Mladen Bestvina

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Mladen Bestvina in 1986

Mladen Bestvina (born 1959[1]) is a Croatian-American mathematician working in the area of geometric group theory. He is a Distinguished Professor in the Department of Mathematics at the University of Utah.

Biographical info[edit]

Mladen Bestvina is a three-time medalist at the International Mathematical Olympiad (two silver medals in 1976 and 1978 and a bronze medal in 1977).[2] He received a B. Sc. in 1982 from the University of Zagreb.[3] He obtained a PhD in Mathematics in 1984 at the University of Tennessee under the direction of John Walsh.[4] He was a visiting scholar at the Institute for Advanced Study in 1987-88 and again in 1990-91.[5] Bestvina had been a faculty member at UCLA, and joined the faculty in the Department of Mathematics at the University of Utah in 1993.[6] He was appointed a Distinguished Professor at the University of Utah in 2008.[6] Bestvina received the Alfred P. Sloan Fellowship in 1988–89[7][8] and a Presidential Young Investigator Award in 1988–91.[9]

Bestvina gave an Invited Address at the International Congress of Mathematicians in Beijing in 2002.[10] He also gave a Unni Namboodiri Lecture in Geometry and Topology at the University of Chicago.[11]

Bestvina served as an Editorial Board member for the Transactions of the American Mathematical Society.[12] He is currently an associate editor of the Annals of Mathematics[13] and an Editorial Board member for Geometric and Functional Analysis,[14] the Journal of Topology and Analysis,[15] Groups, Geometry and Dynamics,[16] Michigan Mathematical Journal,[17] Rocky Mountain Journal of Mathematics,[18] and Glasnik Matematicki.[19]

In 2012 he became a fellow of the American Mathematical Society.[20]

Mathematical contributions[edit]

A 1988 monograph of Bestvina[21] gave an abstract topological characterization of universal Menger compacta in all dimensions; previously only the cases of dimension 0 and 1 were well understood. John Walsh wrote in a review of Bestvina's monograph: 'This work, which formed the author's Ph.D. thesis at the University of Tennessee, represents a monumental step forward, having moved the status of the topological structure of higher-dimensional Menger compacta from one of "close to total ignorance" to one of "complete understanding".'[22]

In a 1992 paper Bestvina and Feighn obtained a Combination Theorem for word-hyperbolic groups.[23] The theorem provides a set of sufficient conditions for amalgamated free products and HNN extensions of word-hyperbolic groups to again be word-hyperbolic. The Bestvina–Feighn Combination Theorem became a standard tool in geometric group theory and has had many applications and generalizations (e.g.[24][25][26][27]).

Bestvina and Feighn also gave the first published treatment of Rips' theory of stable group actions on R-trees (the Rips machine)[28] In particular their paper gives a proof of the Morgan–Shalen conjecture[29] that a finitely generated group G admits a free isometric action on an R-tree if and only if G is a free product of surface groups, free groups and free abelian groups.

A 1992 paper of Bestvina and Handel introduced the notion of a train track map for representing elements of Out(Fn).[30] In the same paper they introduced the notion of a relative train track and applied train track methods to solve[30] the Scott conjecture which says that for every automorphism α of a finitely generated free group Fn the fixed subgroup of α is free of rank at most n. Since then train tracks became a standard tool in the study of algebraic, geometric and dynamical properties of automorphisms of free groups and of subgroups of Out(Fn). Examples of applications of train tracks include: a theorem of Brinkmann[31] proving that for an automorphism α of Fn the mapping torus group of α is word-hyperbolic if and only if α has no periodic conjugacy classes; a theorem of Bridson and Groves[32] that for every automorphism α of Fn the mapping torus group of α satisfies a quadratic isoperimetric inequality; a proof of algorithmic solvability of the conjugacy problem for free-by-cyclic groups;[33] and others.

Bestvina, Feighn and Handel later proved that the group Out(Fn) satisfies the Tits alternative,[34][35] settling a long-standing open problem.

In a 1997 paper[36] Bestvina and Brady developed a version of discrete Morse theory for cubical complexes and applied it to study homological finiteness properties of subgroups of right-angled Artin groups. In particular, they constructed an example of a group which provides a counter-example to either the Whitehead asphericity conjecture or to the Eilenberg−Ganea conjecture, thus showing that at least one of these conjectures must be false. Brady subsequently used their Morse theory technique to construct the first example of a finitely presented subgroup of a word-hyperbolic group that is not itself word-hyperbolic.[37]

Selected publications[edit]

See also[edit]

References[edit]

  1. ^ "Mladen Bestvina". info.hazu.hr (in Croatian). Croatian Academy of Sciences and Arts. Retrieved 2013-03-29. 
  2. ^ "Mladen Bestvina". imo-official.org. International Mathematical Olympiad. Retrieved 2010-02-10. 
  3. ^ Research brochure: Mladen Bestvina, Department of Mathematics, University of Utah. Accessed February 8, 2010
  4. ^ Mladen F. Bestvina, Mathematics Genealogy Project. Accessed February 8, 2010.
  5. ^ Institute for Advanced Study: A Community of Scholars
  6. ^ a b Mladen Bestvina: Distinguished Professor, Aftermath, vol. 8, no. 4, April 2008. Department of Mathematics, University of Utah.
  7. ^ Sloan Fellows. Department of Mathematics, University of Utah. Accessed February 8, 2010
  8. ^ Sloan Research Fellowships, Alfred P. Sloan Foundation. Accessed February 8, 2010
  9. ^ Award Abstract #8857452. Mathematical Sciences: Presidential Young Investigator. National Science Foundation. Accessed February 8, 2010
  10. ^ Invited Speakers for ICM2002. Notices of the American Mathematical Society, vol. 48, no. 11, December 2001; pp. 1343 1345
  11. ^ Annual Lecture Series. Department of Mathematics, University of Chicago. Accessed February 9, 2010
  12. ^ Officers and Committee Members, Notices of the American Mathematical Society, vol. 54, no. 9, October 2007, pp. 1178 1187
  13. ^ Editorial Board, Annals of Mathematics. Accessed February 8, 2010
  14. ^ Editorial Board, Geometric and Functional Analysis. Accessed February 8, 2010
  15. ^ Editorial Board. Journal of Topology and Analysis. Accessed February 8, 2010
  16. ^ Editorial Board, Groups, Geometry and Dynamics. Accessed February 8, 2010
  17. ^ Editorial Board, Michigan Mathematical Journal. Accessed February 8, 2010
  18. ^ Editorial Board, ROCKY MOUNTAIN JOURNAL OF MATHEMATICS. Accessed February 8, 2010
  19. ^ Editorial Board, Glasnik Matematicki. Accessed February 8, 2010
  20. ^ List of Fellows of the American Mathematical Society, retrieved 2012-11-10.
  21. ^ Bestvina, Mladen, Characterizing k-dimensional universal Menger compacta. Memoirs of the American Mathematical Society, vol. 71 (1988), no. 380
  22. ^ John J. Walsh, Review of: Bestvina, Mladen, Characterizing k-dimensional universal Menger compacta. Mathematical Reviews, MR0920964 (89g:54083), 1989
  23. ^ M. Bestvina and M. Feighn, A combination theorem for negatively curved groups. Journal of Differential Geometry, Volume 35 (1992), pp. 85–101
  24. ^ EMINA ALIBEGOVIC, A COMBINATION THEOREM FOR RELATIVELY HYPERBOLIC GROUPS. Bulletin of the London Mathematical Society vol. 37 (2005), pp. 459–466
  25. ^ Francois Dahmani, Combination of convergence groups. Geometry and Topology, Volume 7 (2003), 933–963
  26. ^ I. Kapovich, The combination theorem and quasiconvexity. International Journal of Algebra and Computation, Volume: 11 (2001), no. 2, pp. 185–216
  27. ^ M. Mitra, Cannon–Thurston maps for trees of hyperbolic metric spaces. Journal of Differential Geometry, Volume 48 (1998), Number 1, 135–164
  28. ^ M. Bestvina and M. Feighn. Stable actions of groups on real trees. Inventiones Mathematicae, vol. 121 (1995), no. 2, pp. 287 321
  29. ^ Morgan, John W., Shalen, Peter B., Free actions of surface groups on R-trees. Topology, vol. 30 (1991), no. 2, pp. 143–154
  30. ^ a b Mladen Bestvina, and Michael Handel, Train tracks and automorphisms of free groups. Annals of Mathematics (2), vol. 135 (1992), no. 1, pp. 1–51
  31. ^ P. Brinkmann, Hyperbolic automorphisms of free groups. Geometric and Functional Analysis, vol. 10 (2000), no. 5, pp. 1071–1089
  32. ^ Martin R. Bridson and Daniel Groves. The quadratic isoperimetric inequality for mapping tori of free-group automorphisms. Memoirs of the American Mathematical Society, to appear.
  33. ^ O. Bogopolski, A. Martino, O. Maslakova, E. Ventura, The conjugacy problem is solvable in free-by-cyclic groups. Bulletin of the London Mathematical Society, vol. 38 (2006), no. 5, pp. 787–794
  34. ^ Mladen Bestvina, Mark Feighn, and Michael Handel. The Tits alternative for Out(Fn). I. Dynamics of exponentially-growing automorphisms. Annals of Mathematics (2), vol. 151 (2000), no. 2, pp. 517–623
  35. ^ Mladen Bestvina, Mark Feighn, and Michael Handel. The Tits alternative for Out(Fn). II. A Kolchin type theorem. Annals of Mathematics (2), vol. 161 (2005), no. 1, pp. 1–59
  36. ^ Bestvina, Mladen and Brady, Noel, Morse theory and finiteness properties of groups. Inventiones Mathematicae, vol. 129 (1997), no. 3, pp. 445–470
  37. ^ Brady, Noel, Branched coverings of cubical complexes and subgroups of hyperbolic groups. Journal of the London Mathematical Society (2), vol. 60 (1999), no. 2, pp. 461–480

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