Tutorials Lecturer: Edray Goins
Tutorial Lecture: A Survey of Diophantine Equations
Abstract: There are many beautiful identities involving positive integers. For example, Pythagoras knew 3² + 4² = 5² while Plato knew 3³ + 4³ + 5³ = 6³. Euler discovered 59⁴ + 158⁴ = 133⁴ + 134⁴, and even a famous story involving G. H. Hardy and Srinivasa Ramanujan involves 1³ + 12³ = 9³ + 10³. But how does one find such identities? Around the third century, the Greek mathematician Diophantus of Alexandria introduced a systematic study of integer solutions to polynomial equations. In this talk, we’ll focus on various types of so-called Diophantine Equations, setting up the weekend’s discussion of topics such as Pythagorean Triples, Pell’s Equations, Elliptic Curves, and Fermat’s Last Theorem.
Tutorial Lecture: Pell’s Equations and Continued Fractions
Abstract: In 1657, amateur mathematician Pierre de Fermat wrote to mathematicians William Brouncker and John Wallis asking them if they could find positive integers x and y such that x² = 1 + 61y². Through a series of letters, they worked out a general theory of finding such integers for a more general equation in the form x² = 1 + dy². Today, these are mistakenly called “Pell’s Equations”, after Leonhard Euler learned about them in a book on the subject by Johann Rahn and John Pell. In this talk, we’ll explain the theory of Pell’s Equations, exploring the theory via ideas by Joseph-Louis Lagrange. Along the way, we explore the properties of Continued Fractions.
Tutorial Lecture: From Pendula to Elliptic Curves
Abstract: In 1602, Galileo Galilei became interested in studying the motion of a swinging lamp at a church in Pisa. His studies, along with the invention of the Integral Calculus, led to the introduction of elliptic integrals. In 1718, Count Giulio Fagnano found a curious “duplication formula” by considering a points on a curve called the lemniscate (“pendant ribbon”). These results were subsequently generalized by Leonhard Euler in 1728 to a larger set of curves. Eventually this led to a theory of “elliptic curves”. In this presentation, we give the history and genesis of the subject, culminating with the result that the set of complex points on an elliptic curve form a torus.
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