# Karen Vogtmann

Karen Vogtmann | |
---|---|

Born |
Pittsburg, California | July 13, 1949

Nationality | American |

Fields |
geometric group theory, algebraic K-theory |

Institutions | |

Alma mater | Ph.D., 1977 University of California, Berkeley |

Doctoral advisor | John Bason Wagoner |

Known for | Culler–Vogtmann Outer space |

Notable awards | 2007, Noether Lecture |

**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.

## 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 K-theory.^{[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 currently maintains positions both at Cornell and Warwick.^{[6]}

Vogtmann gave an invited lecture at the International Congress of Mathematicians in Madrid, Spain in August 2006.^{[7]}^{[8]}

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]}^{[9]} 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".^{[10]}

Vogtmann has been the Vice-President of the American Mathematical Society (2003–2006).^{[4]}^{[11]} 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 2013.^{[12]}

Vogtmann is an Editorial Board member of the journal *Algebraic and Geometric Topology*.^{[13]} She is also a member of the ArXiv Advisory Board.^{[14]}

Since 1986 Vogtmann has been a co-organizer of the annual conference called the *Cornell Topology Festival*^{[15]} that usually takes places at Cornell University each May.

On June 21–25, 2010 a 'VOGTMANNFEST' Geometric Group Theory conference in honor of Karen Vogtmann's birthday was held in Luminy, France.^{[16]}

In 2012 she became a fellow of the American Mathematical Society.^{[17]}

## Mathematical contributions

Vogtmann's early work concerned homological properties of orthogonal groups associated to quadratic forms over various fields.^{[18]}^{[19]}

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

^{[20]}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 volume-one

*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 degree-one and degree-two vertices, where

*Γ*is equipped with a volume-one 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*can also be thought of as free and discrete minimal isometric actions

_{n}*F*

_{n}on real trees where the quotient graph has volume one.

By construction the Outer space *X _{n}* is a finite-dimensional 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 Morse-theoretic methods, was that the Outer space

*X*is contractible. Thus the quotient space

_{n}*X*/Out(

_{n}*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 torsion-free, so for any torsion-free subgroup

*H*of Out(

*F*

_{n}) the action of

*H*on

*X*is discrete and free, so that

_{n}*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 2

*n*− 3.

In their 1986 paper Culler and Vogtmann do not assign *X _{n}* a specific name. According to Vogtmann,

^{[21]}the term

*Outer space*for the complex

*X*was later coined by Peter Shalen. In subsequent years the Outer space became a central object in the study of Out(

_{n}*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.

^{[22]}

^{[23]}

^{[24]}

^{[25]}

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

^{[26]}

^{[27]}obtained a number of homological stability results for Out(

*F*

_{n}) and Aut(

*F*

_{n}).

In her papers with Conant,^{[28]}^{[29]}^{[30]} Vogtmann explored the connection found by Maxim Kontsevich between the cohomology of certain infinite-dimensional 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.^{[31]} 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.^{[32]} The fact that the space of phylogenetic trees has "non-positively 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.^{[32]}

## Selected works

- 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, doi:10.1007/BF01388734, MR 0830040 - Hatcher, Allen; Vogtmann, Karen (1998), "Cerf theory for graphs",
*Journal of the London Mathematical Society (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, 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 - 1 2 3 4 Culler, Marc; Vogtmann, Karen (1986), "Moduli of graphs and automorphisms of free groups" (PDF),
*Inventiones Mathematicae*,**84**(1): 91–119, doi:10.1007/BF01388734. - 1 2 3 Karen Vogtmann, 2007 Noether Lecture, Profiles of Women in Mathematics. The Emmy Noether Lectures. Association for Women in Mathematics. Accessed November 28, 2008
- 1 2
*Biographies of Candidates 2007.*Notices of the American Mathematical Society. September 2007, Volume 54, Issue 8, pp. 1043–1057 - 1 2 Karen Vogtmann's Curriculum Vitae
- ↑
- ↑ 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 Sanz-Solé, Javier Soria, Juan Luis Varona and Joan Verdera. European Mathematical Society (EMS), Zürich, 2006. ISBN 978-3-03719-022-7 - ↑ Invited Addresses, Sessions, and Other Activities. AMS 2007 Annual Meeting. American Mathematical Society. Accessed November 28, 2008
- ↑ Karen Vogtmann named 2007 Noether Lecturer. Association for Women in Mathematics press release. May 2, 2006. Accessed November 29, 2008
- ↑ 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
- ↑ Editorial Board. Algebraic and Geometric Topology. Accessed November 28, 2008
- ↑ ArXiv Advisory Board. ArXiv. Accessed November 27, 2008
- ↑ Cornell Topology Festival, grant summary. Cornell University. Accessed November 28, 2008
- ↑ VOGTMANNFEST, conference info. Department of Mathematics, University of Utah. Accessed July 13, 2010
- ↑ List of Fellows of the American Mathematical Society, retrieved 2013-08-29.
- ↑ Karen Vogtmann,
*Spherical posets and homology stability for*. 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 978-0-8218-3838-9; 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*Journal of the Institute of Mathematics of Jussieu, vol. 2 (2003), no. 1, 59–72_{n}have north-south dynamics on compactified Outer space. - ↑ Gilbert Levitt, and Martin Lustig,
*Automorphisms of free groups have asymptotically periodic dynamics.*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 - 1 2 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