Mladen Bestvina
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.
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).^{[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]} and as an associate editor of the Annals of Mathematics.^{[13]} Currently he is an Editorial Board member for Duke Mathematical Journal, ^{[14]}Geometric and Functional Analysis,^{[15]} the Journal of Topology and Analysis,^{[16]} Groups, Geometry and Dynamics,^{[17]} Michigan Mathematical Journal,^{[18]} Rocky Mountain Journal of Mathematics,^{[19]} and Glasnik Matematicki.^{[20]}
In 2012 he became a fellow of the American Mathematical Society.^{[21]}
Mathematical contributions
A 1988 monograph of Bestvina^{[22]} 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".'^{[23]}
In a 1992 paper Bestvina and Feighn obtained a Combination Theorem for word-hyperbolic groups.^{[24]} 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.^{[25]}^{[26]}^{[27]}^{[28]}).
Bestvina and Feighn also gave the first published treatment of Rips' theory of stable group actions on R-trees (the Rips machine)^{[29]} In particular their paper gives a proof of the Morgan–Shalen conjecture^{[30]} 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(F_{n}).^{[31]} In the same paper they introduced the notion of a relative train track and applied train track methods to solve^{[31]} the Scott conjecture which says that for every automorphism α of a finitely generated free group F_{n} 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(F_{n}). Examples of applications of train tracks include: a theorem of Brinkmann^{[32]} proving that for an automorphism α of F_{n} 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^{[33]} that for every automorphism α of F_{n} 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;^{[34]} and others.
Bestvina, Feighn and Handel later proved that the group Out(F_{n}) satisfies the Tits alternative,^{[35]}^{[36]} settling a long-standing open problem.
In a 1997 paper^{[37]} 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.^{[38]}
Selected publications
- Bestvina, Mladen, Characterizing k-dimensional universal Menger compacta. Memoirs of the American Mathematical Society, vol. 71 (1988), no. 380
- Bestvina, Mladen; Feighn, Mark, Bounding the complexity of simplicial group actions on trees. Inventiones Mathematicae, vol. 103 (1991), no. 3, pp. 449–469
- Bestvina, Mladen; Mess, Geoffrey, The boundary of negatively curved groups. Journal of the American Mathematical Society, vol. 4 (1991), no. 3, pp. 469–481
- Mladen Bestvina, and Michael Handel, Train tracks and automorphisms of free groups. Annals of Mathematics (2), vol. 135 (1992), no. 1, pp. 1–51
- M. Bestvina and M. Feighn, A combination theorem for negatively curved groups. Journal of Differential Geometry, Volume 35 (1992), pp. 85–101
- M. Bestvina and M. Feighn. Stable actions of groups on real trees. Inventiones Mathematicae, vol. 121 (1995), no. 2, pp. 287 321
- Bestvina, Mladen and Brady, Noel, Morse theory and finiteness properties of groups. Inventiones Mathematicae, vol. 129 (1997), no. 3, pp. 445–470
- 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
- 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
- Bestvina, Mladen; Bux, Kai-Uwe; Margalit, Dan, The dimension of the Torelli group. Journal of the American Mathematical Society, vol. 23 (2010), no. 1, pp. 61–105
See also
- Real tree
- Artin group
- Out(F_{n})
- Train track map
- Pseudo-Anosov map
- Word-hyperbolic group
- Mapping class group
- Whitehead conjecture
References
- ^ "Mladen Bestvina". info.hazu.hr (in Croatian). Croatian Academy of Sciences and Arts. Retrieved 2013-03-29.
- ^ "Mladen Bestvina". imo-official.org. International Mathematical Olympiad. Retrieved 2010-02-10.
- ^ Research brochure: Mladen Bestvina, Department of Mathematics, University of Utah. Accessed February 8, 2010
- ^ Mladen F. Bestvina, Mathematics Genealogy Project. Accessed February 8, 2010.
- ^ Institute for Advanced Study: A Community of Scholars
- ^ ^{a} ^{b} Mladen Bestvina: Distinguished Professor, Aftermath, vol. 8, no. 4, April 2008. Department of Mathematics, University of Utah.
- ^ Sloan Fellows. Department of Mathematics, University of Utah. Accessed February 8, 2010
- ^ Sloan Research Fellowships, Alfred P. Sloan Foundation. Accessed February 8, 2010
- ^ Award Abstract #8857452. Mathematical Sciences: Presidential Young Investigator. National Science Foundation. Accessed February 8, 2010
- ^ Invited Speakers for ICM2002. Notices of the American Mathematical Society, vol. 48, no. 11, December 2001; pp. 1343 1345
- ^ Annual Lecture Series. Department of Mathematics, University of Chicago. Accessed February 9, 2010
- ^ Officers and Committee Members, Notices of the American Mathematical Society, vol. 54, no. 9, October 2007, pp. 1178 1187
- ^ Editorial Board, Annals of Mathematics. Accessed February 8, 2010
- ^ Duke Mathematical Journal
- ^ Editorial Board, Geometric and Functional Analysis. Accessed February 8, 2010
- ^ Editorial Board. Journal of Topology and Analysis. Accessed February 8, 2010
- ^ Editorial Board, Groups, Geometry and Dynamics. Accessed February 8, 2010
- ^ Editorial Board, Michigan Mathematical Journal. Accessed February 8, 2010
- ^ Editorial Board, ROCKY MOUNTAIN JOURNAL OF MATHEMATICS. Accessed February 8, 2010
- ^ Editorial Board, Glasnik Matematicki. Accessed February 8, 2010
- ^ List of Fellows of the American Mathematical Society, retrieved 2012-11-10.
- ^ Bestvina, Mladen, Characterizing k-dimensional universal Menger compacta. Memoirs of the American Mathematical Society, vol. 71 (1988), no. 380
- ^ John J. Walsh, Review of: Bestvina, Mladen, Characterizing k-dimensional universal Menger compacta. Mathematical Reviews, MR0920964 (89g:54083), 1989
- ^ M. Bestvina and M. Feighn, A combination theorem for negatively curved groups. Journal of Differential Geometry, Volume 35 (1992), pp. 85–101
- ^ EMINA ALIBEGOVIC, A COMBINATION THEOREM FOR RELATIVELY HYPERBOLIC GROUPS. Bulletin of the London Mathematical Society vol. 37 (2005), pp. 459–466
- ^ Francois Dahmani, Combination of convergence groups. Geometry and Topology, Volume 7 (2003), 933–963
- ^ I. Kapovich, The combination theorem and quasiconvexity. International Journal of Algebra and Computation, Volume: 11 (2001), no. 2, pp. 185–216
- ^ M. Mitra, Cannon–Thurston maps for trees of hyperbolic metric spaces. Journal of Differential Geometry, Volume 48 (1998), Number 1, 135–164
- ^ M. Bestvina and M. Feighn. Stable actions of groups on real trees. Inventiones Mathematicae, vol. 121 (1995), no. 2, pp. 287 321
- ^ Morgan, John W., Shalen, Peter B., Free actions of surface groups on R-trees. Topology, vol. 30 (1991), no. 2, pp. 143–154
- ^ ^{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
- ^ P. Brinkmann, Hyperbolic automorphisms of free groups. Geometric and Functional Analysis, vol. 10 (2000), no. 5, pp. 1071–1089
- ^ 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.
- ^ 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
- ^ 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
- ^ 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
- ^ Bestvina, Mladen and Brady, Noel, Morse theory and finiteness properties of groups. Inventiones Mathematicae, vol. 129 (1997), no. 3, pp. 445–470
- ^ 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
- Mladen Bestvina, personal webpage, Department of Mathematics, University of Utah
- Living people
- 1959 births
- Group theorists
- Topologists
- 20th-century American mathematicians
- 21st-century mathematicians
- University of Utah faculty
- American people of Croatian descent
- Croatian mathematicians
- University of Tennessee alumni
- Institute for Advanced Study visiting scholars
- Faculty of Science, University of Zagreb alumni
- Fellows of the American Mathematical Society
- People from Osijek
- International Mathematical Olympiad participants
- Sloan Research Fellows