Day 5 - Friday
Breakfast 8:00-8:45 (Dining Hall)
8:45-9:35
Lisa Naples
Abstract: Suppose that we want to run some errands at various locations throughout town. What is the most efficient way for us to travel from location to location and then return home?
In this course, we will begin by exploring how ideas from graph theory can help us to answer this question.
We will then extend the task by assuming that we need to visit infinitely many locations. We will see that tools from statistics and calculus can sometimes be used to determine efficient routes through sets. We will also explore situations in which these tools can fail to help us find a reasonable route.
Relevant background from graph theory, statistics, and calculus will be introduced in the course.
9:45-10:35
Alex Barrios
The Arithmetic Beyond Arithmetic
Abstract: In 1847, it was uncovered by Kummer that Fermat's Last Theorem is intertwined with prime factorization in new number systems. In this lesson, we visit one of these systems, the Gaussian integers $\mathbb{Z}[i]$, and show Dedekind's 1894 proof that Fermat's Sum of Squares Theorem is a direct consequence from the prime factorization in $\mathbb{Z}[i]$, thus setting the stage for the modern viewpoint of number theory. To tackle questions about the integers, one must venture to other number systems, whose arithmetic controls phenomena that remain invisible within the integers themselves.
Break 10:45-11:15
11:15-12:15
Work time
Lunch 12:15 - 1:30 (Dining Hall)
Free time/leave