You can’t always solve a mathematical problem by reducing it to something you’ve already solved. Sometimes, you need to rethink and reform the original statements. Prof. Justin Moore, mathematics, works to find possible breakthroughs on the generalization of Hindman’s theorems and the classification of uncountable linear orders, in order to better understand the Ramsey Theory of infinite sets.
In 1928, the English mathematician Frank Plumpton Ramsey published a paper in which he proved that when a structure, such as the edges from a graph or the positive integers, is partitioned into a small number of pieces, one of the pieces must exhibit some high amount of order or regularity. According to Moore, this is now known as Ramsey Theory.
An example of Ramsey Theory would be that if the natural numbers are colored red and blue, there is an infinite sequence all of whose finite sums have the same color, Moore said.
Ramsey Theory affects math as varied as algebra, combinatorics, set theory, logic, analysis, and geometry, and has played an important role in several mathematical developments in the last century. However, many parts of Ramsey theory remain limited and undeveloped Moore said.
Moore’s latest research focuses on Hindman’s theorem, a Ramsey-type theorem which asserts “if the natural numbers are partitioned into finitely many pieces, then one of the pieces must contain all of the finite sums of elements of some infinite subset of the natural numbers.” Hindman’s Theorem only applies to certain types of numbers, and these numbers must be associative as in the following case: a+(b+c) = (a+b)+c.
According to Moore, Hindman’s theorem doesn’t work for nonassociative operations only because of trivial reasons. It is possible to modify the statement of Hindman’s theorem to avoid these trivial counterexamples.
“My major contribution to the problem so far has been to realize that it would be solved if one could successfully generalize Hindman’s theorem.” said More, “The question is now whether it is possible to prove the resulting statement.”
Currently, Moore is working to figure out the statement and find the tool to prove it. If he successfully finds a generalization, mathematicians will find themselves staring into a cornucopia of techniques for solving problems in Ramsey theory which will be very different from those that are currently used.
Besides Hindman’s theorem, Moore works on uncountable linear orders. A linear order is a set X whose elements can be compared with one another, and for each element, x < y and y < z implies x < z. Uncountable linear orders also fit under the umbrella of Ramsey theory.
Moore’s interest on Ramsey theory can be traced back to his dissertation about Ramsey theory for uncountable sets at University of Toronto.
“In mathematics one has to first understand the obstructions to giving too strong a solution to a problem, as well as what a problem is really asking,” Moore said.
Moore teaches MATH 4540: Introduction to Differential Geometry, and MATH 7870: Set Theory, in the spring semester.
Original Author: Yidan Xu