Project Overview
Counting Configurations of Points and Lines
Department(s)
Mathematics
Abstract
An n-cap is a set of n points so that no 3 lie on a line. For example, the points (0,0), (1,0), (0,1), and (1,1) form a 4-cap in R^2. In this project, we will study caps not over the reals, but over 3D finite projective space. Since this space has only finitely many points, we can ask the question "how many n-caps are there?" There is a method that reduces this problem to studying special collections of points called hyperfigurations. Unfortunately, this method is computationally complex, so we only know the answer for n-caps up to n = 7. In this project, we will study 8-caps in 3D finite projective space. We will analyze the myriad of hyperfigurations that appear in the reduction using tools from algebra, geometry, and combinatorics. Experience with these topics is not expected, and this project is open to anyone who has taken Math 214 and 250 and is willing and eager to learn.
Student Qualifications
MATH 250 and MATH 214
Coding experience is helpful but not required
Project Length
8 weeks
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