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Piers LAWRENCE Obituary
Memories and Condolences
for Piers LAWRENCE
2 Entries
Rob Corless
July 24, 2025
I first met Piers when I visited the University of Canterbury in
Christchurch in February 2008 and gave a talk there. I mentioned in
my talk that I was looking for graduate students. After my talk,
Piers introduced himself and we chatted. I invited him to apply to
the University of Western Ontario, which he did, and he was
accepted into a Masters program for September of that year. I
should state that Piers had taken an honours undergraduate degree
at UC, and genuine honours degrees were becoming uncommon in
Canada. More, he remembered everything he had been taught. While
working as a tutorial assistant at Western he was at first
disbelieving that our (very good!) Western students did not, in
fact, remember everything that they had been taught, and then he
was appalled. But he got used to it, and became a very good TA and
helped to make the numerical analysis courses at Western very
student-centric.
For research, we started work on a method for evaluating a function
known as the Wright omega function. We submitted a paper on that to
the ACM Transactions on Mathematical Software (TOMS) in May 2010,
which was reviewed, revised, and eventually published in that
journal in April 2012. (That two-year delay is a fairly normal
academic journal turnaround time.) ACM TOMS is an extremely
difficult journal to publish in: one has to get the maths right,
the maths has to be interesting maths, and the software has to be
good enough to publish as well. Piers' work for this was later
incorporated into the Python package SciPy, and is now used
worldwide.
And this was just what he did when enrolled in a Masters' program
for a year, which normally is just for "warming up" to research.
We did not insist on his actually taking a Masters' degree and
indeed he never bothered. We transferred him into the PhD program
by an administrative technique available for exceptional students.
Piers started an entirely new project (he was frankly quite bored
with the Wright omega project, only later becoming interested in it
for real when he found a potential application to machine
learning). Instead, for his PhD he worked on a problem in numerical
linear algebra, namely the numerical stability of a new kind of
companion matrix. He published several papers with me on the
subject, and one entirely on his own in SIAM Journal of Matrix
Analysis (again, one of the top journals in the world). Publishing
on your own while you are still a PhD student is again unusual.
Publishing in such a top journal is basically unheard of. He
defended his PhD in July 2013 and his dissertation is available at
https://ir.lib.uwo.ca/etd/1359/ and has as of this date been
downloaded over a thousand times, almost 1200 in fact in the twelve
years since its publication: about a hundred times a year or about
eight times a month. This is truly remarkable in mathematics, where
the average number of readers of a PhD dissertation might be
expected to be no more than about ten, ever, in total.
Now I will tell the story of what I think was the most impactful
mathematical work Piers did. In spring 2011 I was invited to give a
lecture in the seminar series Teaching and Research in Computer
Science at Western. In then-current events, Benoit B. Mandelbrot
had just died the previous autumn, and since I had met Mandelbrot I
decided to give a lecture on Mandelbrot polynomials. To do so, I
applied our then-new companion matrix to them. This gave
interesting results, but Piers was not satisfied. Late one night
(he afterwards reported that it was about 2:00am, and he never
could explain just how he had thought of it) he discovered
something truly remarkable, which we now call the Mandelbrot
matrix. This is a matrix composed entirely of ones and zeros whose
characteristic polynomial is exactly the Mandelbrot polynomial.
The importance of this invention was not apparent at first. It
seemed to be a simple, specialized trick. It was fun, but it didn't
seem all that useful.
One of my subsequent PhD students, Eunice Y. S. Chan, worked
further on it (actually for her Masters thesis, itself now also
downloaded remarkably frequently) and when I casually mentioned the
result to the famous computer scientist Donald E. Knuth and he
asked whether it could be used for Euclid polynomials (he had to
show me what they were) we found that the technique was much more
general and generally useful than had been immediately apparent.
Indeed Piers' discovery led inexorably, via the work of people who
followed him, to the whole field of what is now known as "Bohemian
matrices" and to a still-as-yet poorly understood but exciting
technique which we call "algebraic linearization".
But Piers by this time had gone on to a very prestigious
postdoctoral fellowship in Belgium, working for Paul Van Dooren and
Marc van Barel, two of the leading researchers in the world in the
field of numerical linear algebra. He published many papers with
them, and with other famous researchers at the University of Carlos
III in Madrid, working out different implications of his PhD work
and indeed going much further. He left that postdoctoral fellowship
and started his consulting business, EigenPoly, but retained an
interest in academic work and I continued to meet him from time to
time at conferences and workshops.
At the workshop celebrating David Watkins' 75th birthday in Leuven
in May 2024, we met in person for the final time and on that
occasion he and Trisha graciously invited me to their home for a
wonderfully memorable family dinner.
I had earlier managed to convince Piers to help with the writing of
what turned out to be our final publication together, called "A
Fractal Eigenvector", which appeared in the American Mathematical
Monthly in 2022. This journal doesn't have the intellectual
prestige of the others mentioned here, but it is actually the
highest impact mathematical journal in the world: it is always
listed in the top fifty journals in the world in JSTOR, and
sometimes in the top twenty. Nobel prize winners have published in
it, and Fields medallists too. Our paper studies the eigenvector
and singular vector associated to the largest eigenvalue of the
Mandelbrot matrix that Piers invented, and the pictures it
generated have fascinated several of the world's leading
mathematicians and computer scientists.
We still don't know half of what we might eventually learn from his
invention of Mandelbrot matrices. Piers had other mathematical
ideas, which he talked to me about occasionally. Sometimes I could
follow what he was talking about, but not always. I'm going to have
to wonder if somewhere in there was an idea that would have put
Mandelbrot matrices in the shade. We'll never know, now. Still, he
has left a mathematical legacy that will ripple for a long, long
time.
Clement Winslow
July 22, 2025
Piers,
This was shattering to hear man. You are one the nicest dudes I've met, definitely are a kind caring soul without showing any judgements towards others. I remember 6th Form/Year 12 Chemistry (with Mr Hicks?) always encouraging and motivating me to keep at it. A good soul, gone to soon.
Love to Your Wife, Children and rest of Your family.
P.S: Happy I got to finally pay ya back that $50, and buy you a drink at the reunion.
Clem.
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