# In memoriam: Peter Fletcher (1939-2019)

In 1987 I went off to Blacksburg, VA, to major in mathematics at Virginia Tech. My goal: to go on to a PhD and become a college professor. I had no idea that this was pretty ambitious for a first-generation student; I merely had the supreme confidence of an 18-year-old who knew he loved math and figured it would all work out. When people at home would ask what it took to get a PhD in math I naively answered, “write a calculus book, I guess.” (See, even then calculus was held up as the end-all-be-all of mathematics to school kids; what could possibly lie beyond the “most advanced math” there was?)

My freshman year was complicated by the fact that I was on an Air Force ROTC scholarship and was therefore a member of the Virginia Tech Corps of Cadets. The Corps is a military-school environment embedded within the larger campus; we wore a common uniform to class everyday, rose early for formation, engaged in physical training and military drill in the afternoons. I didn’t like it very much, but one benefit was the enforced quiet hours in the dorms from 7:00 to 11:00 pm, which meant I had no excuses for not staying up on my homework. That year I took the honors second-year math sequence—multivariable calculus, linear algebra, differential equations—with Bill Floyd (a student of Thurston). I did really well in that class (Bill is an amazing teacher).

Sophomore year was a bit rougher. Foundations of mathematics (set theory, logic, and beginning group theory) introduced me to what it was like to do more advanced math. I worked really hard for that B+. Vector calculus wasn’t so bad. Honors advanced calculus was brutal. Groups and rings: I liked that.

And then, in the fall of 1989, I walked into Introduction to Topology, taught by Professor Peter Fletcher, and my life was changed forever. We spent quite a while on some serious set-theoretic stuff, proving the equivalence of the Axiom of Choice and Zorn’s Lemma and the Well-Ordering Principle (and probably something else, it was 30 years ago, after all). Separation axioms and examples of $$T_3$$ spaces that aren’t $$T_$$ (don’t ask me to produce one). Compact spaces and the Tychonoff Product Theorem and ultrafilters (oh my!). Metric spaces at the end. A pretty standard first topology course, but not in the Munkres style.

From this class alone I got a couple of things that I’ve never forgotten. The first is something Peter said that I still pull out from time to time: “Topology is analysis done right.” This is extremely glib of course, but it’s fun to say. I mean, the Intermediate Value Theorem is nothing but the easily proved assertion that the continuous image of a connected space is connected, but there’s a good bit of work buried in proving that intervals on the real line are connected. The other was Peter’s inimitable style of proving things by contradiction. He would always begin these like this (on the board): Proof: Suppose (ha!) that the conclusion is false (or whatever). It’s that “ha!” that I can still picture in his chalk scrawl (I probably got that from him, too).

Thanks to the magic of AP credits I finished my BS in three years, but I had to stick around to complete ROTC training and so I enrolled in the master’s program in math. To be able to finish this off in only a year I had to pursue the thesis option, and I approached Peter about supervising it. It turned out that he had been thinking about something called “pointless topology” (don’t say it) and suggested I think about it, too. Here’s the idea: a frame is a distributive lattice with a unique minimal element 0, a unique maximal element 1, in which finite meets exist (given elements $$a, b$$, there is a unique element $$a\wedge b$$ less than or equal to both), arbitrary joins exist (given an arbitrary collection of elements $$a_\alpha$$, there is a unique element \(\bigvee a_\alpha) greater than or equal to all of them), and finite meets distribute over arbitrary joins. Example: the lattice of open sets of a topological space. So now let’s see what we can prove about these things themselves, without thinking about them as open sets in a topological space (so “pointless” topology, get it?). Most of the basic notions from topology can be phrased in these terms—compactness and other covering properties, separation axioms, etc. So I proved and wrote up a bunch of standard theorems in this context. For example, a regular Lindelöf frame is normal. Stuff like that. I learned a lot, and then went off to pursue my PhD when I was done.

That Peter would get interested in frames was no surprise. He was one of the world’s experts in quasi-uniform spaces. We all know about metric spaces, and get used to working with the defining properties of a metric on a space: reflexivity, symmetry, and the triangle inequality. Here’s a question: can you capture this same information just using open sets instead of a distance function? That’s roughly what a uniformity is on a space, and a metric determines such a structure (but not necessarily conversely; that is, there are uniform spaces that aren’t metric spaces). In life, we know that distances aren’t always symmetric; indeed, one-way streets can really affect the distance between two points on a map depending on which order you’re trying to get from one to the other. So we have the corresponding notion of quasimetric, and therefore quasiuniformity. Peter developed a lot of the theory of these objects, beginning with his dissertation at UNC-Chapel Hill in the mid-1960s.

I left Virginia Tech in 1991 and Peter took the state up on its offer of an early retirement package. He was in his early 50s at the time, and he had a lot of math left in him. Over the next couple of decades he started doing some work in number theory, a pretty radical switch but one that yielded some dividends.

Peter was a kind and generous man, always willing to talk math or anything else. He played the guitar and could sing pretty well while doing it. He introduced me to Brazilian food, an exotic thing for me at the time. I could always count on Peter for advice and good humor. He passed away at the end of July after a long battle with heart disease and kidney disease.

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