János Komlós (mathematician)

János Komlós (Budapest, 23 May 1942) 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 at San Diego.^[2]

Notable results

• 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 Ajtai and Szemerédi he proved^[3] the ct²/log t upper bound for the Ramsey number R(3,t). The corresponding lower bound was proved by Kim only in 1995, 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

$\frac12n\log n+\frac12n\log\log n+cn$

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

$e^{-e^{-2c}}.$

• 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.^[6]
• Komlós also wrote highly cited papers on sums of random variables,^[7] space-efficient representations of sparse sets,^[8] random matrices,^[9] the Szemerédi regularity lemma,^[10] and derandomization.^[11]

Degrees, awards

Komlós received his Ph.D. in 1967 from Eötvös Loránd University under the supervision of Alfréd Rényi.^[12] 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.^[13]

See also

• Komlós–Major–Tusnády approximation

References

1. Rutgers faculty profile for Komlós.
2. 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. Ajtai, Miklós; Komlós, János; Szemerédi, Endre (1983), "An O(n log n) sorting network", Proc. 15th ACM Symposium on Theory of Computing, pp. 1–9, doi:10.1145/800061.808726; Ajtai, Miklós; Komlós, János; Szemerédi, Endre (1983), "Sorting in c log n parallel steps", Combinatorica 3 (1): 1–19, doi:10.1007/BF02579338.
5. J. Komlós, G. Sárközy, Szemerédi: Blow-Up Lemma, Combinatorica, 17(1997), 109–123.
6. Komlós, J.; Pintz, J.; Szemerédi, E. (1982), "A lower bound for Heilbronn's problem", Journal of the London Mathematical Society 25 (1): 13–24, doi:10.1112/jlms/s2-25.1.13
7. Komlós, J.; Major, P.; Tusnády, G. (1975), "An approximation of partial sums of independent RV'-s, and the sample DF. I", Probability Theory and Related Fields 32 (1–2): 111–131, doi:10.1007/BF00533093.
8. Fredman, Michael L.; Komlós, János; Szemerédi, Endre (1984), "Storing a Sparse Table with O(1) Worst Case Access Time", Journal of the ACM 31 (3): 538, doi:10.1145/828.1884. A preliminary version appeared in 23rd Symposium on Foundations of Computer Science, 1982, doi:10.1109/SFCS.1982.39.
9. Füredi, Zoltán; Komlós, János (1981), "The eigenvalues of random symmetric matrices", Combinatorica 1 (3): 233–241, doi:10.1007/BF02579329.
10. Komlós, János; Simonovits, Miklós (1996), Szemeredi's Regularity Lemma and its applications in graph theory, Technical Report: 96-10, DIMACS.
11. Ajtai, Miklós; Komlós, János; Szemerédi, Endre (1987), "Deterministic simulation in LOGSPACE", Proc. 19th ACM Symposium on Theory of Computing, pp. 132–140, doi:10.1145/28395.28410.
12. János Komlós at the Mathematics Genealogy Project.
13. Rutgers Mathematics Department – Recent Faculty Honors.