“In my field of math, 99 percent of your time is spent pursuing avenues which are ultimately not fruitful. The one percent of the time that it does work, it can be revolutionary,” said Sean Cox, assistant professor of Mathematics at VCU.
Cox is a mathematician whose research focuses on mathematical logic and set theory, an abstract branch of mathematics that deals with collections, called sets, of objects or functions. Along with several of his colleagues in the Mathematics Department, Cox recently hosted a workshop at VCU called Ideal Fest, which brought set theorists from across the world together to work on unsolved, or “open”, problems in his field of study. The event was part of the Open Problems in the Foundations of Mathematics Project, funded by the VCU Presidential Research Quest Fund.
Cox collaborated with Brent Cody, a fellow assistant professors in mathematics at VCU, and Monroe Eskew, a visiting assistant mathematics professor, to organize the workshop, which took place on May 19th-20th. They invited four professors of mathematics and experts in set theory to join them for two days of intense theoretical discussion: James Cummings of Carnegie Mellon University, Paul Larson of Miami University in Ohio, Hiroshi Sakai from Koke University in Japan, and Martin Zeman from the University of California, Irvine.
“We tried to choose a variety of questions, some of which were well-known open problems and others that were more approachable,” Cody said.
The workshop concentrated on a family of open problems about ideals at the foundations of mathematics. Ideals are an organizational tool used by set theorists to classify infinite sets.
“An ideal is a precise way to pretend that two things which aren’t really the same, are the same. It’s a way to ignore differences that are not relevant to the problem that you’re working on,” said Cox.
All the participants at Ideal Fest were experts in the concept of ideals, and had previous experience in solving open problems relating to them. The properties of ideals are notoriously difficult to discover, because many of them have been proven to be unsolvable.
“When we say something’s not provable or not solvable, we don’t just mean that it’s not solvable given the current level of computing or given the current level of human ingenuity. In a technical sense we mean that regardless of how much computing power you have, regardless of how ingenious you are, it doesn’t matter – these problems cannot be solved,” Cox said.
In set theory, the answer to a question isn’t a clear-cut as yes or no – often times the questions that are asked by set theorists are impossible to solve either way because a statement about an ideal can be true sometimes, and false at other times depending on how it’s applied.
While the process of proving whether an open question is true, not true, or unsolvable takes much more time and concentration than two days of collaboration allow, the mathematicians at Ideal Fest made some progress in solving an open question about the principles of an ideal known as the weakly compact ideal.
They did this by comparing the consequence of another more thoroughly understood concept called the non-stationary ideal to the weakly compact ideal, to see if the principles of the non-stationary ideal also applied to the weakly compact ideal. According to Cody, their work indicated that it can, and the weakly compact ideal therefore shares a quality, known as the reflection principle, with the non-stationary ideal.
Future collaboration and work is needed to fully develop their theorem, but these set theorists have taken the first step in solving a mathematical mystery that’s never been unraveled before.
“More important than actually answering the questions is piquing each other’s interest in the questions,” said Cody. “So to make a meeting like this a success, you just need to get the other people interested in what you’re working on and develop potential collaborations.”
Written by Megan Schiffres