George J. Minty Explained

George James Minty Jr. (September 16, 1929, Detroit – August 6, 1986,^[1] Bloomington, Indiana) was an American mathematician, specializing in mathematical analysis and discrete mathematics. He is known for the Klee–Minty cube, the Browder–Minty theorem, the introduction of oriented regular matroids, and the Minty-Vitaver theorem on graph coloring.

Biography

George Minty Jr. grew up in Detroit. His father emigrated, after WW I, from Scotland to work in Detroit's automotive industry as a tool and die maker. George Minty Jr. received his bachelor's degree from Detroit's Wayne State University. After service in the US Army Signal Corps at Fort Monmouth,^[2] he became in 1956 a graduate student in mathematics at the University of Michigan.^[3] There he received in 1959 his PhD with thesis Integrability Conditions for Vector Fields in Banach Spaces supervised by Erich Rothe. As a postdoc in 1959 he was briefly a visiting researcher at Tokyo's Waseda University.^[4]

Minty joined the University of Michigan faculty and was eventually promoted to associate professor before he resigned in 1965.^[3] For the academic year 1964–1965 he was on leave to do research at the Courant Institute.^[5] In 1965 he was awarded a Sloan Research Fellowship.^[6] At Indiana University he was a full professor of mathematics from 1965^[7] until his death in 1986.^[1]

Research

Minty's 1966 paper On the axiomatic foundations of the theories of directed linear graphs, electrical networks and network-programming is important in matroid theory. In that 1966 paper, according to Dominic Welsh:

According to K.-C. Chang:

In 2012 D. Erdős, A. Frank, and K. Kun published a sharpening of Minty's coloring theorem (published in 1962 in The American Mathematical Monthly).^[8]

Selected publications

• Monotone networks. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 257. 1289. 1960. 194–212. 0080-4630. 10.1098/rspa.1960.0144. 1960RSPSA.257..194M. Minty. G. J.. 202574544.
• Monotone (nonlinear) operators in Hilbert space. Duke Math. J.. 29. 3. 341–346. 1962. 10.1215/S0012-7094-62-02933-2. Minty. George J..
• 10.1090/S0002-9939-1962-0143006-9. 0143006. On the simultaneous solution of a certain system of linear inequalities. 1962. Proceedings of the American Mathematical Society. 13. 11. free. Minty. George J..
• A Theorem on n-Coloring the Points of a Linear Graph. The American Mathematical Monthly. 69. 7. 1962. 623–624. 0002-9890. 10.1080/00029890.1962.11989938. Minty. George J..
• 10.1090/S0002-9904-1963-10986-6. Two theorems on nonlinear functional equations in Hilbert space. 1963. Bulletin of the American Mathematical Society. 69. 5. 691–693. free. Minty. George J..
• On a "monotonicity" method for the solution of nonlinear equations in Banach spaces. Proceedings of the National Academy of Sciences. 50. 6. 1963. 1038–1041. 0027-8424. 10.1073/pnas.50.6.1038. free. Minty. G. J.. 16578554. 221269. 1963PNAS...50.1038M.
• A theorem on maximal monotonic sets in Hilbert space. Journal of Mathematical Analysis and Applications. 1965. 11. 434–439. 10.1016/0022-247X(65)90095-8. 2027.42/32073. Minty. George J.. free.
• On the axiomatic foundations of the theories of directed linear graphs, electrical networks and network-programming. Journal of Mathematics and Mechanics. 15. 3. 485–520. 1966. 24901346. 0188102. Minty. George J..
• A theorem on three-coloring the edges of a trivalent graph. Journal of Combinatorial Theory. 2. 2. 1967. 164–167. 10.1016/S0021-9800(67)80097-8. Minty. George J.. free.
• 10.1090/S0002-9904-1967-11732-4. 0212631. On the generalization of a direct method of the calculus of variations. 1967. Bulletin of the American Mathematical Society. 73. 3. 315–322. free. Minty. George J..
• 10.1090/S0002-9904-1970-12466-1. 0254575. On the extension of Lipschitz, Lipschitz-Hölder continuous, and monotone functions. 1970. Bulletin of the American Mathematical Society. 76. 2. 334–340. free. Minty. George J..
• Book: Inequalities III (Proceedings of the Third Symposium on Inequalities held at the University of California, Los Angeles, Calif., September 1–9, 1969, dedicated to the memory of Theodore S. Motzkin). Oved. Shisha. Academic Press. New York-London. 1972. 332165. How good is the simplex algorithm? by Victor Klee and George Minty. 159–175.
  • Abstract of How good is the simplex method?"'
• 10.1007/BF01585531. A "from scratch" proof of a theorem of Rockafellar and Fulkerson. 1974. Mathematical Programming. 7. 368–375. Minty. George J.. 1504975.
• 1974. A finite-dimensional tool-theorem in monotone operator theory. Advances in Mathematics. 12. 1. 1–7. 10.1016/S0001-8708(74)80015-0. free. Minty. George J..
• On variational inequalities for monotone operators, I. Advances in Mathematics. 30. 1. 1978. 1–7. 10.1016/0001-8708(78)90128-7. free. Minty. George J..
• On maximal independent sets of vertices in claw-free graphs. . Series B . 28. 3. 284–304. 1980. 10.1016/0095-8956(80)90074-X. free. Minty. George J..

References

1. Web site: Report of Death, George J. Minty Jr.. Minutes of the Board of Trustees of Indiana University. 6 September 1986.
2. Web site: Who's That mathematician? Paul R. Halmos Collection - Page 35. Mathematical Association of America.
3. Web site: George J. Minty. Faculty History Project, University of Michigan.
4. Appreciation to Referees. Operations Research. 7. 3. 1959. 415–419. 0030-364X. 10.1287/opre.7.3.415. free.
5. News: Leaves of Absence. The Michigan Daily. June 23, 1964. 74. 1. 10.
6. Web site: Past Fellows, Sloan Research Fellows. Alfred P. Sloan Foundation.
7. Web site: Appointments to the Faculties. Minutes of the Board of Trustees of Indiana University. 21 May 1965.
8. Erdős, Dóra. Frank, András. Kun, Krisztián. Sink-Stable Sets of Digraphs. 2012. math.CO. 1205.6071. See page 6 for a statement (Theorem 3.2) of Minty's coloring theorem.