“Jon Folkman”的意思、由来-开放百科全书

Jon Hal Folkman (December 8, 1938 – January 23, 1969)^[2] was an American mathematician, a student of John Milnor, and a researcher at the RAND Corporation.

Schooling

Folkman was a Putnam Fellow in 1960.^[3] He received his Ph.D. in 1964 from Princeton University, under the supervision of Milnor, with a thesis entitled Equivariant Maps of Spheres into the Classical Groups.^[4]

Research

In geometric combinatorics, Folkman is known for his pioneering and posthumously-published studies of oriented matroids; in particular, the Folkman–Lawrence topological representation theorem^[5] is "one of the cornerstones of the theory of oriented matroids".^[6]^[7] In lattice theory, Folkman solved an open problem on the foundations of combinatorics by proving a conjecture of Gian–Carlo Rota; in proving Rota's conjecture, Folkman characterized the structure of the homology groups of "geometric lattices" in terms of the free Abelian groups of finite rank.^[8] In graph theory, he was the first to study semi-symmetric graphs, and he discovered the semi-symmetric graph with the fewest possible vertices, now known as the Folkman graph.^[9] He proved the existence, for every positive h, of a finite K_h + 1-free graph which has a monocolored K_h in every 2-coloring of the edges, settling a problem previously posed by Paul Erdős and András Hajnal.^[10] He further proved that if G is a finite graph such that every set S of vertices contains an independent set of size (|S| − k)/2 then the chromatic number of G is at most k + 2.^[11]

In convex geometry, Folkman worked with his RAND colleague Lloyd Shapley to prove the Shapley–Folkman lemma and theorem: Their results suggest that sums of sets are approximately convex; in mathematical economics their results are used to explain why economies with many agents have approximate equilibria, despite individual nonconvexities.^[12]

In additive combinatorics, Folkman's theorem states that for each assignment of finitely many colors to the positive integers, there exist arbitrarily large sets of integers all of whose nonempty sums have the same color; the name was chosen as a memorial to Folkman by his friends.^[13] In Ramsey theory, the Rado–Folkman–Sanders theorem describes "partition regular" sets.

The Folkman Number F(p, q; r)

Brain cancer and despair

In the late 1960s, Folkman suffered from brain cancer; while hospitalized, Folkman was visited repeatedly by Ronald Graham and Paul Erdős. After his brain surgery, Folkman was despairing that he had lost his mathematical skills. As soon as Folkman received Graham and Erdős at the hospital, Erdős challenged Folkman with mathematical problems, helping to rebuild his confidence.

Folkman later purchased a gun and killed himself. Folkman's supervisor at RAND, Delbert Ray Fulkerson, blamed himself for failing to notice suicidal behaviors in Folkman. Years later Fulkerson also killed himself.^[14]

References

1. ^Jon Hal Folkman at FamilySearch
2. ^Birth and death dates from {{citation|first1=R. L.|last1=Graham|author1-link=R. L. Graham|first2=B. L.|last2=Rothschild|author2-link=Bruce Lee Rothschild|title=Ramsey's theorem for n-parameter sets|journal=Transactions of the American Mathematical Society|volume=159|pages=257–292|url=http://www.math.ucsd.edu/~sbutler/ron/71_04_n_ramsey.pdf|doi=10.2307/1996010|jstor=1996010|year=1971}}{{dead link|date=November 2017 |bot=InternetArchiveBot |fix-attempted=yes }}, and from {{citation|first=Joel|last=Spencer|authorlink=Joel Spencer|title=Optimal ranking of tournaments|year=1971|issue=2|journal=Networks|volume=1|pages=135–138|doi=10.1002/net.3230010204}}, both of which were dedicated to the memory of Folkman.
3. ^Putnam competition results, Mathematical Association of America, retrieved 2010-10-17.
4. ^{{mathgenealogy|name=John Hal Folkman|id=22443}}.
5. ^{{citation | last1 = Folkman | first1 = J. | last2 = Lawrence | first2 = J. | doi = 10.1016/0095-8956(78)90039-4 | issue = 2 | journal = Journal of Combinatorial Theory, Series B | pages = 199–236 | title = Oriented matroids | volume = 25 | year = 1978}}.
6. ^Page 17: {{cite book|last=Björner|first=Anders|last2=Las Vergnas|first2=Michel | author2-link=Michel Las Vergnas | last3=Sturmfels|first3=Bernd|authorlink3=bernd Sturmfels|last4=White|first4=Neil|last5=Ziegler|first5=Günter|authorlink5=Günter M. Ziegler|title=Oriented Matroids|publisher=Cambridge University Press|year=1999|isbn=978-0-521-77750-6}}
7. ^The Folkman-Lawrence representation theorem is called the "Lawrence representation theorem" by Günter M. Ziegler in remark 7.23 on page 211: {{cite book|authorlink=Günter M. Ziegler|last=Ziegler|first=Günter M.|title=Lectures on Polytopes|series=Graduate texts in mathematics|volume=152|publisher=Springer-Verlag|year=1995|location=New York|isbn=0-387-94365-X |id=(paper), }}
8. ^* {{cite book|last=Kung|first=Joseph P. S. (ed.)|chapter=III Enumeration in geometric lattices, 2. Homology|title=A Source book in matroid theory|publisher=Birkhäuser Boston, Inc.|location=Boston, MA|year=1986|isbn=0-8176-3173-9|pages=201–202|mr=890330}}** {{cite news|last=Folkman|first=Jon |title=The homology groups of a lattice|journal=Journal of Mathematics and Mechanics|volume=15|year=1966|pages=631–636|mr=188116}}** {{cite book|last1=Folkman|first1=Jon|chapter=The homology groups of a lattice|pages=243–248|last2=Kung|first2=Joseph P. S. (ed.)|title=A Source book in matroid theory|publisher=Birkhäuser Boston, Inc.|location=Boston, MA|year=1986|isbn=0-8176-3173-9|mr=188116}}**{{cite news|last=Rota|first=Gian-Carlo|authorlink=Gian-Carlo Rota|title=On the foundations of combinatorial theory, I: Theory of Möbius functions|journal=Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete (Probability theory and related fields)|volume=2|year=1964|pages=340–368|doi=10.1007/BF00531932|mr=174487 }}**{{cite book|last1=Rota|first1=Gian-Carlo|authorlink1=Gian-Carlo Rota|chapter=On the foundations of combinatorial theory, I: Theory of Möbius functions|pages=213–242|doi=10.1007/BF00531932|last2=Kung |first2=Joseph P. S. (ed.)|title=A Source book in matroid theory|publisher=Birkhäuser Boston, Inc.|location=Boston, MA|year=1986|isbn=0-8176-3173-9|mr=174487}}
9. ^{{citation | first = J. | last = Folkman | title = Regular line-symmetric graphs | journal = Journal of Combinatorial Theory | volume =3 | year = 1967 | pages = 215–232 | doi = 10.1016/S0021-9800(67)80069-3}}.
10. ^{{citation|first=J.|last=Folkman|title= Graphs with monochromatic complete subgraphs in every edge coloring|journal=SIAM Journal on Applied Mathematics|volume=18|year=1970|pages=19–24|mr=0268080|doi=10.1137/0118004}}.
11. ^J.Folkman: An upper bound on the chromatic number of a graph, in: Combinatorial theory and its application, II (Proc. Colloq., Balatonfüred, 1969), North-Holland, Amsterdam, 1970, 437–457.
12. ^{{citation | last = Starr | first = Ross M. | authorlink = Ross Starr | issue = 1 | journal = Econometrica | pages = 25–38 | title = Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) | volume = 37 | year = 1969 | jstor=1909201 | doi=10.2307/1909201| citeseerx = 10.1.1.297.8498}}.
13. ^Page 81 in {{Citation |authorlink=Ronald Graham |first=R. |last=Graham |first2=B. |last2=Rothschild |authorlink3=Joel Spencer |first3=J. H. |last3=Spencer |title=Ramsey Theory |publisher=John Wiley and Sons |location=New York |year=1990 |isbn=0-471-50046-1 |edition=2nd }}.
14. ^¹{{citation|title=The man who loved only numbers: the story of Paul Erdős and the search for mathematical truth| first=Paul|last=Hoffman|authorlink=Paul Hoffman (science writer)|publisher=Hyperion|year=1998 |isbn=978-0-7868-6362-4|pages=109–110}}.