Masters Thesis Defense, Nate Josephs: 'Finding Rational Solutions on a Nonsingular Cubic Surface in P^3'

Tuesday, April 28, 2015
4:15 PM - 5:50 PM (ET)
ESC 618
Event Type
Lecture
Contact
Christopher Rasmussen, crasmussen@wesleyan.edu
Department
Academic
Link
https://eaglet.wesleyan.edu/MasterCalendar/EventDetails.aspx?EventDetailId=48377

Abstract: The purpose of this talk is to present a strategy for parametrizing the rational points on a nonsingular, homogeneous cubic surfaces. The particular Diophantine equation I will consider is X^3 + Y^3 + YZ^2 + W^3 = 0. The strategy will be to find a family of singular cubic curves on our hypersurface with which to sweep through rational solutions, not unlike the standard parametrization of the circle. In this talk I will parametrize the unit circle, as well as a cubic curve. I will then give a brief introduction to algebraic geometry. I will define varieties, the central object of algebraic geometry, and explore the ideal-variety correspondence. This discussion will culminate with the statement of the Nullstellensatz. Having introduced the basics of algebraic geometry, I will proceed to explain the strategy with which I found a 2-parameter set of solutions to the aforementioned polynomial.

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