Jump to content

László Pyber

From Wikipedia, the free encyclopedia

László Pyber (born 8 May 1960 in Budapest) is a Hungarian mathematician. He is a researcher at the Alfréd Rényi Institute of Mathematics, Budapest. He works in combinatorics and group theory.

Biography

[edit]

Pyber received his Ph.D. from the Hungarian Academy of Sciences in 1989 under the direction of László Lovász and Gyula O.H. Katona with the thesis Extremal Structures and Covering Problems.[1]

In 2007, he was awarded the Academics Prize by the Hungarian Academy of Sciences.[2]

In 2017, he was the recipient of an ERC Advanced Grant.[3]

Mathematical contributions

[edit]

Pyber has solved a number of conjectures in graph theory. In 1985, he proved the conjecture of Paul Erdős and Tibor Gallai that edges of a simple graph with n vertices can be covered with at most n-1 circuits and edges.[4] In 1986, he proved the conjecture of Paul Erdős that a graph with n vertices and its complement can be covered with n2/4+2 cliques.[5]

He has also contributed to the study of permutation groups. In 1993, he provided an upper bound for the order of a 2-transitive group of degree n not containing An avoiding the use of the classification of finite simple groups.[6] Together with Tomasz Łuczak, Pyber proved the conjecture of McKay that for every ε>0, there is a constant C such that C randomly chosen elements invariably generate the symmetric group Sn with probability greater than 1-ε.[7]

Pyber has made fundamental contributions in enumerating finite groups of a given order n. In 1993, he proved[8] that if the prime power decomposition of n is n=p1g1pkgk and μ=max(g1,...,gk), then the number of groups of order n is at most

In 2004, Pyber settled several questions in subgroup growth by completing the investigation of the spectrum of possible subgroup growth types.[9]


In 2011, Pyber and Andrei Jaikin-Zapirain obtained a surprisingly explicit formula for the number of random elements needed to generate a finite d-generator group with high probability.[10] They also explored related questions for profinite groups and settled several open problems.

In 2016, Pyber and Endre Szabó proved that in a finite simple group L of Lie type, a generating set A of L either grows, i.e., |A3||A|1+ε for some ε depending only on the Lie rank of L, or A3=L.[11] This implies that diameters of Cayley graphs of finite simple groups of bounded rank are polylogarithmic in the size of the group, partially resolving a well-known conjecture of László Babai.

References

[edit]
  1. ^ "László Pyber – The Mathematics Genealogy Project".
  2. ^ "Akadémiai Díj". February 2016.
  3. ^ "Growth in Groups and Graph Isomorphism Now".
  4. ^ Pyber, László (1985). "An Erdös-Gallai conjecture". Combinatorica. 5: 67–79. doi:10.1007/BF02579444. S2CID 30972963.
  5. ^ Pyber, László (1986). "Clique convering of graphs". Combinatorica. 6 (4): 393–398. doi:10.1007/BF02579265. S2CID 40109732.
  6. ^ Pyber, László (1993). "On the orders of doubly transitive permutation groups, elementary estimates". Journal of Combinatorial Theory, Series A. 62 (2): 361–366. doi:10.1016/0097-3165(93)90053-B.
  7. ^ Pyber and Łuczak (1993). "On Random Generation of the Symmetric Group". Combinatorics, Probability and Computing. 2 (4): 505–512. doi:10.1017/S0963548300000869. S2CID 34255045.
  8. ^ Pyber, László (1993). "Enumerating finite groups of given order". Annals of Mathematics. 137 (1): 203–220. doi:10.2307/2946623. JSTOR 2946623.
  9. ^ Pyber, László (2004). "Groups of intermediate subgroup growth and a problem of Grothendieck". Duke Mathematical Journal. 121: 169–188. doi:10.1215/S0012-7094-04-12115-3.
  10. ^ Jaikin-Zapirain and Pyber (2011). "Random generation of finite and profinite groups and group enumeration". Annals of Mathematics. 173 (2): 769–814. doi:10.4007/annals.2011.173.2.4. hdl:10486/662154.
  11. ^ Pyber and Szabo (2014). "Growth in finite simple groups of Lie type". Journal of the American Mathematical Society. 29: 95–146. arXiv:1001.4556. doi:10.1090/S0894-0347-2014-00821-3. S2CID 51800011.
[edit]