Matching theory in group settings (Spring 2021)

This research project examines several important problems in matching theory in group settings. These are problems arising from both combinatorics and linear algebra, with the common theme of determining how large or small certain finite subsets of prime numbers can be under certain restrictions. For example, given a set of primes, whether or not it satisfies a certain combinatorial property, called acyclic matching property? Many of these problems have fascinated mathematicians for decades, while others are fueled by the more recent development of combinatorial number theory. We will tackle these problems using a wide range of mathematical tools and techniques from linear algebra, additive number theory, field theory and combinatorics. One of the main goals of this project is to study the interplay between combinatorial and algebraic methods, and to further develop these methods by studying some of the most central open problems in the field. We will also continue to encourage undergraduate, and graduate students to work in combinatorial/algebraic research, and continue to teach courses which cover the latest results and methods used in combinatorics and linear algebra.

The first area in this project examines counting problems in the area of matching theory. Given an abelian group and two finite subsets of it, a matching from one subset to another one is a certain type of bijection, correlated to existence of perfect matching in a balanced bipartite graph. One of the major goals of this project is to obtain new bounds for the number of matchings from a given set to another set of the same cardinality in the context of abelian groups. We will continue a sequence of recent works on characterizing primes with respect to certain subfamilies of matchings. The second area of this project explores various extensions of the matching theory, and its applications in field theory. Our problems include characterizing separable fields and estimating dimension of primitive subspaces in a field extension.

For more information contact Mohsen Aliabadi (aliabadi@iastate.edu).

People:

• Mohsen Aliabadi (Postdoc)

• Shira Zerbib (Faculty)

Pre-requisites:

• Basic knowledge in abstract algebra (groups, rings and finite fields)

• Basic knowledge in combinatorics.

• Basic knowledge in linear algebra.

• Basic knowledge in combinatorial number theory. (optional)

• Mathematical maturity of an advanced undergraduate student.