Mladen Bestvina

Mladen Bestvina

Mladen Bestvina 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.

Contents

Biographical info

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).[1] He received a B. Sc. in 1982 from the University of Zagreb.[2] He obtained a PhD in Mathematics in 1984 at the University of Tennessee under the direction of John Walsh.[3] Bestvina had been a faculty member at UCLA, and joined the faculty in the Department of Mathematics at the University of Utah in 1993.[4] He was appointed a Distinguished Professor at the University of Utah in 2008.[4] Bestvina received the Alfred P. Sloan Fellowship in 1988–89[5][6] and a Presidential Young Investigator Award in 1988–91.[7]

Bestvina gave an Invited Address at the International Congress of Mathematicians in Beijing in 2002.[8] He also gave a Unni Namboodiri Lecture in Geometry and Topology at the University of Chicago.[9]

Bestvina served as an Editorial Board member for the Transactions of the American Mathematical Society.[10] He is currently an associate editor of the Annals of Mathematics[11] and an Editorial Board member for Geometric and Functional Analysis,[12] the Journal of Topology and Analysis,[13] Groups, Geometry and Dynamics,[14] Michigan Mathematical Journal,[15] Rocky Mountain Journal of Mathematics,[16] and Glasnik Matematicki.[17]

Mathematical contributions

A 1988 monograph of Bestvina[18] 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".[19]

In a 1992 paper Bestvina and Feighn obtained a Combination Theorem for word-hyperbolic groups.[20] 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.[21][22][23][24] ).

Bestvina and Feighn also gave the first published treatment of Rips' theory of stable group actions on R-trees (the Rips machine)[25] In particular their paper gives a proof of the Morgan–Shalen conjecture[26] 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).[27] In the same paper they introduced the notion of a relative train track and applied train track methods to solve[27] 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[28] 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[29] 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;[30] and others.

Bestvina, Feighn and Handel later proved that the group Out(Fn) satisfies the Tits alternative,[31][32] settling a long-standing open problem.

In a 1997 paper[33] 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.[34]

Selected publications

See also

References

  1. ^ "Mladen Bestvina". imo-official.org. International Mathematical Olympiad. http://www.imo-official.org/participant_r.aspx?id=68. Retrieved 2010-02-10. 
  2. ^ Research brochure: Mladen Bestvina, Department of Mathematics, University of Utah. Accessed February 8, 2010
  3. ^ Mladen F. Bestvina, Mathematics Genealogy Project. Accessed February 8, 2010.
  4. ^ a b Mladen Bestvina: Distinguished Professor, Aftermath, vol. 8, no. 4, April 2008. Department of Mathematics, University of Utah.
  5. ^ Sloan Fellows. Department of Mathematics, University of Utah. Accessed February 8, 2010
  6. ^ Sloan Research Fellowships, Alfred P. Sloan Foundation. Accessed February 8, 2010
  7. ^ Award Abstract #8857452. Mathematical Sciences: Presidential Young Investigator. National Science Foundation. Accessed February 8, 2010
  8. ^ Invited Speakers for ICM2002. Notices of the American Mathematical Society, vol. 48, no. 11, December 2001; pp. 1343 1345
  9. ^ Annual Lecture Series. Department of Mathematics, University of Chicago. Accessed February 9, 2010
  10. ^ Officers and Committee Members, Notices of the American Mathematical Society, vol. 54, no. 9, October 2007, pp. 1178 1187
  11. ^ Editorial Board, Annals of Mathematics. Accessed February 8, 2010
  12. ^ Editorial Board, Geometric and Functional Analysis. Accessed February 8, 2010
  13. ^ Editorial Board. Journal of Topology and Analysis. Accessed February 8, 2010
  14. ^ Editorial Board, Groups, Geometry and Dynamics. Accessed February 8, 2010
  15. ^ Editorial Board, Michigan Mathematical Journal. Accessed February 8, 2010
  16. ^ Editorial Board, ROCKY MOUNTAIN JOURNAL OF MATHEMATICS. Accessed February 8, 2010
  17. ^ Editorial Board, Glasnik Matematicki. Accessed February 8, 2010
  18. ^ Bestvina, Mladen, Characterizing k-dimensional universal Menger compacta. Memoirs of the American Mathematical Society, vol. 71 (1988), no. 380
  19. ^ John J. Walsh, Review of: Bestvina, Mladen, Characterizing k-dimensional universal Menger compacta. Mathematical Reviews, MR0920964 (89g:54083), 1989
  20. ^ M. Bestvina and M. Feighn, A combination theorem for negatively curved groups. Journal of Differential Geometry, Volume 35 (1992), pp. 85–101
  21. ^ EMINA ALIBEGOVIC, A COMBINATION THEOREM FOR RELATIVELY HYPERBOLIC GROUPS. Bulletin of the London Mathematical Society vol. 37 (2005), pp. 459–466
  22. ^ Francois Dahmani, Combination of convergence groups. Geometry and Topology, Volume 7 (2003), 933–963
  23. ^ I. Kapovich, The combination theorem and quasiconvexity. International Journal of Algebra and Computation, Volume: 11 (2001), no. 2, pp. 185–216
  24. ^ M. Mitra, Cannon–Thurston maps for trees of hyperbolic metric spaces. Journal of Differential Geometry, Volume 48 (1998), Number 1, 135–164
  25. ^ M. Bestvina and M. Feighn. Stable actions of groups on real trees. Inventiones Mathematicae, vol. 121 (1995), no. 2, pp. 287 321
  26. ^ Morgan, John W., Shalen, Peter B., Free actions of surface groups on R-trees. Topology, vol. 30 (1991), no. 2, pp. 143–154
  27. ^ 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
  28. ^ P. Brinkmann, Hyperbolic automorphisms of free groups. Geometric and Functional Analysis, vol. 10 (2000), no. 5, pp. 1071–1089
  29. ^ 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.
  30. ^ 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
  31. ^ 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
  32. ^ 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
  33. ^ Bestvina, Mladen and Brady, Noel, Morse theory and finiteness properties of groups. Inventiones Mathematicae, vol. 129 (1997), no. 3, pp. 445–470
  34. ^ 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

External links


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Bass–Serre theory — is a part of the mathematical subject of group theory that deals with analyzing the algebraic structure of groups acting by automorphisms on simplicial trees. The theory relates group actions on trees with decomposing groups as iterated… …   Wikipedia

  • Rips machine — In geometric group theory, the Rips machine is a method of studying the action of groups on R trees. It was introduced in unpublished work of Eliyahu Rips in about 1991. An R tree is a uniquely arcwise connected metric space in which every arc is …   Wikipedia

  • Artin group — In mathematics, an Artin group (or generalized braid group) is a group with a presentation of the form:Biglangle x 1,x 2,ldots,x n Big| langle x 1, x 2 angle^{m {1,2=langle x 2, x 1 angle^{m {2,1, ldots , langle x {n 1}, x n angle^{m {n… …   Wikipedia

  • Eilenberg-Ganea conjecture — The Eilenberg Ganea conjecture is a claim in algebraic topology. It was formulated by Samuel Eilenberg and Tudor Ganea in 1957, in a short, but influential paper. It states that if a group G has cohomological dimension 2, then it has a 2… …   Wikipedia

  • Whitehead conjecture — The Whitehead conjecture is a claim in algebraic topology. It was formulated by J. H. C. Whitehead in 1941. It states that every connected subcomplex of a two dimensional aspherical CW complex is aspherical.In 1997, Mladen Bestvina and Noel Brady …   Wikipedia

  • Annals of Mathematics — Infobox Journal title = Annals of Mathematics editor = Jean Bourgain, David Gabai, Nick Katz, Peter Sarnak, Yakov Sinai, Gang Tian discipline = mathematics language = English abbreviation = Ann. of Math. publisher = Princeton University and the… …   Wikipedia

  • John R. Stallings — John Robert Stallings is a mathematician known for his seminal contributions to geometric group theory and 3 manifold topology. Stallings is a Professor Emeritus in the Department of Mathematics and the University of California at Berkeley. [… …   Wikipedia

  • Tits alternative — In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups. It states that every such group is either virtually solvable (i.e. has a solvable subgroup of finite… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”