Kathreen Yanit, directed by Dr. Brian Wissman


Abstract               Presentation


Approximating solutions to nonlinear scalar conservation laws using the wave front tracking method

Cody Kalā, Raven Showels, and Amanda Urquiza, directed by Dr. Brian Wissman


Abstract               Presentation

    The Interns topics introduced twelve undergraduates to shock waves and conservation laws and the abelian sandpile model. The Interns Program was directed by Dr. Brian Wissman and Dr. Luis García-Puente, with assistance from Anastasia Chavez.

Course Descriptions

The Abelian Sandpile Model - The abelian sandpile model developed by Bak, Tang, and Wiesenfeld (1987) and later generalized by Dhar (1990) is a mathematical model for dynamical systems that naturally evolve toward critical states, exemplifying the complex behavior known as self-organized criticality. This model has been widely studied and used by various disciplines, including physics, computer science, and economics. The abelian sandpile model has a rich algebraic, geometric, and combinatorial structure.

    In this seminar, we will focus on the combinatorial and dynamical aspects of this model. Specifically, we will study certain combinatorial invariants associated to the lengths of sandpile avalanches. During the course, the students will work on a sequence of elementary computational and theoretical results that will guide them towards the discovery of a general theorem. This will help students develop a better intuition and a more rigorous mathematical thinking process.


Shock Waves and Conservation Laws - Conservation Laws form a system of one or more first order non-linear partial differential equations that model a diverse number of phenomena including Gas Dynamics, Traffic Flow, and Shallow Water Waves.  During the evolution of these systems density waves tend to “break” leading to the formation of a shockwave; a discontinuous solution.  Both exact and approximate methods to analyze and continue a discontinuous solution after the formation of a shock are discussed.  Major topics include Method of Characteristics, Rankine-Hugoniot Jump Conditions, Riemann problems, Entropy Solutions, Finite Volume Methods, and Godunov’s Method.