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Macaulay Rings and Macaulay Posets 

This is a brief introduction to the 2024 Polymath Jr project that will be run by Alexandra Seceleanu

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This project is in combinatorial commutative algebra - a field that connects the study of algebraic structures called rings to combinatorics. Examples of rings are the set of integers or the set of polynomials. Both of these come with a notion of divisibility which turns them into a partially ordered set. Partially ordered sets can be represented by graphs called Hasse diagrams, which are combinatorial counterparts to these rings.

 

British mathematician Francis Sowerby Macaulay made important contributions to the study of rings. One of his most significant contributions was figuring out how to count elements of a given degree in a polynomial ring and in some distinguished subsets of the polynomial ring. This turns out to be related to the problem of finding sets of vertices in a graph that have minimum boundary, which gives a different, combinatorial path to the problem.

 

In this project we will consider the same type of problems that F.S. Macaulay did, but in other rings, which are related to the ring of polynomials in the same manner that the integers modulo n are related to all integers. This corresponds on the combinatorics side to considering nice classes of graphs.

 

There are no required pre-requisites for this project, however having taken a course in abstract algebra will be very helpful.

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