The Anti-Pigeonhole Conjecture
A conjecture about faculty behavior
“Dr. Kibzwang” source
Colin Potts is Vice Provost for Undergraduate Education at Georgia Tech. His job includes being a member of the President’s Cabinet—our president, not the real one—and he is charged with academic policies and changes to such policies. He is also a College of Computing colleague and fellow chess fan.
Today I want to state a conjecture about the behavior of faculty that arose when Tech tried to change a policy.
I am currently at Georgia Tech, but this conjecture applies I believe to all institutions, all faculty. Ken mostly agrees. Potts recently supplied a wonderful example of the conjecture in action—I will get to that after I formally state it. Perhaps we should call it Potts’s Conjecture?
The conjecture is easy to state:
Conjecture 1 Let be any issue and let be any collection of distinct faculty members. Then during a long enough period of email exchanges among the above faculty on at least opinions will be voiced.
You can see why I refer to it as an anti-pigeonhole principle. Right?
I have tried to prove the conjecture—I view it as a kind of Arrow’s Paradox. I have failed so far to obtain a formal proof of it. The conjecture does have the interesting corollary:
Corollary 2 Let be any issue and let be any collection of distinct faculty members. Then during a long enough period of email exchanges on the issue some faculty member will voice at least two different opinions.
A weaker version that we will cleverly call The Weak Conjecture is the following:
Conjecture 3 Let be any issue and let be any collection of distinct faculty members. Then during a long enough period of email exchanges on the issue at least opinions will be voiced.
The point is that the total number of opinions is unbounded. Strong or weak, we can call it the Conjecture.
Of course, being mathematicians we want proofs not examples. But as in areas like number theory, one is often led to good conjectures by observations. In any event simple tests of conjectures are useful to see if they are plausible enough to try to prove.
Here is the policy change that has been suggested. You are free to skip this or go here for even more detail. The point is that this is the issue .
Per the proposal, starting in fall 2015, classes would not meet on the Wednesday before Thanksgiving, giving students an additional day for their break. A change implemented as a pilot this spring will continue to stand, which eliminated finals being held during the last exam session on the Friday before Commencement to prevent finals overlapping with graduation festivities. Starting the next academic year, it was approved to extend the individual course withdrawal deadline by two weeks, allowing students more time to evaluate whether to drop a class.
In Spring 2016, the current Dead Week would be replaced with Final Instructional Class Days and Reading Periods. The new schedule would designate Monday and Tuesday of the penultimate week of the semester as Final Instructional Class Days, followed by a day and a half of reading period, and administering the first final on Thursday afternoon. Finals would be broken up by that weekend and resume Monday, with an additional reading period the next Tuesday morning. Finals would finish that Thursday, allowing Friday for conflict periods and a day between exams and Commencement.
The final recommendation would extend the length of Monday/Wednesday/Friday classes during spring and fall semesters from 50 to 55 minutes. Breaks between classes would extend from to 10 to 15 minutes.
Pretty exciting, no? No.
The result of Potts announcing the above was a storm of emails from our faculty members. As you would expect, given the Conjecture, this quickly led to a vast number of opinions. The number of opinions seems easily to follow the Conjecture.
Ken analogizes this kind of policy tuning for a university to finding a regional optimum in the landscape of a multivariable function . A proposal like Potts’s, with so many little changed parts, resembles a step in simulated annealing where one periodically jumps out of a well to test for better conditions in another. He is not surprised that such a ‘jump’ would bring multiple reactions from faculty.
Even so, however, one would expect there to be a gradient in the new region so that opinions could converge to the bottom of the new well. This is a different matter: a helpful gradient should be in force after a jump.
April is the month when US undergraduates have been informed of all their college acceptances and in many—fortunate—cases must make a choice. Ken has a front-row seat this year. From comparing various colleges and universities with widely different policies, and noting the market incentive to diversify, he has come to a conjecture of his own:
Conjecture 4 There is no gradient: for any university , is defined only on a set of measure zero.
To all appearances, this conjecture implies the others. Is it capable of being proved? Again you—our readers—are best placed to furnish input for a proof.
Do you believe any of the conjectures? I hope we get lots of opinions…
Ken and I are divided: he thinks we will not get many, I think we will get a lot, and we both think that we may get just a couple. But in my opinion it is possible that …
[added to first paragraph; format fixes]