János Komlós (mathematician) explained

János Komlós (born 23 May 1942, in Budapest) is a Hungarian-American mathematician, working in probability theory and discrete mathematics. He has been a professor of mathematics at Rutgers University^[1] since 1988. He graduated from the Eötvös Loránd University, then became a fellow at the Mathematical Institute of the Hungarian Academy of Sciences. Between 1984 - 1988 he worked at the University of California, San Diego.^[2]

Notable results

• Komlós' theorem: He proved that every L¹-bounded sequence of real functions contains a subsequence such that the arithmetic means of all its subsequences converge pointwise almost everywhere. In probabilistic terminology, the theorem is as follows. Let ξ₁,ξ₂,... be a sequence of random variables such that E[ξ₁],E[ξ₂],... is bounded. Then there exist a subsequence ξ'₁, ξ'₂,... and a random variable β such that for each further subsequence η₁,η₂,... of ξ'₀, ξ'₁,... we have (η₁+...+η_n)/n → β a.s.
• With Miklós Ajtai and Endre Szemerédi he proved^[3] the ct²/log t upper bound for the Ramsey number R(3,t). The corresponding lower bound was established by Jeong Han Kim only in 1995, and this result earned him a Fulkerson Prize.
• The same team of authors developed the optimal Ajtai–Komlós–Szemerédi sorting network.^[4]
• Komlós and Szemerédi proved that if G is a random graph on n vertices with

edges, where c is a fixed real number, then the probability that G has a Hamiltonian circuit converges to

• With Gábor Sárközy and Endre Szemerédi he proved the so-called blow-up lemma which claims that the regular pairs in Szemerédi's regularity lemma are similar to complete bipartite graphs when considering the embedding of graphs with bounded degrees.^[5]
• Komlós worked on Heilbronn's problem; he, János Pintz and Szemerédi disproved Heilbronn's conjecture.
• Komlós also wrote highly cited papers on sums of random variables,^[6] space-efficient representations of sparse sets,^[7] random matrices,^[8] the Szemerédi regularity lemma,^[9] and derandomization.^[10]

Degrees, awards

Komlós received his Ph.D. in 1967 from Eötvös Loránd University under the supervision of Alfréd Rényi.^[11] In 1975, he received the Alfréd Rényi Prize, a prize established for researchers of the Alfréd Rényi Institute of Mathematics. In 1998, he was elected as an external member to the Hungarian Academy of Sciences.^[12]

See also

• Komlós–Major–Tusnády approximation

Notes and References

1. Web site: Komlos, Janos .
2. http://math.ucsd.edu/about/history/ UCSD Maths Dept history
3. M. Ajtai, J. Komlós, E. Szemerédi: A note on Ramsey numbers, J. Combin. Theory Ser. A, 29(1980), 354 - 360.

4. .

5. J. Komlós, G. Sárközy, Szemerédi: Blow-Up Lemma, Combinatorica, 17(1997), 109 - 123.
6. .
7. . A preliminary version appeared in 23rd Symposium on Foundations of Computer Science, 1982, .
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12. http://sites.math.rutgers.edu/people/faculty-honors.html Rutgers Mathematics Department – Recent Faculty Honors