Karen Vogtmann
Karen Vogtmann  

Born 
Pittsburg, California 
July 13, 1949
Nationality  American 
Alma mater  Ph.D., 1977 University of California, Berkeley 
Known for  Culler–Vogtmann Outer space 
Awards 

Scientific career  
Fields 
geometric group theory, algebraic Ktheory 
Institutions  
Doctoral advisor  John Bason Wagoner 
Doctoral students  Martin Bridson 
Karen Vogtmann (born July 13, 1949 in Pittsburg, California^{[1]}) is an American mathematician working primarily in the area of geometric group theory. She is known for having introduced, in a 1986 paper with Marc Culler,^{[2]} an object now known as the Culler–Vogtmann Outer space. The Outer space is a free group analog of the Teichmüller space of a Riemann surface and is particularly useful in the study of the group of outer automorphisms of the free group on n generators, Out(F_{n}). Vogtmann is a Professor of Mathematics at Cornell University and The University of Warwick.
Contents
Biographical data
Vogtmann was inspired to pursue mathematics by a National Science Foundation summer program for high school students at the University of California, Berkeley.^{[3]}
She received a B.A. from the University of California, Berkeley in 1971. Vogtmann then obtained a PhD in Mathematics, also from the University of California, Berkeley in 1977.^{[4]} Her PhD advisor was John Wagoner and her doctoral thesis was on algebraic Ktheory.^{[3]}
She then held positions at University of Michigan, Brandeis University and Columbia University.^{[5]} Vogtmann has been a faculty member at Cornell University since 1984, and she became a Full Professor at Cornell in 1994.^{[5]} In September 2013, she also joined the University of Warwick. She is currently a Professor of Mathematics at Warwick, and a Goldwin Smith Professor of Mathematics Emeritus at Cornell.^{[6]}
Vogtmann has been the VicePresident of the American Mathematical Society (2003–2006).^{[4]}^{[7]} She has been elected to serve as a member of the Board of Trustees of the American Mathematical Society for the period February 2008 – January 2018.^{[8]}^{[9]}
Vogtmann is a former Editorial Board member (20062006) of the journal Algebraic and Geometric Topology and a former Associate Editor of Bulletin of the American Mathematical Society.^{[10]} She is currently an Associate Editor of the Journal of the American Mathematical Society,^{[11]} an Editorial Board member Geometry & Topology Monographs book series,^{[12]} and a Consulting Editor for the Proceedings of the Edinburgh Mathematical Society.^{[13]}
She is also a member of the ArXiv Advisory Board.^{[14]}
Since 1986 Vogtmann has been a coorganizer of the annual conference called the Cornell Topology Festival^{[15]} that usually takes places at Cornell University each May.
Awards, honors and other recognition
Vogtmann gave an invited lecture at the International Congress of Mathematicians in Madrid, Spain in August 2006.^{[16]}^{[17]}
She gave the 2007 annual AWM Noether Lecture titled "Automorphisms of Free Groups, Outer Space and Beyond" at the annual meeting of American Mathematical Society in New Orleans in January 2007.^{[3]}^{[18]} Vogtmann was selected to deliver the Noether Lecture for "her fundamental contributions to geometric group theory; in particular, to the study of the automorphism group of a free group".^{[19]}
On June 21–25, 2010 a 'VOGTMANNFEST' Geometric Group Theory conference in honor of Karen Vogtmann's birthday was held in Luminy, France.^{[20]}
In 2012 she became a fellow of the American Mathematical Society.^{[21]}
Karen Vogtmann received the Royal Society Wolfson Research Merit Award in 2014.^{[22]} She also received the Humboldt Research Award from the Humboldt Foundation in 2014.^{[23]}^{[24]} Vogtmann was a Clay Senior Scholar in 2016.^{[25]}
Karen Vogtmann gave a plenary talk at the 2016 European Congress of Mathematics in Berlin.^{[26]}^{[27]}
In 2018 she won the Pólya Prize of the London Mathematical Society "for her profound and pioneering work in geometric group theory, particularly the study of automorphism groups of free groups".^{[28]}
Mathematical contributions
Vogtmann's early work concerned homological properties of orthogonal groups associated to quadratic forms over various fields.^{[29]}^{[30]}
Vogtmann's most important contribution came in a 1986 paper with Marc Culler called "Moduli of graphs and automorphisms of free groups".^{[2]} The paper introduced an object that came to be known as Culler–Vogtmann Outer space. The Outer space X_{n}, associated to a free group F_{n}, is a free group analog^{[31]} of the Teichmüller space of a Riemann surface. Instead of marked conformal structures (or, in an equivalent model, hyperbolic structures) on a surface, points of the Outer space are represented by volumeone marked metric graphs. A marked metric graph consists of a homotopy equivalence between a wedge of n circles and a finite connected graph Γ without degreeone and degreetwo vertices, where Γ is equipped with a volumeone metric structure, that is, assignment of positive real lengths to edges of Γ so that the sum of the lengths of all edges is equal to one. Points of X_{n} can also be thought of as free and discrete minimal isometric actions F_{n} on real trees where the quotient graph has volume one.
By construction the Outer space X_{n} is a finitedimensional simplicial complex equipped with a natural action of Out(F_{n}) which is properly discontinuous and has finite simplex stabilizers. The main result of Culler–Vogtmann 1986 paper,^{[2]} obtained via Morsetheoretic methods, was that the Outer space X_{n} is contractible. Thus the quotient space X_{n} /Out(F_{n}) is "almost" a classifying space for Out(F_{n}) and it can be thought of as a classifying space over Q. Moreover, Out(F_{n}) is known to be virtually torsionfree, so for any torsionfree subgroup H of Out(F_{n}) the action of H on X_{n} is discrete and free, so that X_{n}/H is a classifying space for H. For these reasons the Outer space is a particularly useful object in obtaining homological and cohomological information about Out(F_{n}). In particular, Culler and Vogtmann proved^{[2]} that Out(F_{n}) has virtual cohomological dimension 2n − 3.
In their 1986 paper Culler and Vogtmann do not assign X_{n} a specific name. According to Vogtmann,^{[32]} the term Outer space for the complex X_{n} was later coined by Peter Shalen. In subsequent years the Outer space became a central object in the study of Out(F_{n}). In particular, the Outer space has a natural compactification, similar to Thurston's compactification of the Teichmüller space, and studying the action of Out(F_{n}) on this compactification yields interesting information about dynamical properties of automorphisms of free groups.^{[33]}^{[34]}^{[35]}^{[36]}
Much of Vogtmann's subsequent work concerned the study of the Outer space X_{n}, particularly its homotopy, homological and cohomological properties, and related questions for Out(F_{n}). For example, Hatcher and Vogtmann^{[37]}^{[38]} obtained a number of homological stability results for Out(F_{n}) and Aut(F_{n}).
In her papers with Conant,^{[39]}^{[40]}^{[41]} Vogtmann explored the connection found by Maxim Kontsevich between the cohomology of certain infinitedimensional Lie algebras and the homology of Out(F_{n}).
A 2001 paper of Vogtmann, joint with Billera and Holmes, used the ideas of geometric group theory and CAT(0) geometry to study the space of phylogenetic trees, that is trees showing possible evolutionary relationships between different species.^{[42]} Identifying precise evolutionary trees is an important basic problem in mathematical biology and one also needs to have good quantitative tools for estimating how accurate a particular evolutionary tree is. The paper of Billera, Vogtmann and Holmes produced a method for quantifying the difference between two evolutionary trees, effectively determining the distance between them.^{[43]} The fact that the space of phylogenetic trees has "nonpositively curved geometry", particularly the uniqueness of shortest paths or geodesics in CAT(0) spaces, allows using these results for practical statistical computations of estimating the confidence level of how accurate particular evolutionary tree is. A free software package implementing these algorithms has been developed and is actively used by biologists.^{[43]}
Selected works
 Vogtmann, Karen (1981), "Spherical posets and homology stability for On,n" (PDF), Topology, 20 (2): 119–132, doi:10.1016/00409383(81)90032x, MR 0605652
 Culler, Marc; Vogtmann, Karen (1986), "Moduli of graphs and automorphisms of free groups" (PDF), Inventiones Mathematicae, 84 (1): 91–119, Bibcode:1986InMat..84...91C, doi:10.1007/BF01388734, MR 0830040
 Hatcher, Allen; Vogtmann, Karen (1998), "Cerf theory for graphs", Journal of the London Mathematical Society, Series 2, 58 (3): 633–655, doi:10.1112/s0024610798006644, MR 1678155
 Billera, Louis J.; Holmes, Susan P.; Vogtmann, Karen (2001), "A Grove of Evolutionary Trees", Advances in Applied Mathematics, 27 (4): 733–767, doi:10.1006/aama.2001.0759, MR 1867931
 Conant, James; Vogtmann, Karen (2004), "Morita classes in the homology of automorphism groups of free groups" (PDF), Geometry & Topology, 8: 1471–1499, arXiv:math/0406389, doi:10.2140/gt.2004.8.1471, MR 2119302
See also
References
 ^ Biographies of Candidates 2002. Notices of the American Mathematical Society. September 2002, Volume 49, Issue 8, pp. 970–981
 ^ ^{a} ^{b} ^{c} ^{d} Culler, Marc; Vogtmann, Karen (1986), "Moduli of graphs and automorphisms of free groups" (PDF), Inventiones Mathematicae, 84 (1): 91–119, Bibcode:1986InMat..84...91C, doi:10.1007/BF01388734.
 ^ ^{a} ^{b} ^{c} Karen Vogtmann, 2007 Noether Lecture, Profiles of Women in Mathematics. The Emmy Noether Lectures. Association for Women in Mathematics. Accessed November 28, 2008
 ^ ^{a} ^{b} Biographies of Candidates 2007. Notices of the American Mathematical Society. September 2007, Volume 54, Issue 8, pp. 1043–1057
 ^ ^{a} ^{b} Karen Vogtmann's Curriculum Vitae
 ^ [1]
 ^ 2002 Election results. Notices of the American Mathematical Society. February 2003, Volume 50 Issue 2, p. 281
 ^ 2007 Election Results. Notices of the American Mathematical Society. February 2008, Volume 55, Issue 2, p. 301
 ^ 2012 Election Results, Notices of the American Mathematical Society, February 2013, Volume 60, Issue 2, p. 256
 ^ CURRICULUM VITAE  Karen Vogtmann, University of Warwick. Accessed September 14, 2017
 ^ Editorial Board, Journal of the American Mathematical Society. Accessed September 14, 2017.
 ^ Editorial Board, Geometry & Topology Monographs. Accessed September 14, 2017
 ^ Editorial Board, Proceedings of the Edinburgh Mathematical Society. Accessed September 14, 2017.
 ^ ArXiv Advisory Board. ArXiv. Accessed November 27, 2008
 ^ Cornell Topology Festival, grant summary. Cornell University. Accessed November 28, 2008
 ^ ICM 2006 – Invited Lectures. Abstracts, International Congress of Mathematicians, 2006.
 ^ Karen Vogtmann, The cohomology of automorphism groups of free groups. International Congress of Mathematicians. Vol. II, 1101–1117, Invited lectures. Proceedings of the congress held in Madrid, August 22–30, 2006. Edited by Marta SanzSolé, Javier Soria, Juan Luis Varona and Joan Verdera. European Mathematical Society (EMS), Zürich, 2006. ISBN 9783037190227
 ^ Invited Addresses, Sessions, and Other Activities. AMS 2007 Annual Meeting. American Mathematical Society. Accessed November 28, 2008
 ^ Karen Vogtmann named 2007 Noether Lecturer. Archived 20080516 at the Wayback Machine. Association for Women in Mathematics press release. May 2, 2006. Accessed November 29, 2008
 ^ VOGTMANNFEST, conference info. Department of Mathematics, University of Utah. Accessed July 13, 2010
 ^ List of Fellows of the American Mathematical Society, retrieved 20130829.
 ^ Royal Society announces new round of esteemed Wolfson Research Merit Awards, The Royal Society press release, 09 May 2014. Accessed 14 September 2017.
 ^ Awards: since March 2013, Alexander von Humboldt Foundation. Accessed September 14, 2017
 ^ Karen Vogtmann receives Humboldt Research Award, Math Matters. Department of Mathematics, Cornell University, December 2014; p. 2
 ^ Karen Vogtmann: Recent Senior Scholars, Clay Mathematics Institute. Accessed September 14, 2017
 ^ 7ECM Plenary Talks, 7th European Congress of Mathematics, July 18–22, 2016. The quadrennial Congress of the European Mathematical Society. Accessed September 14, 2017
 ^ Editorial: 7th European Congress of Mathematics, Newsletter of the European Mathematical Society, June 2015, issue 96, p. 3
 ^ "Prizes of the London Mathematical Society" (PDF), Mathematics People, Notices of the American Mathematical Society, 65 (9): 1122, October 2018
 ^ Karen Vogtmann, Spherical posets and homology stability for O n , n {\displaystyle O_{n,n}} . Topology, vol. 20 (1981), no. 2, pp. 119–132.
 ^ Karen Vogtmann, A Stiefel complex for the orthogonal group of a field. Commentarii Mathematici Helvetici, vol. 57 (1982), no. 1, pp. 11–21
 ^ Benson Farb. Problems on Mapping Class Groups and Related Topics. American Mathematical Society, 2006. ISBN 9780821838389; p. 335
 ^ Karen Vogtmann, Automorphisms of free groups and Outer space. Geometriae Dedicata, vol. 94 (2002), pp. 1–31; Quote from p. 3: "Peter Shalen later invented the name Outer space for X_{n}".
 ^ M. Bestvina, M. Feighn, M. Handel, Laminations, trees, and irreducible automorphisms of free groups. Geometric and Functional Analysis, vol. 7 (1997), no. 2, 215–244
 ^ Gilbert Levitt and Martin Lustig, Irreducible automorphisms of F_{n} have northsouth dynamics on compactified Outer space. Journal of the Institute of Mathematics of Jussieu, vol. 2 (2003), no. 1, 59–72
 ^ Gilbert Levitt, and Martin Lustig, Automorphisms of free groups have asymptotically periodic dynamics.^{[permanent dead link]} Crelle's journal, vol. 619 (2008), pp. 1–36
 ^ Vincent Guirardel, Dynamics of Out(Fn) on the boundary of Outer space. Annales Scientifiques de l'École Normale Supérieure (4), vol. 33 (2000), no. 4, 433–465.
 ^ Allen Hatcher, and Karen Vogtmann. Cerf theory for graphs. Journal of the London Mathematical Society (2), vol. 58 (1998), no. 3, pp. 633–655.
 ^ A. Hatcher, and K. Vogtmann, Homology stability for outer automorphism groups of free groups. Algebraic and Geometric Topology, vol. 4 (2004), pp. 1253–1272
 ^ James Conant, and Karen Vogtmann. On a theorem of Kontsevich. Algebraic and Geometric Topology, vol. 3 (2003), pp. 1167–1224
 ^ James Conant, and Karen Vogtmann, Infinitesimal operations on complexes of graphs. Mathematische Annalen, vol. 327 (2003), no. 3, pp. 545–573.
 ^ James Conant, and Karen Vogtmann, Morita classes in the homology of automorphism groups of free groups. Geometry & Topology, vol. 8 (2004), pp. 1471–1499
 ^ Louis J. Billera, Susan P. Holmes, and Karen Vogtmann. Geometry of the space of phylogenetic trees. Advances in Applied Mathematics, vol. 27 (2001), no. 4, pp. 733–767
 ^ ^{a} ^{b} Julie Rehmeyer. A Grove of Evolutionary Trees. Science News. May 10, 2007. Accessed November 28, 2008
External links
 Karen Vogtmann's webpage at Cornell University
 Karen Vogtmann at the Mathematics Genealogy Project
 Cornell Topology Festival
 20thcentury American mathematicians
 21stcentury American mathematicians
 American women mathematicians
 Topologists
 Group theorists
 Cornell University faculty
 1949 births
 University of California, Berkeley alumni
 Living people
 Fellows of the American Mathematical Society
 University of Michigan faculty
 Academics of the University of Warwick