Karen VogtmannFRS (born July 13, 1949 inPittsburg, California^[1]) is an American mathematician working primarily in the area ofgeometric group theory. She is known for having introduced, in a 1986 paper withMarc Culler,^[2] an object now known as theCuller–Vogtmann Outer space. The Outer space is afree group analog of theTeichmüller space of aRiemann surface and is particularly useful in the study of thegroup ofouter automorphisms of the free group onn generators, Out(F_n). Vogtmann is a professor of mathematics atCornell University andthe University of Warwick.

Karen Vogtmann FRS
Born	(1949-07-13)July 13, 1949 (age 75) Pittsburg, California
Nationality	American
Alma mater	Ph.D., 1977University of California, Berkeley
Known for	Culler–Vogtmann Outer space
Awards	2006,International Congress of Mathematicians invited lecture 2007,Noether Lecture 2014,Royal Society Wolfson Research Merit Award 2014,Humboldt Prize 2016,European Congress of Mathematics plenary talk
Scientific career
Fields	geometric group theory algebraic K-theory
Institutions	Cornell University University of Warwick
Thesis	Homology stability for 0n,n (1977)
Doctoral advisor	John Bason Wagoner
Doctoral students	Debra Boutin Martin Bridson

Vogtmann was inspired to pursue mathematics by aNational Science Foundation summer program for high school students at theUniversity of California, Berkeley.^[3]

She received a B.A. from theUniversity of California, Berkeley in 1971. Vogtmann then obtained a PhD in mathematics, also from theUniversity of California, Berkeley in 1977.^[4] Her PhD advisor was John Wagoner and her doctoral thesis was onalgebraic K-theory.^[3]

She then held positions atUniversity of Michigan,Brandeis University andColumbia University.^[5] Vogtmann has been a faculty member atCornell University since 1984, and she became a full professor at Cornell in 1994.^[5] In September 2013, she also joined theUniversity of Warwick. She is married to the mathematicianJohn Smillie. The couple moved in 2013 to England and settled inKenilworth.^[6] She is currently a professor of mathematics at Warwick, and a Goldwin Smith Professor of Mathematics Emeritus at Cornell.^[5]

Vogtmann has been the vice-president of theAmerican 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 (2006–2016) of the journalAlgebraic and Geometric Topology and a former associate editor ofBulletin of the American Mathematical Society.^[5] She is currently an associate editor of theJournal of the American Mathematical Society,^[10] an editorial board memberGeometry & Topology Monographs book series,^[11] and a consulting editor for theProceedings of the Edinburgh Mathematical Society.^[12]

She is also a member of theArXiv advisory board.^[13]

Since 1986 Vogtmann has been a co-organizer of the annual conference called theCornell Topology Festival^[14] that usually takes places atCornell University each May.

Vogtmann gave aninvited lecture at the International Congress of Mathematicians inMadrid, Spain, in August 2006.^[15][16]

She gave the 2007 annualAWMNoether Lecture titled "Automorphisms of Free Groups, Outer Space and Beyond" at the annual meeting ofAmerican Mathematical Society inNew Orleans in January 2007.^[3][17] Vogtmann was selected to deliver theNoether Lecture for "her fundamental contributions togeometric group theory; in particular, to the study of the automorphism group of a free group".^[18]

On June 21–25, 2010 a 'VOGTMANNFEST' Geometric Group Theory conference in honor of Vogtmann's birthday was held inLuminy, France.^[19]

In 2012 she became a fellow of theAmerican Mathematical Society.^[20] She became a member of theAcademia Europaea in 2020.^[21] She was elected to theAmerican Academy of Arts and Sciences in 2023.^[22]

Vogtmann received theRoyal Society Wolfson Research Merit Award in 2014.^[23] She also received theHumboldt Research Award from theHumboldt Foundation in 2014.^[24][25]She was namedMSRI Clay Senior Scholar in 2016 and Simons Professor for 2016-2017.^[26][27]

Vogtmann gave a plenary talk at the 2016European Congress of Mathematics in Berlin.^[28][29]

In 2018 she won thePólya Prize of theLondon Mathematical Society "for her profound and pioneering work in geometric group theory, particularly the study of automorphism groups of free groups".^[30]

In May 2021 she was elected aFellow of the Royal Society.^[31]

In 2022 she was elected to theNational Academy of Sciences (NAS).^[32]

Vogtmann's early work concernedhomological properties oforthogonal groups associated toquadratic forms over variousfields.^[33][34]

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 asCuller–Vogtmann Outer space. The Outer spaceX_n, associated to afree groupF_n, is a free group analog^[35] of theTeichmüller space of aRiemann surface. Instead of markedconformal structures (or, in an equivalent model, hyperbolic structures) on a surface, points of the Outer space are represented by volume-onemarked metric graphs. Amarked metric graph consists of ahomotopy equivalence between a wedge ofn circles and a finite connected graphΓ without degree-one and degree-two vertices, whereΓ is equipped with a volume-one metric structure, that is, assignment of positive real lengths to edges ofΓ so that the sum of the lengths of all edges is equal to one. Points ofX_n can also be thought of as free and discrete minimal isometric actionsF_n onreal trees where the quotient graph has volume one.

By construction the Outer spaceX_n is a finite-dimensionalsimplicial complex equipped with a natural action ofOut(F_n) which is properly discontinuous and has finite simplex stabilizers. The main result of Culler–Vogtmann 1986 paper,^[2] obtained via Morse-theoretic methods, was that the Outer spaceX_n is contractible. Thus thequotient spaceX_n /Out(F_n) is "almost" aclassifying space forOut(F_n) and it can be thought of as a classifying space overQ. Moreover, Out(F_n) is known to bevirtually torsion-free, so for any torsion-freesubgroupH of Out(F_n) the action ofH onX_n is discrete and free, so thatX_n/H is a classifying space forH. For these reasons the Outer space is a particularly useful object in obtaininghomological andcohomological 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 assignX_n a specific name. According to Vogtmann,^[36] the termOuter space for the complexX_n was later coined byPeter Shalen. In subsequent years the Outer space became a central object in the study ofOut(F_n). In particular, the Outer space has a natural compactification, similar toThurston's compactification of theTeichmüller space, and studying the action of Out(F_n) on this compactification yields interesting information about dynamical properties ofautomorphisms offree groups.^{[37][38][39][40]}

Much of Vogtmann's subsequent work concerned the study of the Outer spaceX_n, particularly its homotopy, homological and cohomological properties, and related questions for Out(F_n). For example, Hatcher and Vogtmann^[41][42] obtained a number of homological stability results for Out(F_n) and Aut(F_n).

In her papers with Conant,^[43][44][45] Vogtmann explored the connection found byMaxim Kontsevich between the cohomology of certain infinite-dimensionalLie algebras and the homology of Out(F_n).

A 2001 paper of Vogtmann, joint withLouis Billera andSusan P. Holmes, used the ideas ofgeometric group theory andCAT(0) geometry to study the space ofphylogenetic trees, that is trees showing possible evolutionary relationships between different species.^[46] Identifying precise evolutionary trees is an important basic problem inmathematical 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.^[47] The fact that the space ofphylogenetic trees has "non-positively curved geometry", particularly the uniqueness of shortest paths orgeodesics inCAT(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.^[47]

• Vogtmann, Karen (1981),"Spherical posets and homology stability for O_n,n"(PDF),Topology,20 (2):119–132,doi:10.1016/0040-9383(81)90032-x,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,S2CID 122869546
• 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,S2CID 15231486
• Billera, Louis J.;Holmes, Susan P.;Vogtmann, Karen (2001)."A Grove of Evolutionary Trees".Advances in Applied Mathematics.27 (4):733–767.CiteSeerX 10.1.1.29.3424.doi:10.1006/aama.2001.0759.MR 1867931. Archived fromthe original on February 27, 2012. RetrievedNovember 29, 2008.
• Conant, James;Vogtmann, Karen (2004),"Morita classes in the homology of automorphism groups of free groups"(PDF),Geometry & Topology,8 (3):1471–1499,arXiv:math/0406389,Bibcode:2004math......6389C,doi:10.2140/gt.2004.8.1471,MR 2119302,S2CID 3131626

• Geometric group theory
• Teichmüller space
• Mapping class group
• Train track map

• Karen Vogtmann's webpage atCornell University
• Karen Vogtmann at theMathematics Genealogy Project
• Cornell Topology Festival