Wednesday

Mini-course 1. From Here to There: Exploring Routes in Graphs and Sets

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.

Mini-course 2. How Fermat's Last Theorem influenced modern number theory.

Alex Barrios

Fermat's Little Theorem

Abstract: Among Fermat's many claims was a remarkable observation about prime numbers: whenever p is prime and does not divide an integer a, the remainder of a^{p-1} is always 1! For decades, mathematicians searched for a proof, which was finally supplied by Leibniz in 1683. In this lecture, we look at Gauss's elegant 1801 proof and point to Leibniz's earlier 1683 argument as a contrasting approach.

Informal conversation

Luda Korobenko

Talk. Eulerian graphs and Elizabethan embroidery

Jane Butterfield

Abstract: Blackwork embroidery was the height of fashion in 17th century England, and is popular again in this century. It also has some very interesting mathematical properties! Together we'll explore some "embroiderable" patterns, and model these patterns as a special type of Eulerian circuit through a special type of multgraph. Using this graph theory model will lead us to a remarkably simple characterization of embroiderable patterns (originally proved by Holden 2005).

No previous knowledge of embroidery or graph theory is necessary.

Talk. Studying differential equations without solving them: what, why, and how?

Luda Korobenko

Abstract: In this talk I will try to explain what it means to study PDEs (Partial Differential Equations) without actually solving them and why you might want to do it. I will concentrate on the particular type of PDEs which I mostly work with, called (degenerate) elliptic equations, covering some questions that people try to answer, and methods they (and myself in particular) use to do it. 

Worktime