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Mathematical Sciences

Mathematics Senior Thesis Presentations

Date: 3:30pm - 5:00pm PDT April 29, 2015 Location: John Howard 254

John Howard 254

Things Fall Apart: The Spontaneous Blowup of Water
Sam Stewart ’15

Major in Mathematics

If you throw a rock into a calm pond, you will see a splash, some ripples, and then the surface of the pond will slowly become smooth again. Regardless of how hard you throw the rock, the pond will eventually become smooth (perhaps after a few hours). The wave equation is a mathematical model that describes such behavior. However, minor changes to the equation produces models with intriguing behavior. Under these models, the pond water can spontaneously explode, even after a gentle toss. How forcefully can we throw the rock without the water exploding? What is the correct way to we measure “force”? Can we predict if the water will explode from the size of the splash? How can we simulate such explosions computationally? How can we understand such explosions mathematically?   In this presentation, I examine such a “pathological” version of the wave equation and present numerical answers to some of these questions. 

Mean Curvature Flow of Tori of Revolution

Colin Gavin ’15

Major in Mathematics and Physics

Mean curvature flow is the gradient flow of the volume functional on embedded surfaces. As a nonlinear system of parabolic equations, its behavior is quite complicated, but generally solutions become more spherical over time as their volume decreases. The evolution of tori under this flow is of interest because their non-trivial topology prevents them from becoming round. This leads to the formation of a variety of singularities. In this talk, I will focus on tori of revolution, which reduces the problem to a version of planar curve shortening flow. From this viewpoint, the possible singularities can be classified and, in some cases, their asymptotic behavior can be determined. I will give a brief overview of my results, and discuss some of the methods in differential geometry and partial differential equations that I used.

An Introduction to Alexandrov Spaces

Isaac Goldstein ’16

pre-thesis report for Honors in Mathematics

Alexandrov Spaces are a special kind of metric space invented and defined by some of the most influential geometers of the twentieth century. This talk will focus on mathematically defining Alexandrov Spaces and Gromov-Hausdorff distance, one of the main tools used in generating Alexandrov Spaces.