Mladen Bestvina achieved three medals at the International Mathematical Olympiad between 1976 and 1978, establishing an early trajectory in academic mathematics. Since obtaining his PhD from the University of Tennessee in 1984, he has contributed extensively to geometric group theory and topology, currently serving as a Distinguished Professor in the Department of Mathematics at the University of Utah.
Academic Progression and Honors
Born in Osijek in 1959, Bestvina completed his undergraduate education at the University of Zagreb in 1982 before moving to the United States for doctoral studies. His research career includes appointments as a visiting scholar at the Institute for Advanced Study during the late 1980s and early 1990s. Beyond his faculty role at Utah, which began in 1993, he has received the Presidential Young Investigator Award and the Alfred P. Sloan Fellowship. In 2013, he was named a Fellow of the American Mathematical Society and serves as a correspondent member of the Croatian Academy of Science and Art.
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His 1988 monograph provided an abstract topological characterization of universal Menger compacta across all dimensions. This study transformed the field's understanding of higher-dimensional Menger compacta, moving the area from a state of limited knowledge to a comprehensive framework. This work originated from his doctoral thesis conducted under the supervision of John Walsh.
Contributions to Geometric Group Theory
Bestvina has co-authored significant developments in group theory, including the 1992 Combination Theorem for word-hyperbolic groups with Mark Feighn, which established criteria for amalgamated free products and HNN extensions. Together with Michael Handel, he introduced train track maps to analyze the automorphisms of free groups, providing a solution to the Scott conjecture. Additionally, their collaborative research settled the long-standing open problem regarding the Tits alternative for the group Out(Fn).
Applications of Discrete Morse Theory
In a 1997 paper with Noel Brady, Bestvina applied discrete Morse theory to cubical complexes to investigate the homological finiteness properties of subgroups of right-angled Artin groups. Their construction produced an example that disproved either the Whitehead asphericity conjecture or the Eilenberg-Ganea conjecture, demonstrating that at least one of these two long-standing mathematical propositions must be false.
Fast facts
- Born: 1959, Osijek
- Citizenship: United States
- PhD: University of Tennessee, 1984
- Field: Geometric group theory and topology
- Academic Home: University of Utah
- Fellow of the American Mathematical Society: 2013
- International Mathematical Olympiad medals: 1976, 1977, 1978
Questions readers ask
What is the Bestvina-Feighn Combination Theorem?
It is a mathematical theorem providing sufficient conditions for amalgamated free products and HNN extensions of word-hyperbolic groups to remain word-hyperbolic.
What role do train track maps play in Bestvina's work?
Introduced with Michael Handel in 1992, train track methods represent elements of Out(Fn) and have become a standard tool for studying the dynamical and algebraic properties of free group automorphisms.
Achievements
- Held posts at University of Utah
- Fields: group theory and topology
