Tom Brown (mathematician)

Tom Brown
Born	Thomas Craig Brown 1938 (age 85–86) Portland, Oregon, U.S.
Alma mater	Reed College (BS) Washington University in St. Louis (Ph.D.)
Known for	Brown's Lemma
Scientific career
Fields	Mathematics Ramsey Theory
Institutions	Simon Fraser University
Thesis	On Semigroups which are Unions of Periodic Groups  (1964)
Doctoral advisor	Earl Edwin Lazerson

Thomas Craig Brown (born 1938) is an American-Canadian mathematician, Ramsey Theorist, and Professor Emeritus at Simon Fraser University.^[1]

Collaborations

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] 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 A Density Version of a Geometric Ramsey Theorem,^[4] he and Joe P. Buhler showed 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 ${\displaystyle p\neq 2}$".

In Descriptions of the characteristic sequence of an irrational,^[5] 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 ${\displaystyle f_{n}=1}$.

He has collaborated with Paul Erdős, including Quasi-Progressions and Descending Waves^[6] and Quantitative Forms of a Theorem of Hilbert.^[7]

References

1. "Tom Brown Professor Emeritus at SFU". Retrieved 10 November 2020.
2. Jensen, Gary R.; Krantz, Steven G. (2006). 150 Years of Mathematics at Washington University in St. Louis. American Mathematical Society. p. 15. ISBN 978-0-8218-3603-3.
3. Brown, T. C. (1971). "An interesting combinatorial method in the theory of locally finite semigroups" (PDF). Pacific Journal of Mathematics. 36 (2): 285–289. doi:10.2140/pjm.1971.36.285.
4. Brown, T. C.; Buhler, J. P. (1982). "A Density version of a Geometric Ramsey Theorem" (PDF). Journal of Combinatorial Theory. Series A. 32: 20–34. doi:10.1016/0097-3165(82)90062-0.
5. Brown, T. C. (1993). "Descriptions of the Characteristic Sequence of an Irrational" (PDF). Canadian Mathematical Bulletin. 36: 15–21. doi:10.4153/CMB-1993-003-6.
6. Brown, T. C.; Erdős, P.; Freedman, A. R. (1990). "Quasi-Progressions and Descending Waves". Journal of Combinatorial Theory. Series A. 53: 81–95. doi:10.1016/0097-3165(90)90021-N.
7. Brown, T. C.; Chung, F. R. K.; Erdős, P. (1985). "Quantitative Forms of a Theorem of Hilbert" (PDF). Journal of Combinatorial Theory. Series A. 38 (2): 210–216. doi:10.1016/0097-3165(85)90071-8.

External links

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