A footnote to Robert H. Mountjoy

In an earlier post titled Mathematical romantic? I mentioned some papers I inherited of one of my mathematical hero’s Andre Weil with his signature. In fact, I was fortunate enough to go to dinner with him once in Princeton in the mid-to-late 1980s – a very gentle, charming person with a deep love of mathematics. I remember he said he missed his wife, Eveline, who passed away in 1986. (They were married in 1937.)

All this is simply to motivate the question, why did I get these papers? First, as mentioned in the post, I was given Larry Goldstein‘s old office and he either was kind enough to gift me his old preprints or left them to be thrown away by the next inhabitant of his office. BTW, if you haven’t heard of him, Larry was a student of Shimura, when became a Gibbs Fellow at Yale, then went to the University of Maryland at COllege Park in 1969. He wrote lots of papers (and books) on number theory, eventually becoming a full professor, but eventually settled into computers and data science work. He left the University of Maryland about the time I arrived in the early 1980s to create some computer companies that he ran.

This motivates the question: How did Larry get these papers of Weil? I think Larry inherited them from Mountjoy (who died before Larry arrived at UMCP, but more on him later). This motivates the question, who is Mountjoy and how did he get them?

I’ve done some digging around the internet and here’s what I discovered.

The earliest mention I could find is when he was listed as a recipient of an NSF Fellowship in “Annual Report of the National Science Foundation: 1950-1953” under Chicago, Illinois, Mathematics, 1953. So he was a grad student at the University of Chicago in 1953. Andre Weil was there at the time. (He left sometime in 1958.) Mountjoy could have gotten the notes of Andre Weil then. Just before Weil left Chicago, Walter Lewis Baily arrived (in 1957, to be exact). This is important because in May 1965 the Notices of the AMS reported that reported:

Mountjoy, Robert Harbison
Abelian varieties attached to representations of discontinuous groups (S. Mac Lane and W. L. Baily)

(His thesis was published posthumously in American Journal of Mathematics Vol. 89 (1967)149-224.) This thesis is in a field studied by Weil and Baily but not Saunders.

But we’re getting ahead of ourselves. The 1962 issue of Maryland Magazine had this:

Mathematics Grant
A team of University of Maryland mathematics researchers have received a grant of $53,000 from the National Science Foundation to continue some technical investigations they started two years ago. The mathematical study they are directing is entitled “Problems in Geometric Function Theory.” The project is under the direction of Dr. James Hummel. Dr. Mischael Zedek. and Prof. Robert H. Mountjoy, all of the Mathematics Department. They are assisted by four graduate-student researchists. The$53,000 grant is a renewal of an original grant which was made two years ago.

We know he was working at UMCP in 1962.

The newspaper Democrat and Chronicle, from Rochester, New York, on Wednesday, May 25, 1965 (Page 40) published the news that Robert H. Mountjoy “Died suddenly at Purcellville, VA, May 23, 1965”. I couldn’t read the rest (it’s behind a paywall but I could see that much). The next day, they published more: “Robert H. Mountjoy, son-in-law of Mr and Mrs Allen P Mills of Brighton, was killed in a traffic crash in Virgina. Mountjoy, about 30, a mathematics instructors at the University of Maryland, leaves a widow Sarah Mills Mountjoy and a 5-month old son Alexander, and his parents Mr and Mrs Lucius Mountjoy of Chicago.”

It’s so sad. The saying goes “May his memory be a blessing.” I never met him, but from what I’ve learned of Mountjoy, his memory is indeed a blessing.

The Riemann-Hurwitz formula for regular graphs

A little over 10 years ago, M. Baker and S. Norine (I’ve also seen this name spelled Norin) wrote a terrific paper on harmonic morphisms between simple, connected graphs (see “Harmonic morphisms and hyperelliptic graphs” – you can find a downloadable pdf on the internet of you google for it). Roughly speaking, a harmonic function on a graph is a function in the kernel of the graph Laplacian. A harmonic morphism between graphs is, roughly speaking, a map from one graph to another that preserves harmonic functions.

They proved quite a few interesting results but one of the most interesting, I think, is their graph-theoretic analog of the Riemann-Hurwitz formula. We define the genus of a simple connected graph $\Gamma = (V,E)$ to be

${\rm genus}(\Gamma) = |E| - |V | + 1.$

It represents the minimum number of edges that must be removed from the graph to make it into a tree (so, a tree has genus 0).

Riemann-Hurwitz formula (Baker and Norine): Let $\phi:\Gamma_2\to \Gamma_1$ be a harmonic morphism from a graph $\Gamma_2 = (V_2,E_2)$ to a graph $\Gamma_1 = (V_1, E_1)$. Then

${\rm genus}(\Gamma_2)-1 = {\rm deg}(\phi)({\rm genus}(\Gamma_1)-1)+\sum_{x\in V_2} [m_\phi(x)+\frac{1}{2}\nu_\phi(x)-1].$

I’m not going to define them here but $m_\phi(x)$ denotes the horizontal multiplicity and $\nu_\phi(x)$ denotes the vertical multiplicity.

I simply want to record a very easy corollary to this, assuming $\Gamma_2 = (V_2,E_2)$ is $k_2$-regular and $\Gamma_1 = (V_1, E_1)$ is $k_1$-regular.

Corollary: Let $\Gamma_2 \rightarrow \Gamma_1$ be a non-trivial harmonic morphism from a connected $k_2$-regular graph
to a connected $k_1$-regular graph.
Then

$\sum_{x\in V_2}\nu_\phi(x) = k_2|V_2| - k_1|V_1|\deg (\phi).$

The number-theoretic side of J. Barkley Rosser

By chance, I ran across a reference to a paper of J Barkey Rosser and it brought back fond memories of days long ago when I would browse the stacks in the math dept library at the University of Washington in Seattle. I remember finding papers describing number-theoretic computations of Rosser and Schoenfeld. I knew nothing about Rosser. Was he a number theorist?

J. Barkley Rosser, taken at Math meeting in Denver

Here’s all I could glean from different sources on the internet:
J. Barkley Rosser was born in Jacksonville, Florida in 1907. He earned both his Bachelor of Science (1929) and his Master of Science (1931) from the University of Florida. Both degrees were in physics. He obtained his Ph.D. in mathematics (in fact, mathematical logic) from Princeton University in 1934, under the supervision of Alonso Church. After getting his Ph.D., Rosser taught at Princeton, Harvard, and Cornell and spent the latter part of his career at the University of Wisconsin-Madison. As a logician, Rosser is known for his part in the Church-Rosser Theorem and the Kleene–Rosser Paradox in lambda calculus. Moreover, he served as president of the Association for Symbolic Logic. As an applied mathematician, he served as president of the Society of Industrial and Applied Mathematics (otherwise known as SIAM). While at the University of Wisconsin-Madison, he served as the director of the U.S. Army Mathematics Research Center. He continued to lecture well into his late 70s, and passed away at his home in Madison in 1989. He has a son, J. Barkley Rosser Jr, who’s an economist at James Madison University.

Lowell Schoenfeld spent his early years in New York City, graduating Cum Laude from the College of the City of New York in 1940. He went on to MIT to earn a Master’s. He received his Ph.D. in 1944 from the University of Pennsylvania under the direction of Hans Rademacher. (During his years in graduate school, he seems to have worked for the Philadelphia Navy Yard as well, writing reports on aircraft navigational computers.) After positions at Temple University and Harvard, he moved to the University of Illinois, where he met his future wife. He met Josephine M. Mitchell when she was a tenured Associate Professor and he was an untenured Assistant Professor. After they married, the University would no longer allow Mitchell to teach, so the couple both resigned their positions and eventually settled at Pennsylvania State University. They spent about 10 years there but in 1968 the couple moved to the University of Buffalo, where they remained until their retirements in the 1980s.

As far as I can tell, these are the papers they wrote together, all in analytic number theory:

[1] Rosser, J. Barkley; Schoenfeld, Lowell. “Approximate formulas for some functions of prime numbers”. Illinois J. Math. 6 (1962), no. 1, 64–94.
[2] Rosser, J. Barkley; Schoenfeld, Lowell; J.M. Yohe. “Rigorous Computation and the Zeros of the Riemann Zeta-Function,” 1969
[3] Rosser, J. Barkley; Schoenfeld, Lowell. “Sharper Bounds for the Chebyshev Functions $\theta (x)$ and $\psi (x)$” Mathematics of Computation Vol. 29, No. 129 (Jan., 1975), pp. 243-269
[4] Rosser, J. Barkley; Schoenfeld, Lowell. “Approximation of the Riemann Zeta-Function” 1971.

I haven’t seen a copy of the papers [2] and [4] in years but I’m guessing these are what I looked at as a teenager in Seattle, years ago, wandering through the stacks at the UW.

Rosser also wrote papers on topics in recreational mathematics, such as magic squares. Two such papers were co-written with R.J. Walker from Cornell University, who’s more well-known for his textbook Algebraic Curves:

Rosser, Barkley; Walker, R. J. “The algebraic theory of diabolic magic squares,” Duke Math. J. 5 (1939), no. 4, 705–728
Rosser, Barkley; Walker, R. J. “On the transformation group for diabolic magic squares of order four,” Bull. Amer. Math. Soc. 44 (1938), no. 6, 416–420.

Diabolic magic squares, also called pan-diagonal magic squares, are $n\times n$ squares of integers $1, 2, ..., n^2$ whose rows all add to a constant C, whose columns all add to C, whose diagonals both add to C, and whose “broken diagonals” all add to C. An example was given by the German artist Albrecht Durer in the 1514 engraving called Melencolia I: (where C=34):