# Jesse Douglas – yet another Beautiful Mind?

Nobel prize winner John Nash struggled with mental illness for most of his life. His struggles were described Sylvia Nasar’s well-known biography, A Beautiful Mind, as well as a film of the same name starring Russell Crowe. Now, John Nash is practically a household name, honored for his contributions to the mathematics of game theory and economics.

While Jesse Douglas isn’t nearly as well-known as Nash, and didn’t win a Nobel Prize, he was (with Lars Ahlfors in 1936) the first recipient of the Fields Medal. According to one source, he too struggled with mental illness. However, before talking about the life of Jesse Douglas, let’s talk about his mathematical research.

Jesse Douglas in his 40s.

#### His Mathematics

Jesse Douglas solved a geometrical problem formulated in the 1700’s now called Plateau’s problem.  Plateau’s problem (also known as the soap bubble problem) is to show the existence of a minimal surface with a given boundary. Plateau was a a Belgian physicist who lived in the 1800s, known for his experiments with soap films. A minimal surface is a surface that locally minimizes its area. For example, the catenoid (see below) obviously doesn’t have minimum area for its (disconnected) boundary, but any piece of it bounded by a (connected) closed curve does have minimum area.

His research was not universally recognized at first. According to Reid [R76]:

Arriving in the year 192901930 at the end of a European tour as a National Research Fellow, he proposed to talk about his not-yet-published work on Plateau’s Problem at the weekly meeting of the mathematische Gesellschaft [at Gottingen University]. The problem had been around for a long time. Many outstanding mathematicians, including Riemann himself, had worked on it. The members of the Gesellschaft simply did not believe that an American had solved it. … When he had finished his presentation, some of the members of the Gesellschaft took him severely to task on almost every detail of the proof. Let left Gottingen deeply offended …

Douglas solved this geometrical problem in his paper “Solution of the problem of Plateau,” Trans. Amer. Math. Soc. 33 (1931)263–321. Gray and Micallef [GM07] do an excellent job of describing Jesse’s mathematical approach to the problem. Jesse continued working on generalizing and refining aspects of his research on this problem for the next 10 years or so. Some of his papers in this field include

• A method of numerical solution of the problem of Plateau, Annals of Mathematics 29(1927 – 1928)180-188
• Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931) 263–321.
• Systems of K-dimensional manifolds in an N-dimensional space, Mathematische Annalen 105 (1931) 707-733.
• The Problem of Plateau for Two Contours, Studies in Applied Mathematics 10(1931)315-359.
• One-sided minimal surfaces with a given boundary, Trans. Am Math Soc 34(1932)731–756.
• A Jordan space curve no arc of which can form part of a contour which bounds a finite area, Annals of Mathematics 35(1934) 100-103.
• Green’s function and the problem of Plateau, American Journal of Mathematics. 61 (1939) 545–589.
• The most general form of the problem of Plateau, American Journal of Mathematics. 61 (1939)590–608.
• Solution of the inverse problem of the calculus of variations, Proceedings of the National Academy of Sciences. 25(1939) 631–637.
• The analytic prolongation of a minimal surface across a straight line, Proc Natl Acad Sci USA. 25(1939)375-377.
• The higher topological form of Plateau’s problem, Annali della Scuola Normale Superiore di Pisa , 8 (1939)195-218.
• Minimal surfaces of higher topological structure, Annals of Mathematics 40(1939) 205-298 .
• Theorems in the inverse problem of the calculus of variations, Proc Natl Acad Sci USA, 26(1940) 215–221.

He also wrote papers in other fields in mathematics, for example, the theory of finite groups. For example, in 1951 he published a series of short articles On finite groups with two independent generators, all in the Proc. Natl. Acad. Sci. USA. Also, in 1953, he published a series of short articles On the basis theorem for finite abelian groups, again, all in the Proc. Natl. Acad. Sci. USA.

#### His Life

Jesse’s father, Louis, was a Polish immigrant who moved to Canada sometime in the late 1800s. The family name was changed at Canadian Immigration and I don’t know his original name. At some point, Louis married Sarah Kommel (I don’t know when the move occurred, nor when and where they met) and moved to New York City. I think both Louis and Sarah were Jewish. Jesse was born in NYC on July 3rd, 1897. According to a geneology website, Sarah’s parents were Leazar Louis Kommel (1851-1923) and Chaya Lande, who died in 1906. Jesse’s mother Sarah passed away in 1939, a year before Jesse married (as we will see below).

Jesse was educated at public schools in New York City. According to Steinhardt [St92], Jesse had a photographic memory. After graduation from high school, he entered the City College of New York and won the Belden Medal for excellence in mathematics in his first year at City College of New York, the youngest recipient at the time. He graduated with a B.Sc. cum laude in February 1916 at the age of 18. Then he entered Columbia University to undertake research under the supervision of Edward Kasner. He took part in Kasner’s seminar on differential geometry and it was there that Jesse first heard of Plateau’s Problem. He submitted his doctoral thesis On Certain Two-Point Properties of General Families of Curves in 1920 (some references say 1921). After getting his PhD, he was an instructor at Columbia from 1920 to 1926. (Was this in the statistics department or the mathematics department? I don’t know.)

He won a National Research Fellowship and as a result, traveled widely for four years, visiting Princeton and Harvard in 1927, Chicago in 1928, and Paris from 1928 to 1930, with trips to Gottingen, Hamburg, and Rome. Jesse was offered an Assistant Professor position at MIT in January 1930. He took leave of absence for a term in 1932. After his promotion to Associate Professor at MIT in 1934, he almost at once took leave of absence again to be a Research Worker at the Institute for Advanced Study in Princeton from 1934 to 1935.  After returning to MIT, Jesse was awarded the first Fields Medal, together with Lars Ahlfors, in 1936. Once again, Jesse took leave of absence from MIT in 1936, and on 1 July 1938, he resigned. According to Aronson [A13], Jesse “became ill” at this time and Levinson was eventually hired as his replacement.

This may seem like the romantic life of a world-class mathematician, but it really is very unusual to be “homeless” 18 years after your get your PhD degree. Something is going on. According to Hersh [H10]:

Douglas’s name is almost forgotten today. He is a rather tragic figure, one of several important mathematicians gravely handicapped by what are now called bipolar, and used to be called manic-depressive, symptoms. He had a junior position at M.I.T., which he lost as a result of inability to perform consistently in the classroom.

According to Steinhardt [St92],

… the volatility of this mix of great gifts and sharp emotions [in which his “over-sized sensitivity surfaced not infrequently in intense feelings articulated vigorously to colleagues”] in Douglas’ makeup worked to the disadvantage of the great American mathematician’s external fortunes.

I don’t know what he did professionally in the 1939-1940 period, but he married Jessie Nayler on June 30th, 1940 (they had one son Lewis Philip Douglas). The husband named Jesse and the wife named Jessie! He often taught summer courses at Columbia University, as well as courses at Yeshiva University. The following year, Jesse was a Guggenheim fellow at Columbia University from 1940 to 1941. He taught at Brooklyn College from 1942 to 1946, as an assistant professor. His teaching during these war years went well according to a letter of commendation (did he teach students from the US Military Academy during this time?), and he received a Distinguished Service Award. In 1945, Jesse wrote an application letter for a teaching position saying [St92]:

This semester I have a teaching load of 19 hours per week at Brooklyn College, including two advances courses: complex variables and calculus of variations.

It is no surprise that he published nothing in the period 1942-1950. According to Micallef [M13],

There is a blank in Douglas’s career from 1946 to 1950, and the later part of Douglas’ life seems to have been troubled. His marriage ended in divorce in 1950.

In 1951, Jesse published a string of papers in a completely different field, finite group theory. Perhaps the end of Jesse’s marriage to Jessie caused him to reinvent himself, mathematically?

According to some sources, Douglas harbored resentment towards his research “competition” Rado (who he strongly beieved should not be credited with solving Plateau’s Problem [St92]) and Courant, as well as J.F. Ritt at Columbia, who may have blocked his appointment to a permanent position at his alma mater. (Sadly, anti-Semitic prejudices common of those times may have played a role in Jesse’s troubles in this regard.) Never-the-less, he is remembered as a helpful, sympathetic, and approachable colleague [St92].

His resentment of Courant was unfortunate because some 10 years earlier, in March 1935, Courant invited Jesse to speak at NYU. According to Reid [R76]:

Courant’s invitation to Douglas to speak at NYU was to a certain extent an olive branch. … Remembering this past history [Jesse’s reception at Gottingen 5 years earlier], Courant did his best to make Douglas’ lecture at NYU an event.

Clearly, Courant appreciated Jesse’s work and tried to treat him with due respect.

Other than Jesse’s mathematical research, this does not appear to have been a happy period in Jesse’s life. According to O’Conner and Robertson [OCR],

His [ex-]wife, Jessie Douglas died in 1955, the year in which Douglas was appointed professor of mathematics at the City College of New York. He remained in that post for the final ten years of his life, living in Butler Hall, 88 Morningside Drive in New York.

Steinhardt [St92] recalls the difficulty that Abraham Schwartz had in having CCNY give tenure at the full professor level to Jesse, when he was 1958. Except for the logician Emil Post, there was no one at CCNY at the time close to Jesse’s stature as a researcher. Ironically, a number of CCNY mathematicians “felt threatened by the appointment”, presumably fearing Jesse would require them to work harder.

There are a number of anecdotes of Jesse Douglas and his interactions with students. They aren’t universally flattering and I hesitate to repeat the negative ones, as I don’t know how typical they are. A number of sources suggest that when he was feeling well, he could be an impressive, charismatic, self-confident expositor and a lucid  teacher. While I don’t know the teaching load at Columbia University the time (where he belonged), I do know that, compared to today’s teaching loads, they were often quite heavy (see for example Parikh’s biography, The Unreal Life of Oscar Zariski). Are these unflattering reports of Jesse’s interactions with students merely due to his oppressive teaching load and inability to handle that volume of students? I don’t know. Perhaps he held others to the same very high standards that he held himself in his own mathematical research.

#### References

[A13] D.G. Aronson, Norman Levinson: 1912-1975, Biographical memoirs, Nat. Acad. Sci., 2013.

[GM07] Jeremy Gray, Mario Micallef, The work of Jesse Douglas on minimal surfaces, 2013. (appeared in Bull AMS 2008)  pdf

[H10] R. Hersh, Under-represented then over-represented, College J. Math, 2010.

[M13] M. Micallef, The work of Jesse Douglas on Minimal Surfaces, talk at Universidad de Granada, 2013-02-07.

[OCR] John J. O’Connor and Edmund F. Robertson, Jesse Douglas.

[R76] C. Reid, Courant, Springer-Verlag, 1976.

[St92] F. Steinhardt, “Jesse Douglas as teacher and colleague”, in The Problem of Plateau, ed. T. Rassias, World Scientific, 1992.

# Calculus on graphs

In these notes, I tried to cover enough material to get a feeling for “calculus on graphs”, with applications to sports rankings and the Friendship Theorem. Here’s a list of the topics.

1 . Introduction
2. Examples
3. Basic definitions
3.1 Diameter, radius, and all that
3.2 Treks, trails, paths
3.3 Maps between graphs
3.4 Colorings
3.5 Transitivity
4.1 Definition
4.2 Basic results
4.3 The spectrum
4.4 Application to the Friendship Theorem
4.5 Eigenvector centrality and the Keener ranking
4.6 Strongly regular graphs
4.7  Orientation on a graph
5. Incidence matrix
5.1 The unsigned incidence matrix
5.2 The oriented case
5.3 Cycle space and cut space
6. Laplacian matrix
6.1 The Laplacian spectrum
7  Hodge decomposition for graphs
7.1 Abstract simplicial complexes
7.2 The Bjorner complex and the Riemann hypothesis
7.3 Homology groups
8. Comparison graphs
8.1 Comparison matrices
8.2 HodgeRank
8.3 HodgeRank example

# Simple unsolved math problem, 8

Sylver coinage is a game for 2 players invented by John H. Conway.

The two players take turns naming positive integers that are not the sum of non-negative multiples of any previously named integers. The player who is forced to name 1 loses.

James Joseph Sylvester proved the following fact.

Lemma: If a and b are relatively prime positive integers, then (a – 1)(b – 1) – 1 is the largest number that is not a sum of nonnegative multiples of a and b.

Therefore, if a and b have no common prime factors and are the first two moves, this formula gives an upper bound on the next number that can still be played.

R. L. Hutchings proved the following fact.

Theorem: If the first player selects any prime number $p>3$ as a first move then he/she has a winning strategy.

Very little is known about the subsequent winning moves. That is, a winning strategy exists but it’s not know what it is!

Unsolved problem:Are there any non-prime winning opening moves in Sylver coinage?

For further info, Sicherman maintains a Sylver coinage game webpage.

# Sports ranking methods, 3

This is the third of a series of expository posts on matrix-theoretic sports ranking methods. This post discusses the random walker ranking.

We follow the presentation in the paper by Govan and Meyer (Ranking National Football League teams using Google’s PageRank). The table of “score differentials” based on the table in a previous post is:

$\begin{tabular}{c|cccccc} \verb+x\y+ & Army & Bucknell & Holy Cross & Lafayette & Lehigh & Navy \\ \hline Army & 0 & 0 & 1 & 0 & 0 & 0 \\ Bucknell & 2 & 0 & 0 & 2 & 3 & 0 \\ Holy Cross & 0 & 3 & 0 & 4 & 14 & 0 \\ Lafayette & 10 & 0 & 0 & 0 & 0 & 0 \\ Lehigh & 2 & 0 & 0 & 11 & 0 & 0 \\ Navy & 11 & 14 & 8 & 22 & 6 & 0 \\ \end{tabular}$
This leads to the following matrix:

$M_0=\left(\begin{array}{cccccc} 0 & 0 & 1 & 0 & 0 & 0 \\ 2 & 0 & 0 & 2 & 3 & 0 \\ 0 & 3 & 0 & 4 & 14 & 0 \\ 10 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 11 & 0 & 0 \\ 11 & 14 & 8 & 22 & 6 & 0 \\ \end{array}\right) .$

The edge-weighted score-differential graph associated to $M_0$ (regarded as a weighted adjacency matrix) is in the figure below.

This matrix $M_0$ must be normalized to create a (row) stochastic matrix:

$M = \left(\begin{array}{cccccc} 0 & 0 & 1 & 0 & 0 & 0 \\ {2}/{7} & 0 & 0 /{7} /{7} & 0 \\ 0 /{7} & 0 /{21} /{3} & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ {2}/{13} & 0 & 0 /{13} & 0 & 0 \\ {11}/{61} /{61} /{61} /{61} /{61} & 0 \\ \end{array}\right) .$

Next, to insure it is irreducible, we replace $M$ by $A=(M+J)/2$, where $J$ is the $6\times 6$ doubly stochastic matrix with every entry equal to $1/6$:

$A=\left(\begin{array}{cccccc} {1}/{12} & 1/{12} & 7/{12} & 1/{12} & 1/{12} & 1/{12} \\ {19}/{84} & 1/{12} & 1/{12} & 19/{84} & 25/{84} & 1/{12} \\ {1}/{12} & 13/{84} & 1/{12} & 5/{28} & 5/{12} & 1/{12} \\ {7}/{12} & 1/{12} & 1/{12} & 1/{12} & 1/{12} & 1/{12} \\ {25}/{156} & 1/{12} & 1/{12} & 79/{156} & 1/{12} & 1/{12} \\ {127}/{732} & 145/{732} & 109/{732} & 193/{732} & 97/{732} & 1/{12} \end{array}\right).$

Let

${\bf v}_0 = \left( \frac{1}{6} , \frac{1}{6} , \frac{1}{6} , \frac{1}{6} , \frac{1}{6} , \frac{1}{6}\right).$

The ranking determined by the random walker method is the reverse of the left eigenvector of $A$ associated to the largest eigenvalue $\lambda_{max}=1$ (by reverse, I mean that the vector ranks the teams from worst-to-best, not from best-to-worst, as we have seen in previous ranking methods).
In other words, the vector

${\bf r}^*=\lim_{n\to \infty}{\bf v}_0A^n.$

This is approximately

${\bf r}^* \cong \left(0.2237\dots ,\,0.1072\dots ,\,0.2006\dots ,\,0.2077\dots ,\,0.1772\dots ,\,0.0833\dots \right).$

Its reverse gives the ranking:

Army $<$ Lafayette $<$ Bucknell $<$ Lehigh $<$ Holy Cross $<$ Navy.

This gives a prediction failure rate of $13.3\%$.

# Memories of TS Michael, by Thomas Quint

TS Michael passed away on November 22, 2016, from cancer. I will miss him as a colleague and a kind, wise soul. Tom Quint has kindly allowed me to post these reminiscences that he wrote up.

Well, I guess I could start with the reason TS and I met in the first place. I was a postdoc at USNA in about 1991 and pretty impressed with myself. So when USNA offered to continue my postdoc for two more years (rather than give me a tenure track position), I turned it down. Smartest move I ever made, because TS got the position and so we got to know each other.

We started working w each other one day when we both attended a talk on “sphere of influence graphs”. I found the subject moderately interesting, but he came into my office all excited, and I couldn’t get rid of him — wouldn’t leave until we had developed a few research ideas.

Interestingly, his position at USNA turned into a tenure track, while mine didn’t. It wasn’t until 1996 that I got my position at U of Nevada.

Work sessions with him always followed the same pattern. As you may or may not know, T.S. a) refused to fly in airplanes, and b) didn’t drive. Living across the country from each other, the only way we could work together face-to-face was: once each summer I would fly out to the east coast to visit my parents, borrow one of their cars for a week, and drive down to Annapolis. First thing we’d do is go to Whole Foods, where he would load up my car with food and other supplies, enough to last at least a few months. My reward was that he always bought me the biggest package of nigiri sushi we could find — not cheap at Whole Foods!

It was fun, even though I had to suffer through eight million stories about the USNA Water Polo Team.

Oh yes, and he used to encourage me to sneak into one of the USNA gyms to work out. We figured that no one would notice if I wore my Nevada sweats (our color is also dark blue, and the pants also had a big “N” on them). It worked.

Truth be told, TS didn’t really have a family of his own, so I think he considered the mids as his family. He cared deeply about them (with bonus points if you were a math major or a water polo player :-).

One more TS anecdote, complete with photo.  Specifically, TS was especially thrilled to find out that we had named our firstborn son Theodore Saul Quint.  Naturally, TS took to calling him “Little TS”.  At any rate, the photo below is of “Big TS” holding “Little TS”, some time in the Fall of 2000.

TS Michael in 2000.

# Simple unsolved math problem, 7

Everyone’s heard of the number $\pi =$ 3.141592…, right?

Robert Couse-Baker / CC BY http://2.0 / Flickr: 29233640@N07

And you probably know that $\pi$ is not a rational number (i.e., a quotient of two integers, like 7/3). Unlike a rational number, whose decimal expansion is eventually periodic, if you look at the digits of $\pi$ they seem random,

3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482…

But are they really? No one really knows. There’s a paper that explores the statistics of these digits using the first 22.4 trillion digits of $\pi$. Does any finite sequence of k digits (say, for example, the 4-digit sequence 2016) occur just as often as any other sequence of the same length (say, 1492), for each k? When the answer is yes, the number is called ‘normal.’ That is, a normal number is a real number whose infinite sequence of digits is distributed uniformly in the sense that each digit has the same natural density 1/10, also all possible k-tuples of digits are equally likely with density 1/k, for any integer $k>1$.

The following simple problem is unsolved:

Conjecture: $\pi$ is normal.

# Simple unsolved math problem, 6

If you know a little point-set topology, below is an unsolved math problem whose statement is relatively simple.

Probably everyone has at least seen the Mandelbrot set in some form, as it’s a popular object of mathematical artists. Here’s a picture from Wikipedia:

The formal definition is as follows. Let $f_c (z)=z^2+c$, where $c\in \mathbb{C}$ is a complex number. The Mandelbrot set $X$ is the complex plot of the set of complex numbers $c$ for which the sequence of iterates

$f_c (0), f_c (f_c (0)), f_c (f_c (f_c (0))), \dots,$

remains bounded in absolute value.
We say $X$ is locally connected if every point $x\in X$ admits a neighborhood basis consisting entirely of open, connected sets.

Conjecture: The Mandelbrot set $X$ is locally connected.

# A tribute to TS Michael

I’ve known TS for over 20 years as a principled colleague and a great teacher.

TS at the USNA in Dec 2015.

However, we really never spoke much except for the past five-to-ten years or so. For a period, I wrote a lot about error-correcting codes and we’d talk occasionally about our common interests (for example, I found his paper “The rigidity theorems of Hamada and Ohmori, revisited” fascinating). However, once I became interested in graph theory, we spoke as often as I could corner him. He taught me a lot and only know I realize how lucky I was to have him as a colleague.

I remember many times, late on a Friday, when we’d talk for an hour or two about chess, mathematics, “office politics” (he always knew more than me), and allergies. Here’s one of his favorite chess problems:

Mate in 549 moves. This problem was discovered by a team of chess engame experts at Lomonosov University, Moscow, August 2012.

Maybe this says more about me than him, but when it was just the two of us, we rarely talked about families or relationships. None-the-less, he always treated me like a good friend. One of my favorite memories was when my wife and I were shopping at the plaza where his condo building was located (it’s a big plaza). Elva and I were walking store-to-store when we spotted TS. He was walking to distract himself from his discomfort. At the time, doctors didn’t know what his problems were and he suspected allergies. I have a number of food sensitivities and he was a welcomed fountain of medical knowledge about these issues. (In fact, his hints have really helped me a lot, health-wise.) In any case, TS and Elva and I spoke for 30 minutes or so about health and family. I remember how gracious and thoughtful he was, skillfully steering the conversation into non-technical matters for Elva’s benefit. I ran into him another time while waiting for Elva, who was in a nearby doctor’s office (I told you this was a big shopping plaza). TS generously waited with me until Elva was ready to be picked up. What we chatted about is lost in the cobwebs of my memory but I remember vividly where we sat and the kind of day it was. TS had such a kind heart.

As I said, TS taught me a lot about graph theory. Whether in-between classes or when I was lucky enough to spot him late in the day, he’d kindly entertain my naive (usually false) conjectures and speculations about strongly regular graphs. I never heard him speak in anything but the kindest terms. He’d never say “that’s just plain wrong” or “idiotic” (even if it was) but instead teach me the correct way to think about it in a matter in which I could see myself how my speculations were wrong-headed. My upcoming book with Caroline Melles is indebted to his insight and suggestions.

Even after he left Maryland to spend his remaining days with his family in California, TS emailed encouragement and suggestions about an expository paper I was writing to help connect my matrix theory students with the methods of ranking sports teams. While he was very helpful and provided me with his excellent insights as usual, in truth, I used the work on the paper as an excuse to keep up with his health status. I’m relatively ignorant of medical issues and tried to stay optimistic until it’s totally unrealistic. As sad as it was, we was always frank and honest with me about his prognosis.

He’s gone now, but as a teacher, researcher, and as a kind soul, TS is unforgettable.

A list of TS’s publications:

1. T. S. Michael, Tournaments, book chapter in Handbook of Linear Algebra, 2nd ed, CRC Press, Boca Raton, 2013.
2. T. S. Michael, Cycles of length 5 in triangle-free graphs: a sporadic counterexample to a characterization of equality, Bulletin of the Institute of Combinatorics and Its Applications, 67 (2013) 6–8.
3. T. S. Michael and Val Pinciu, Guarding orthogonal prison yards: an upper bound,
Congressus Numerantium, 211 (2012) 57–64.
4. Ilhan Hacioglu and T. S. Michael, The p-ranks of residual and derived skew Hadamard designs,
Discrete Mathematics, 311 (2011) 2216-2219.
5. T. S. Michael, Guards, galleries, fortresses, and the octoplex, College Math Journal, 42 (2011) 191-200. (This paper won a Polya Award)
6. Elizabeth Doering, T. S. Michael, and Bryan Shader, Even and odd tournament matrices with minimum rank over finite fields, Electronic Journal of Linear Algebra, 22 (2011) 363-377.
7. Brenda Johnson, Mark E. Kidwell, and T. S. Michael, Intrinsically knotted graphs have at least 21 edges, Journal of Knot Theory and Its Ramifications, 19 (2010) 1423-1429.
8. T. S. Michael, How to Guard an Art Gallery and Other Discrete Mathematical Adventures. Johns Hopkins University Press, Baltimore, 2009.
9. T. S. Michael and Val Pinciu, Art gallery theorems and triangulations, DIMACS Educational Module Series, 2007, 18 pp (electronic 07-1)
10. T. S. Michael and Thomas Quint, Sphericity, cubicity, and edge clique covers of graphs, Discrete Applied Mathematics, 154 (2006) 1309-1313.
11. T. S. Michael and Val Pinciu, Guarding the guards in art galleries, Math Horizons, 14 (2006), 22-23, 25.
12. Richard J. Bower and T. S. Michael, Packing boxes with bricks, Mathematics Magazine, 79 (2006), 14-30.
13. T. S. Michael and Thomas Quint, Optimal strategies for node selection games: skew matrices and symmetric games, Linear Algebra and Its Applications 412 (2006) 77-92.
14. T. S. Michael, Ryser’s embedding problem for Hadamard matrices, Journal of Combinatorial Designs 14 (2006) 41-51.
15. Richard J. Bower and T. S. Michael, When can you tile a box with translates of two given rectangular bricks?, Electronic Journal of Combinatorics 11 (2004) Note 7, 9 pages.
16. T. S. Michael and Val Pinciu, Art gallery theorems for guarded guards, Computational Geometry 26 (2003) 247-258.
17. T. S. Michael, Impossible decompositions of complete graphs into three Petersen subgraphs, Bulletin of the Institute of Combinatorics and Its Applications 39 (2003) 64-66.
18. T. S. Michael and William N. Traves, Independence sequences of well-covered graphs: non-unimodality and the roller-coaster conjecture, Graphs and Combinatorics 19 (2003) 403-411.
19. T. S. Michael and Thomas Quint, Sphere of influence graphs and the L-Infinity metric, Discrete Applied Mathematics 127 (2003) 447-460.
20. T. S. Michael, Signed degree sequences and multigraphs, Journal of Graph Theory 41 (2002) 101-105.
21. T. S. Michael and Val Pinciu, Multiply guarded guards in orthogonal art galleries, Lecture Notes in Computer Science 2073, pp 753-762, in: Proceedings of the International Conference on Computer Science, San Francisco, Springer, 2001.
22. T. S. Michael, The rigidity theorems of Hamada and Ohmori, revisited, in Coding Theory and Cryptography: From the Geheimschreiber and Enigma to Quantum Theory. (Annapolis, MD, 1998), 175-179, Springer, Berlin, 2000.
23. T. S. Michael and Thomas Quint, Sphere of influence graphs in general metric spaces, Mathematical and Computer Modelling, 29 (1999) 45-53.
24. Suk-Geun Hwang, Arnold R. Kraeuter, and T. S. Michael, An upper bound for the permanent of a nonnegative matrix, Linear Algebra and Its Applications 281 (1998), 259-263.
* First Corrections: Linear Algebra and Its Applications 300 (1999), no. 1-3, 1-2
25. T. S. Michael and W. D. Wallis, Skew-Hadamard matrices and the Smith normal form, Designs, Codes, and Cryptography, 13 (1998) 173-176.
26. T. S. Michael, The p-ranks of skew Hadamard designs, Journal of Combinatorial Theory, Series A, 73 (1996) 170-171.
27. T. S. Michael, The ranks of tournament matrices, American Mathematical Monthly, 102 (1995) 637-639.
28. T. S. Michael, Lower bounds for graph domination by degrees, pp 789-800 in Graph Theory, Combinatorics, and Algorithms: Proceedings of the Seventh Quadrennial International Conference on the Theory and Applications of Graphs, Y. Alavi and A. Schwenk (eds.), Wiley, New York, 1995.
29. T. S. Michael and Thomas Quint, Sphere of influence graphs: a survey, Congressus Numerantium, 105 (1994) 153-160.
30. T. S. Michael and Thomas Quint, Sphere of influence graphs: edge density and clique size, Mathematical and Computer Modelling, 20 (1994) 19-24.
31. T. S. Michael and Aaron Stucker, Mathematical pitfalls with equivalence classes, PRIMUS, 3 (1993) 331-335.
32. T. S. Michael, The structure matrix of the class of r-multigraphs with a prescribed degree sequence, Linear Algebra and Its Applications, 183 (1993) 155-177.
33. T. S. Michael, The decomposition of the complete graph into three isomorphic strongly regular graphs, Congressus Numerantium, 85 (1991) 177-183.
34. T. S. Michael, The structure matrix and a generalization of Ryser’s maximum term rank formula, Linear Algebra and Its Applications, 145 (1991) 21-31.
35. Richard A. Brualdi and T. S. Michael, The class of matrices of zeros, ones and twos with prescribed row and column sums, Linear Algebra and Its Applications, 114(115) (1989) 181-198.
36. Richard A. Brualdi and T. S. Michael, The class of 2-multigraphs with a prescribed degree sequence, Linear and Multilinear Algebra, 24 (1989) 81-102.
37. Richard A. Brualdi, John L. Goldwasser, and T. S. Michael, Maximum permanents of matrices of zeros and ones, Journal of Combinatorial Theory, Series A, 47 (1988) 207-245.

# Memories of TS Michael, by Bryan Shader

TS Michael passed away on November 22, 2016, from cancer. I will miss him as a colleague and a kind, wise soul.

TS Michael in December 2015 at the USNA

Bryan Shader has kindly allowed me to post these reminiscences that he wrote up.

Memories of TS Michael, by Bryan Shader

Indirect influence
TS indirectly influenced my choice of U. Wisconsin-Madison for graduate school. My senior year as an undergraduate, Herb Ryser gave a talk at my school. After the talk I was able to meet Ryser and asked for advice on graduate schools. Herb indicated that one of his very good undergraduate students had chosen UW-Madison and really liked the program. I later found out that the person was TS.

Back in the dark ages, universities still did registration by hand. This meant that for a couple of days before each semester the masses of students would wind their way through a maze of stations in a large gymnasium. For TS’s first 4 years, he would invariably encounter a road block because someone had permuted the words in his name (Todd Scott Michael) on one of the forms. After concretely verifying the hatcheck probabilities and fearing that this would cause some difficulties in graduating, he legally changed his name to TS Michael.

Polyominoes & Permanents
I recall many stories about how TS’s undergraduate work on polyominoes affected
his life. In particular, he recalled how once he started working on tilings on
polyominoes, he could no longer shower, or swim without visualizing polynomino
tilings on the wall’s or floor’s tiling. We shared an interest and passion for permanents (the permanent is a function of a matrix much like the determinant and plays a critical role in combinatorics). When working together we frequently would find that we both couldn’t calculate the determinant of a 3 by 3 matrix correctly, because we were calculating the permanent rather than the determinant.

Presentations and pipe-dreams
TS and I often talked about how best to give a mathematical lecture, or
presentation at a conference. Perhaps this is not at all surprising, as our common advisor (Richard Brualdi) is an incredible expositor, as was TS’s undergraduate advisor (Herb Ryser, our mathematical grandfather). TS often mentioned how Herb Ryser scripted every moment of a lecture; he knew each word he would write on the board and exactly where it would be written. TS wasn’t quite so prescriptive–but before any presentation he gave he would go to the actual room of the presentation a couple of times and run through the talk. This would include answering questions from the “pretend” audience. After being inspired by TS’s talks, I adopted this preparation method.
TS and I also fantasized about our talks ending with the audience lifting us up on their shoulders and carrying us out of the room in triumph! That is never happened to either of us (that I know of), but to have it, as a dream has always been a good motivation.

Mathematical heritage
TS was very interested in his mathematical heritage, and his mathematics brothers and sisters. TS was the 12th of Brandi’s 37 PhD students; I was the 15th. In 2005, TS and I organized a conference (called the Brualidfest) in honor of Richard Brualdi. Below I attach some photos of the design for the T-shirt.

t-shirt design for Brualdi-Fest, 1

The first image shows a biclique partition of K_5; for each color the edges of the given color form a complete bipartite graph; and each each of the completed graph on 5 vertices is in exactly one of these complete bipartite graph. This is related to one of TS’s favorite theorem: the Graham-Pollak Theorem.

t-shirt design for Bruldi-Fest, 2

The second image (when the symbols are replaced by 1s) is the incidence matrix of the projective plane of order 2; one of TS’s favorite matrices.

Here’s a photo of the Brualdi and his students at the conference:

From L to R they are: John Mason (?), Thomas Forreger, John Goldwasser, Dan Pritikin, Suk-geun Hwang, Han Cho, T.S. Michael, B. Shader, Keith Chavey, Jennifer Quinn, Mark Lawrence, Susan Hollingsworth, Nancy Neudauer, Adam Berliner, and Louis Deaett.

Here’s a picture for a 2012 conference:

From bottom to top: T.S. Michael (1988), US Naval Academy, MD; Bryan Shader (1990), University of Wyoming, WY; Jennifer Quinn (1993), University of Washington, Tacoma, WA; Nancy Neudauer (1998), Pacific University, OR; Susan Hollingsworth (2006), Edgewood College, WI; Adam Berliner (2009), St. Olaf College, MN; Louis Deaett (2009), Quinnipiac University, CT; Michael Schroeder (2011), Marshall University, WV; Seth Meyer (2012), Kathleen Kiernan (2012).

Here’s a caricature of TS made by Kathy Wilson (spouse of mathematician
Richard Wilson) at the Brualdifest:

TS Michael, by Kathy Wilson

Long Mathematical Discussions
During graduate school, TS and I would regularly bump into each other as we
were coming and going from the office. Often this happened as we were crossing University Avenue, one of the busiest streets in Madison. The typical conversation started with a “Hi, how are you doing? Have you considered X?” We would then spend the next 60-90 minutes on the street corner (whether it was a sweltering 90 degrees+, or a cold, windy day) considering X. In more recent years, these conversations have moved to hotel lobbies at conferences that we attend together. These discussions have been some of the best moments of my life, and through them I have become a better mathematician.

Here’s a photo of T.S. Michael with Kevin van der Meulen at the Brualdi-fest.

I’m guessing they are in the midst of one of those “Have you considered X?” moments that TS is famous for.

Mathematical insight
TS has taught me a lot about mathematics, including:

•  How trying to generalize a result can lead to better understanding of the original result.
•  How phrasing a question appropriately is often the key to a mathematical breakthrough
• Results that are surprising (e.g. go against ones intuition), use an elegant proof (e.g. bring in matrices in an unexpected way), and are aesthetically pleasing are worth pursing (as Piet Hein said “Problems worthy of attack, prove their worth by fighting back.”)
•  The struggle to present the proof of a result in the simplest, most self-contained way is important because often it produces a better understanding. If you can’t say something in a clean way, then perhaps you really don’t understand it fully.

TS’ mathematics fathers are:
Richard Brualdi ← Herb Ryser ← Cyrus MacDuffee ← Leonard Dickson ← E.H. Moore ← H. A. Newton ← Michel Chasles ← Simeon Poisoon ← Joseph Lagrange ← Leonhard Euler ← Johann Bernoulli.

# Simple unsolved math problem, 5

This is now almost completely solved! Kaisa Matomäki, Maksym Radziwill, Xuancheng Shao, Joni Teräväinen, and Terrance Tao solved the conjecture below in the “interior” of Pascal’s triangle (see T. Tao’s blog post for further details, with the link to the paper and a discussion).

It seems everyone’s heard of Pascal’s triangle. However, if you haven’t then it is an infinite triangle of integers with 1‘s down each side and the inside numbers determined by adding the two numbers above it:

First 6 rows of Pascal’s triangle

The first 6 rows are depicted above. It turns out, these entries are the binomial coefficients that appear when you expand $(x+y)^n$ and group the terms into like powers $x^{n-k}y^k$:

First 6 rows of Pascal’s triangle, as binomial coefficients.

The history of Pascal’s triangle pre-dates Pascal, a French mathematician from the 1600s, and was known to scholars in ancient Persia, China, and India.

Starting in the mid-to-late 1970s, British mathematician David Singmaster was known for his research on the mathematics of the Rubik’s cube. However, in the early 1970’s, Singmaster made the following conjecture [1].

Conjecture: If $N(a)$ denotes the number of times the number $a > 1$ appears in Pascal’s triangle then $N(a) \leq 12$ for all $a>1$.

In fact, there are no known numbers $a>1$ with $N(a)>8$ and the only number greater than one with $N(a)=8$ is a=3003.

References:

[1] Singmaster, D. “Research Problems: How often does an integer occur as a binomial coefficient?”, American Mathematical Monthly, 78(1971) 385–386.