A short thread on the philosophy of "pathology" in mathematics. This is the serious version of the tweet below. 1/n
As far as I can tell there are two things a mathematician may mean if they're talking about pathology:
(A) if you drop some hypothesis (characteristic zero, noetherian, finitely generated, etc) things become much wilder, and
(B) something you hoped was true but isn't.
2/n
(A) if you drop some hypothesis (characteristic zero, noetherian, finitely generated, etc) things become much wilder, and
(B) something you hoped was true but isn't.
2/n
Mumford's four "Pathologies in Algebraic Geometry" papers are a great example of type (B): in each paper, he finds some odd or unexpected behavior, some of which contradicted conjectures or assertions by other mathematicians. 3/n
Each of these papers has birthed new avenues of research: Mumford's work on Hilbert schemes gave rise to Vakil's work on "Murphy's law," his paper on 0-cycles is related to the Bloch-Beilinson conjectures, and Hodge-to-de Rham degeneration in char p is still actively studied. 4/n
In my view the lesson here is that a type (B) "pathology" is actually an opportunity -- it means you've discovered something new. When something you hoped was true turns out to be untrue, it means you've found a gap in your intuition. 5/n
What about type (A) pathologies? E.g. the so-called bad behavior of non-Noetherian rings, imperfect fields, and so on? I think that in some sense we view this behavior as pathological because of the pedagogical choices to which we've been exposed. 6/n
For example, a Galois theory course might restrict most of its attention to perfect fields; an algebraic geometry course might assume all rings discussed are Noetherian from the get-go. In my view these choices are reasonable. 7/n
But I think it's really important to be open to these so-called pathological objects. Indeed, I think that this openness is key to math research. Some examples follow:
8/n
8/n
Mori's proof use of "bend-and-break" to prove results about the existence of rational curves on complex varieties wouldn't have been possible without the bravery to reduce modulo p. 9/n
Scholze's and other's ongoing work on perfectoid spaces, the pro-étale site, and so on relies crucially on the willingness to work with certain (very special) non-Noetherian or non-finitely-generated rings. 10/n
Even imperfect fields are more or less unavoidable as soon as one works in characteristic p>0 -- the function field of a curve over any field of positive characteristic is imperfect. 11/n
(Sorry that so many of my examples are in algebra or algebraic geometry -- that's my background. I'm sure someone else could give you persuasive examples in other fields; feel free to post them in the replies.) 12/n
The point I wanted to make is that embracing so-called pathologies -- at least when they arise in the course of one's research -- can be extremely powerful. 13/n, n=13
