Mladen Bestvina (born 1959) is a

Croatian-American Croatian Americans or Croat Americans ( hr, Američki Hrvati) are Americans who have full or partial Croatian ancestry. In 2012, there were 414,714 American citizens of Croat or Croatian descent living in the United States as per revised 2010 Un ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

working in the area of

geometric group theory Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such group (mathematics), groups and topology, topological and geometry, geometric pro ...

. He is a Distinguished Professor in the Department of Mathematics at the

University of Utah The University of Utah (U of U, UofU, or simply The U) is a public research university in Salt Lake City, Utah. It is the flagship institution of the Utah System of Higher Education. The university was established in 1850 as the University of De ...

Biographical info

Mladen Bestvina is a three-time medalist at the

International Mathematical Olympiad The International Mathematical Olympiad (IMO) is a mathematical olympiad for pre-university students, and is the oldest of the International Science Olympiads. The first IMO was held in Romania in 1959. It has since been held annually, except i ...

(two silver medals in 1976 and 1978 and a bronze medal in 1977). He received a B. Sc. in 1982 from the

University of Zagreb The University of Zagreb ( hr, Sveučilište u Zagrebu, ; la, Universitas Studiorum Zagrabiensis) is the largest Croatian university and the oldest continuously operating university in the area covering Central Europe south of Vienna and all of ...

. He obtained a PhD in Mathematics in 1984 at the

University of Tennessee The University of Tennessee (officially The University of Tennessee, Knoxville; or UT Knoxville; UTK; or UT) is a public land-grant research university in Knoxville, Tennessee. Founded in 1794, two years before Tennessee became the 16th state, ...

under the direction of John Walsh. He was a visiting scholar at the

Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...

in 1987-88 and again in 1990–91. Bestvina had been a faculty member at

UCLA The University of California, Los Angeles (UCLA) is a public land-grant research university in Los Angeles, California. UCLA's academic roots were established in 1881 as a teachers college then known as the southern branch of the California St ...

, and joined the faculty in the Department of Mathematics at the

in 1993.Mladen Bestvina: Distinguished Professor
''Aftermath'', vol. 8, no. 4, April 2008. Department of Mathematics,

. He was appointed a Distinguished Professor at the

in 2008. Bestvina received the Alfred P. Sloan Fellowship in 1988–89 and a

Presidential Young Investigator Award The Presidential Young Investigator Award (PYI) was awarded by the National Science Foundation of the United States Federal Government. The program operated from 1984 to 1991, and was replaced by the NSF Young Investigator (NYI) Awards and Presiden ...

in 1988–91. Bestvina gave an

invited address at the International Congress of Mathematicians This is a list of International Congresses of Mathematicians Plenary and Invited Speakers. Being invited to talk at an International Congress of Mathematicians has been called "the equivalent, in this community, of an induction to a hall of fame." ...

Beijing } Beijing ( ; ; ), alternatively romanized as Peking ( ), is the capital of the People's Republic of China. It is the center of power and development of the country. Beijing is the world's most populous national capital city, with over 21 ...

in 2002. He also gave a Unni Namboodiri Lecture in Geometry and Topology at the

University of Chicago The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park neighborhood. The University of Chicago is consistently ranked among the b ...

. Bestvina served as an Editorial Board member for the

Transactions of the American Mathematical Society The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 p ...

and as an associate editor of the

Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the ...

. Currently he is an editorial board member for

Duke Mathematical Journal ''Duke Mathematical Journal'' is a peer-reviewed mathematics journal published by Duke University Press. It was established in 1935. The founding editors-in-chief were David Widder, Arthur Coble, and Joseph Miller Thomas Joseph Miller Thomas (16 ...

Geometric and Functional Analysis ''Geometric and Functional Analysis'' (''GAFA'') is a mathematical journal published by Birkhäuser, an independent division of Springer-Verlag. The journal is published approximately bi-monthly. The journal publishes papers on broad range of top ...

Geometry and Topology In mathematics, geometry and topology is an umbrella term for the historically distinct disciplines of geometry and topology, as general frameworks allow both disciplines to be manipulated uniformly, most visibly in local to global theorems in Ri ...

, the ''Journal of Topology and Analysis'', ''Groups, Geometry and Dynamics'', ''Michigan Mathematical Journal'', ''Rocky Mountain Journal of Mathematics'', and ''Glasnik Matematicki''. In 2012 he became a fellow of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

Mathematical contributions

A 1988 monograph of Bestvina 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

, 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".' In a 1992 paper Bestvina and Feighn obtained a ''Combination Theorem'' for

word-hyperbolic group In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a ''word hyperbolic group'' or ''Gromov hyperbolic group'', is a finitely generated group equipped with a word metric satisfying certain properties abstra ...

s. The theorem provides a set of sufficient conditions for amalgamated free products and

HNN extension In mathematics, the HNN extension is an important construction of combinatorial group theory. Introduced in a 1949 paper ''Embedding Theorems for Groups'' by Graham Higman, Bernhard Neumann, and Hanna Neumann, it embeds a given group ''G'' into an ...

s of word-hyperbolic groups to again be word-hyperbolic. The Bestvina–Feighn Combination Theorem became a standard tool in

and has had many applications and generalizations (e.g.). Bestvina and Feighn also gave the first published treatment of Rips' theory of stable group actions on R-trees (the

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 ...

) In particular their paper gives a proof of the Morgan–Shalen conjecture that a

finitely generated group In algebra, a finitely generated group is a group ''G'' that has some finite generating set ''S'' so that every element of ''G'' can be written as the combination (under the group operation) of finitely many elements of ''S'' and of inverses of s ...

''G'' admits a free isometric

action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...

on an R-tree if and only if ''G'' is a

free product In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, and i ...

of surface groups,

free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...

s and

free abelian group In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...

s. A 1992 paper of Bestvina and

Handel George Frideric (or Frederick) Handel (; baptised , ; 23 February 1685 – 14 April 1759) was a German-British Baroque composer well known for his operas, oratorios, anthems, concerti grossi, and organ concertos. Handel received his training i ...

introduced the notion of a

train track map In the mathematical subject of geometric group theory, a train track map is a continuous map ''f'' from a finite connected graph to itself which is a homotopy equivalence and which has particularly nice cancellation properties with respect to iter ...

for representing elements of Out(''F''_''n'').Mladen Bestvina, and Michael Handel
''Train tracks and automorphisms of free groups.''

(2), vol. 135 (1992), no. 1, pp. 1–51 In the same paper they introduced the notion of a ''relative train track'' and applied train track methods to solve the ''Scott conjecture'' which says that for every automorphism ''α'' of a finitely generated

''F_n'' the fixed subgroup of ''α'' is free of

rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * H ...

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 BrinkmannP. Brinkmann
''Hyperbolic automorphisms of free groups.''

, vol. 10 (2000), no. 5, pp. 1071–1089 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 GrovesMartin R. Bridson and Daniel Groves
''The quadratic isoperimetric inequality for mapping tori of free-group automorphisms.''
Memoirs of the American Mathematical Society, to appear. that for every automorphism ''α'' of ''F_n'' the mapping torus group of ''α'' satisfies a quadratic

isoperimetric inequality In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...

; a proof of algorithmic solvability of the

conjugacy problem In abstract algebra, the conjugacy problem for a group ''G'' with a given presentation is the decision problem of determining, given two words ''x'' and ''y'' in ''G'', whether or not they represent conjugate elements of ''G''. That is, the probl ...

for free-by-cyclic groups;O. Bogopolski, A. Martino, O. Maslakova, E. Ventura
''The conjugacy problem is solvable in free-by-cyclic groups.''

Bulletin of the London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical ...

, vol. 38 (2006), no. 5, pp. 787–794 and others. Bestvina, Feighn and Handel later proved that the group Out(''F_n'') satisfies the

Tits alternative In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups. Statement The theorem, proven by Tits, is stated as follows. Consequences A linear group is not am ...

,Mladen Bestvina, Mark Feighn, and Michael Handel
''The Tits alternative for Out(F_n). I. Dynamics of exponentially-growing automorphisms.''

(2), vol. 151 (2000), no. 2, pp. 517–623Mladen Bestvina, Mark Feighn, and Michael Handel.
''The Tits alternative for Out(F_n). II. A Kolchin type theorem.''

(2), vol. 161 (2005), no. 1, pp. 1–59 settling a long-standing open problem. In a 1997 paper Bestvina and Brady developed a version of

discrete Morse theory Discrete Morse theory is a combinatorial adaptation of Morse theory developed by Robin Forman. The theory has various practical applications in diverse fields of applied mathematics and computer science, such as configuration spaces, homology compu ...

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

that is not itself word-hyperbolic.Brady, Noel, ''Branched coverings of cubical complexes and subgroups of hyperbolic groups''.

Journal of the London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical S ...

(2), vol. 60 (1999), no. 2, pp. 461–480

Selected publications

*Bestvina, Mladen
''Characterizing ''k''-dimensional universal Menger compacta''.
Memoirs of the

, vol. 71 (1988), no. 380 *Bestvina, Mladen; Feighn, Mark
''Bounding the complexity of simplicial group actions on trees''.

Inventiones Mathematicae ''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors ...

, vol. 103 (1991), no. 3, pp. 449–469 *Bestvina, Mladen; Mess, Geoffrey
''The boundary of negatively curved groups''.

Journal of the American Mathematical Society The ''Journal of the American Mathematical Society'' (''JAMS''), is a quarterly peer-reviewed mathematical journal published by the American Mathematical Society. It was established in January 1988. Abstracting and indexing This journal is abstr ...

, vol. 4 (1991), no. 3, pp. 469–481 *Mladen Bestvina, and Michael Handel
''Train tracks and automorphisms of free groups.''

(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 The ''Journal of Differential Geometry'' is a peer-reviewed scientific journal of mathematics published by International Press on behalf of Lehigh University in 3 volumes of 3 issues each per year. The journal publishes an annual supplement in book ...

, Volume 35 (1992), pp. 85–101 *M. Bestvina and M. Feighn
''Stable actions of groups on real trees.''

, vol. 121 (1995), no. 2, pp. 287 321 *Bestvina, Mladen and Brady, Noel
''Morse theory and finiteness properties of groups''.

, vol. 129 (1997), no. 3, pp. 445–470 *Mladen Bestvina, Mark Feighn, and Michael Handel
''The Tits alternative for Out(F_n). I. Dynamics of exponentially-growing automorphisms.''

(2), vol. 151 (2000), no. 2, pp. 517–623 *Mladen Bestvina, Mark Feighn, and Michael Handel.
''The Tits alternative for Out(F_n). II. A Kolchin type theorem.''

(2), vol. 161 (2005), no. 1, pp. 1–59 *Bestvina, Mladen; Bux, Kai-Uwe; Margalit, Dan
''The dimension of the Torelli group''.

, vol. 23 (2010), no. 1, pp. 61–105

References

External links

Mladen Bestvina, personal webpage
Department of Mathematics,

{{DEFAULTSORT:Bestvina, Mladen Living people 1959 births Group theorists Topologists 20th-century American mathematicians 21st-century American mathematicians University of Utah faculty Yugoslav emigrants to the United States 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

Biographical info

Mathematical contributions

Selected publications

See also

References

External links