Distinct volume subsets (Spring 2022)

Combinatorial geometry deals with arrangements of geometric objects and with discrete properties of these objects. There are several fascinating problems in the area, which while often easy to state, have deep underlying theory and connections to all kinds of mathematics. A number of questions study properties of sets of points in the plane. For example, one of the most important questions in the area is the following: Given a set of n points in the plane, how many pairs of points can be at distance one from each other? This question has been open for seventy years, and no progress has been made in the last forty years. So we will not be working on this question!

What we will do is consider different variants of this question. Given a set of n points in the plane, how large of a subset of these points can we find such that all distances among pairs of points are distinct. Or we could consider triples of points and ask for the areas of the resulting triangles to be distinct. Or we could look at point sets in higher dimensions. There are infinitely many variations!

For the first few weeks, we will learn techniques from algebra and probability that will help us tackle these questions. We hope to learn a lot of interesting mathematics and hopefully publish a paper while we are at it.

For more information contact Abdul Basit (abasit@iastate.edu)

People:

• Abdul Basit (Postdoc)

• Sam Potter (Undergrad)

Pre-requisites:

• Experience with proofs (in courses or research projects)

• Experience with combinatorics (MATH 304) is desirable, but not necessary