Scientific

Over fifty years ago Richard Kenneth Guy joined the then Department of Mathematics, Statistics and Computer Science at the nascent University of Calgary. Although he retired from the University in 1982, he continued, even in his last year, to come in to the University every day and work on the mathematics that he loved. In this talk I will provide a glimpse into the life and research of this most remarkable man. In doing this, I will recount several of the important events of Richard’s life and briefly discuss some of his mathematical contributions.

About Dr. Williams

: Dr. Hugh Williams is internationally recognized as an expert in computational number theory and its applications to cryptography. Shortly after obtaining his Ph.D. in 1969 from the Department of Applied Analysis and Computer Science at the University of Waterloo, he joined the newly established Department of Computer Science at the University of Manitoba, where he was promoted to the rank of Full Professor in 1979. He also served there as Associate Dean of Science for Research Development for seven years (1994-2001). He moved to the University of Calgary in 2001 to take up the iCORE Chair for Algorithmic Number Theory and Cryptography (2001-2013) and retired as Emeritus Professor of Mathematics and Statistics in 2016. Dr. Williams has authored over 150 refereed journal papers, 30 refereed conference papers and 20 books or book chapters, and from 1983-85 held a national Killam Research Fellowship. In February 2009, Dr, Williams was selected for a six year term as the inaugural Director of the Tutte Institute for Mathematics and Computing (TIMC), a highly classified research facility established by the federal government. In 2016, he was appointed Professor Emeritus in Mathematics and Statistics at the University of Calgary.

TBA

Even in the title of one of his papers, Richard Guy called the elliptic curve with equation $y^2 = x^3 - 4x + 4$ his favorite. During the CNTA-XIV meeting in Calgary in 2016, I recalled some of his reasons for this (with Richard listening from the front row). The story as well as a few additional developments will also be the topic of the present lecture.

An account of how a great Guy and his Brown coauthor created a 300-page book entitled "The Unity of Combinatorics" out of a 30-page paper from 1995 of the same name. The latter was an outline of a proposed lecture series, whose purpose was to feature the many connections within the vast area of combinatorics, thereby dispelling the then prevalent notion that combinatorics is just a bag of tricks. In writing the book, we took this notion and ran with it --- and how!

I'll talk about a number of these connections and some topics that seem almost magical, including Beatty sequences, Conway worms, games played with turtles instead of coins, and a way of viewing the non-negative integers as a field. The book begins with a child playing with colored blocks on his living-room rug and ends with a description of the Miracle Octad Generator. Finally, I'll talk about working with this gentlemanly giant of the world of numbers and sequences and patterns and games.

I will discuss a lemma which is usually attributed to Atkinson, is very useful and apparently has been rediscovered multiple times. The lemma says that if (X, B, mu, T) is an ergodic system and f is in L^1(mu) with mean zero, then the Birkhoff sums of f do not diverge almost surely. Moreover the sums switch signs in a wide sense, infinitely many times. I will give the proof and discuss some applications of this result. Then I will discuss higher dimensional analogues, where the situation is significantly more complicated.