For faculty members who have a strong interest in teaching undergraduates, PCMI’s Undergraduate Faculty Program offers the opportunity to renew excitement about mathematics, talk with peers about new teaching approaches, address and learn about some challenging research questions, and interact with the broader mathematical community. Each year the theme of the UFP bridges the research and education themes of the Summer Session. Participants are not expected to be experts in the summer's research topic.
The 28th Annual PCMI Summer Session will be held July 1 – 21, 2018.
Alex Iosevich, University of Rochester
The study of finite point configurations touches upon many areas of modern mathematics. The basic questions include the existence of geometric patterns, the frequency of their repetition and the presence of a variety of congruence types. I have asked all three of my kids, at one point or another, how to arrange many points on a piece of paper so that the distance one arises as many times as possible. Then I asked them to arrange the points so that they determine as many equilateral triangles as possible. When the kids got older, they were somewhat shocked to discover that these problems are studied by many mathematicians and that in general very little is known about them! We are going to continue this quest in the PCMI program next summer in a variety of discrete, continuous and arithmetic settings. Starting from the elementary definitions, we are going to quickly build up the skills needed to study a variety of research problems that have proven to be a rich source of mathematical ideas in the past ten years.
The program will begin with the discussions of the classical problems of geometric combinatorics, such as the Erdos distance problem, the Szemeredi-Trotter incidence theorem and their variants. We will then explore the possible formulations of these problems in vector spaces over finite fields while slowly developing the arithmetic and character sum tools that we will need as our program develops. Throughout this process we will be solving a multitude of small but interesting problems that will quickly take us to the cutting edge of what is currently known. The transition from learning to original research will never be announced, it will happen in the course of our investigations.
No prior knowledge beyond high school mathematics, basic calculus and elementary linear algebra will be assumed. Lectures will be heavily supplemented with problem sessions and discussions, both individual and in small groups. The participants will be continually encouraged to make the material their own by infusing the discussions with ideas, conjectures and suggestions.