Tom Brown (mathematician)

Tom Brown was born in 1938 in Portland. His father was born in 1905 in a logging camp on the Columbia River in Oregon, not far from Portland and his mother was born in 1909 in a small town in southern Wyoming. Shortly after the end of WW2, Brown’s family moved to Beaverton, a small town in the US about an hours’ drive from Portland. Brown entered a two-room school in 3rd grade where the principal was also the janitor and the school-bus driver. The new class had already mastered subtraction of one-digit numbers, which he found completely baffling at the time. Brown’s 3rd grade teacher patiently spent hours with him to catch up in mathematics. Without her assistance to surpass his difficulties, Brown would have never become a mathematician. In the 12th grade, he realized his interest and talent in mathematics, however, Brown spent most of his time playing the trumpet, and considered a career in music.

Tom Brown graduated in 1956 with a 4-year “General Motors scholarship” which could be used at any college or university in the US. He chose the California Institute of Technology, because of an article he read in Time magazine. After one year, Brown transferred to Reed College in Portland because of limited course offerings at Caltech. He took a wide range of subjects like Russian, ancient Greek, philosophy, music, psychology. Because the U.S.S.R. had launched Sputnik in 1957 the United States poured millions into higher education and Brown received an NDEA (National Defence Education Act) Fellowship for 3 years of graduate studies. He chose the Washington University in St. Louis, Missouri because a senior math student told him his appreciation of mathematics classes there. Brown received his Ph.D in 1964 and took a 1-year appointment as an instructor at Reed College. At Reed, he learned of an exchange program between the United States and the USSR which involved only about 25 US students and 25 Soviet students. Brown signed up and spent the next academic year at Kiev State University.

Tom Brown learned how to juggle in the early 70s when he saw Ron Graham juggling at a math meeting. He received occasional lessons from him afterwards, learning to juggle 3 balls at first and then 5 later. Brown mentioned the difficulty of learning how to juggle is as hard as learning to ride a bicycle, but you never lose this ability. He recalled that he was once standing with Ron side-by-side, with his right hand and Ron’s left hand, acted like a single person juggling 5 balls. When asked about his interest in Mathematics, he responded: “Currently, I think there are three main reasons I like math. One, it’s beautiful. Two, it “exists” independent of the physical world and independent of humanity – at least, that’s what many or most mathematicians like to believe. Three, it’s thrilling to discover something “new” in this invisible world.”

While Brown was in Kiev, he learned from a Canadian student the existence of Simon Fraser University which had opened recently in 1965. After his exchange program ended, he applied for an assistant professor and spent 1966–2003 at Simon Fraser University. Brown took many leaves of absence from Simon Fraser University to teach at the Bosphorus University in Istanbul, Cairo, Nairobi, and various other places in Europe and Turkey.

As a mathematician, Brown’s primary focus in his research is in the field of Ramsey Theory. When completing his Ph. D, his thesis was 'On Semigroups which are Unions of Periodic Groups'[2] Today, much of his research and interest is in van der Waerden’s theorem and arithmetic progression. In 1963 as a graduate student, he showed that if the positive integers are finitely colored, then some color class is piece-wise syndetic.[3] In semi-group literature, this result is now known as “Brown’s Lemma.”

One of Brown’s most cited papers is A Density Version of a Geometric Ramsey Theorem.[4] In it he and Joe P. Buhler show that “for every ${\displaystyle \varepsilon >0}$ there is an ${\displaystyle n(\varepsilon )}$ such that if ${\displaystyle n=dim(V)\geq n(\varepsilon )}$ then any subset of ${\displaystyle V}$ with more than ${\displaystyle \varepsilon |V|}$ elements must contain 3 collinear points” where ${\displaystyle V}$ is an ${\displaystyle n}$-dimensional affine space over the field with ${\displaystyle p^{d}}$ elements, and ${\displaystyle p\neq 2}$".

Another commonly cited piece of research by Brown is Descriptions of the characteristic sequence of an irrational,[5] which has been cited approximately one hundred times. In this paper, Brown discusses the following idea: Let ${\displaystyle \alpha }$ be a positive irrational real number. The characteristic sequence of ${\displaystyle \alpha }$ is ${\displaystyle f(\alpha )=f_{1}f_{2}\ldots }$; where ${\displaystyle f_{n}=[(n+1)\alpha ][\alpha ]}$.” From here he discusses “the various descriptions of the characteristic sequence of α which have appeared in the literature” and refines this description to “obtain a very simple derivation of an arithmetic expression for ${\displaystyle [n\alpha ]}$.” He then gives some conclusions regarding the conditions for ${\displaystyle [n\alpha ]}$ which are equivalent to ${\displaystyle f_{n}=1}$.

One of Brown’s most notable collaborations was with notable Ramsey Theorist, Paul Erdős. He has noted that “talking with Erdӧs, or just overhearing him talking with others, was always exciting and even exhilarating. At any big math meeting, if he was 2 talking to one or two people, there would be 6 or 8 people trying to listen in, or trying to ask him a question. His memory was phenomenal.” He has collaborated on various papers with Erdős, including Quasi-Progressions and Descending Waves[6] and Quantitative Forms of a Theorem of Hilbert.[7]

• Archive of papers published by Tom Brown
• 2003 INTEGERS conference dedicated to Tom Brown