Karen Vogtmann
Encyclopedia
Karen Vogtmann is a U.S. mathematician working primarily in the area of geometric group theory
. She is known for having introduced, in a 1986 paper with Marc Culler
, 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(Fn)
. Vogtmann is a Professor of Mathematics at Cornell University
.
summer program for high school students at the University of California, Berkeley
.
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. Her PhD advisor was John Wagoner and her doctoral thesis was on algebraic K-theory
.
She then held positions at University of Michigan
, Brandeis University
and Columbia University
. Vogtmann has been a faculty member at Cornell University
since 1984, and she became a Full Professor at Cornell in 1994.
Vogtmann gave an Invited Lecture at the International Congress of Mathematicians
in Madrid, Spain in August 2006.
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. Vogtmann was selected to deliever the Noether Lecture
for "her fundamental contributions to
geometric group theory; in particular, to the study of the automorphism group of a free group".
Vogtmann has been the Vice-President of the American Mathematical Society
(2003–2006). 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.
Vogtmann is an Editorial Board member of the journal Algebraic and Geometric Topology. She is also a member of the ArXiv
Advisory Board.
Since 1986 Vogtmann has been a co-organizer of the annual conference called the Cornell Topology Festival 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.
properties of orthogonal group
s associated to quadratic form
s over various fields
.
Vogtmann's most important contribution came in a 1986 paper with Marc Culler called "Moduli of graphs and automorphisms of free groups". The paper introduced an object that came to be known as Culler–Vogtmann Outer space. The Outer space Xn, associated to a free group
Fn, is a free group analog 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 Xn can also be thought of as free and discrete minimal isometric actions Fn on real tree
s where the quotient graph has volume one.
By construction the Outer space Xn is a finite-dimensional simplicial complex
equipped with a natural action of Out(Fn)
which is properly discontinuous and has finite simplex stabilizers. The main result of Culler–Vogtmann 1986 paper, obtained via Morse-theoretic methods, was that the Outer space Xn is contractible. Thus the quotient space
Xn /Out(Fn) is "almost" a classifying space
for Out(Fn)
and it can be thought of as a classifying space over Q
. Moreover, Out(Fn) is known to be virtually
torsion-free, so for any torsion-free subgroup
H of Out(Fn) the action of H on Xn is discrete and free, so that Xn/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(Fn). In particular, Culler and Vogtmann proved that Out(Fn) has virtual cohomological dimension 2n − 3.
In their 1986 paper Culler and Vogtmann do not assign Xn a specific name. According to Vogtmann, the term Outer space for the complex Xn was later coined by Peter Shalen
. In subsequent years the Outer space became a central object in the study of Out(Fn)
. In particular, the Outer space has a natural compactification, similar to Thurston
's compactification of the Teichmüller space
, and studying the action of Out(Fn) on this compactification yields interesting information about dynamical properties of automorphism
s of free group
s.
Much of Vogtmann's subsequent work concerned the study of the Outer space Xn, particularly its homotopy, homological and cohomological properties, and related questions for Out(Fn). For example, Hatcher and Vogtmann obtained a number of homological stability results for Out(Fn) and Aut(Fn).
In her papers with Conant, Vogtmann explored the connection found by Maxim Kontsevich
between the cohomology of certain infinite-dimensional Lie algebra
s and the homology of Out(Fn).
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 tree
s, that is trees showing possible evolutionary relationships between different species. 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. The fact that the space of phylogenetic tree
s 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.
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 groups and topological and geometric properties of spaces on which these groups act .Another important...
. She is known for having introduced, in a 1986 paper with Marc Culler
Marc Culler
Marc Edward Culler is an American mathematician who works in geometric group theory and low-dimensional topology. A native Californian, Culler did his undergraduate work at the University of California at Santa Barbara and his graduate work at Berkeley where he graduated in 1978. He is now at the...
, an object now known as the Culler–Vogtmann Outer space. The Outer space is a free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
analog of the Teichmüller space
Teichmüller space
In mathematics, the Teichmüller space TX of a topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism...
of a Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
and is particularly useful in the study of the group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
of outer automorphisms of the free group on n generators, Out(Fn)
Out(Fn)
In mathematics, Out is the outer automorphism group of a free group on n generators. These groups play an important role in geometric group theory.-Outer space:...
. Vogtmann is a Professor of Mathematics at Cornell University
Cornell University
Cornell University is an Ivy League university located in Ithaca, New York, United States. It is a private land-grant university, receiving annual funding from the State of New York for certain educational missions...
.
Biographical data
Vogtmann was inspired to pursue mathematics by a National Science FoundationNational Science Foundation
The National Science Foundation is a United States government agency that supports fundamental research and education in all the non-medical fields of science and engineering. Its medical counterpart is the National Institutes of Health...
summer program for high school students at the University of California, Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
.
She received a B.A. from the University of California, Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
in 1971. Vogtmann then obtained a PhD in Mathematics, also from the University of California, Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
in 1977. Her PhD advisor was John Wagoner and her doctoral thesis was on algebraic K-theory
Algebraic K-theory
In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....
.
She then held positions at University of Michigan
University of Michigan
The University of Michigan is a public research university located in Ann Arbor, Michigan in the United States. It is the state's oldest university and the flagship campus of the University of Michigan...
, Brandeis University
Brandeis University
Brandeis University is an American private research university with a liberal arts focus. It is located in the southwestern corner of Waltham, Massachusetts, nine miles west of Boston. The University has an enrollment of approximately 3,200 undergraduate and 2,100 graduate students. In 2011, it...
and Columbia University
Columbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
. Vogtmann has been a faculty member at Cornell University
Cornell University
Cornell University is an Ivy League university located in Ithaca, New York, United States. It is a private land-grant university, receiving annual funding from the State of New York for certain educational missions...
since 1984, and she became a Full Professor at Cornell in 1994.
Vogtmann gave an Invited Lecture at the International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
in Madrid, Spain in August 2006.
She gave the 2007 annual AWM
Association for Women in Mathematics
The Association for Women in Mathematics is a professional society whose mission is to encourage women and girls to study and to have active careers in the mathematical sciences. Equal opportunity and the equal treatment of women and girls in the mathematical sciences are promoted. The AWM was...
Noether Lecture
Noether Lecture
The Association for Women in Mathematics annually presents the Noether Lectures to honor women who have made fundamental and sustained contributions to the mathematical sciences. These one-hour expository lectures are presented at the Joint Mathematics Meetings each January...
titled "Automorphisms of Free Groups, Outer Space and Beyond" at the annual meeting of American Mathematical Society
American Mathematical Society
The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
in New Orleans in January 2007. Vogtmann was selected to deliever the Noether Lecture
Noether Lecture
The Association for Women in Mathematics annually presents the Noether Lectures to honor women who have made fundamental and sustained contributions to the mathematical sciences. These one-hour expository lectures are presented at the Joint Mathematics Meetings each January...
for "her fundamental contributions to
geometric group theory; in particular, to the study of the automorphism group of a free group".
Vogtmann has been the Vice-President of the American Mathematical Society
American Mathematical Society
The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
(2003–2006). 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.
Vogtmann is an Editorial Board member of the journal Algebraic and Geometric Topology. She is also a member of the ArXiv
ArXiv
The arXiv |Chi]], χ) is an archive for electronic preprints of scientific papers in the fields of mathematics, physics, astronomy, computer science, quantitative biology, statistics, and quantitative finance which can be accessed online. In many fields of mathematics and physics, almost all...
Advisory Board.
Since 1986 Vogtmann has been a co-organizer of the annual conference called the Cornell Topology Festival that usually takes places at Cornell University
Cornell University
Cornell University is an Ivy League university located in Ithaca, New York, United States. It is a private land-grant university, receiving annual funding from the State of New York for certain educational missions...
each May.
On June 21–25, 2010 a 'VOGTMANNFEST' Geometric Group Theory conference in honor of Karen Vogtmann's birthday was held in Luminy
Faculté des Sciences de Luminy
Faculté des Sciences de Luminy is the formation and research unity of the Université de la Méditerranée, the faculty work around six departments....
, France.
Mathematical contributions
Vogtmann's early work concerned homologicalHomology (mathematics)
In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...
properties of orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
s associated to quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
s over various fields
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
.
Vogtmann's most important contribution came in a 1986 paper with Marc Culler called "Moduli of graphs and automorphisms of free groups". The paper introduced an object that came to be known as Culler–Vogtmann Outer space. The Outer space Xn, associated to a free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
Fn, is a free group analog of the Teichmüller space
Teichmüller space
In mathematics, the Teichmüller space TX of a topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism...
of a Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
. 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 Xn can also be thought of as free and discrete minimal isometric actions Fn on real tree
Real tree
A real tree, or an \mathbb R-tree, is a metric space such thatfor any x, y in M there is a unique arc from x to y and this arc is a geodesic segment. Here by an arc from x to y we mean the image in M of a topological embedding f from an interval [a,b] to M such that f=x and f=y...
s where the quotient graph has volume one.
By construction the Outer space Xn is a finite-dimensional simplicial complex
Simplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
equipped with a natural action of Out(Fn)
Out(Fn)
In mathematics, Out is the outer automorphism group of a free group on n generators. These groups play an important role in geometric group theory.-Outer space:...
which is properly discontinuous and has finite simplex stabilizers. The main result of Culler–Vogtmann 1986 paper, obtained via Morse-theoretic methods, was that the Outer space Xn is contractible. Thus the quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
Xn /Out(Fn) is "almost" a classifying space
Classifying space
In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG by a free action of G...
for Out(Fn)
Out(Fn)
In mathematics, Out is the outer automorphism group of a free group on n generators. These groups play an important role in geometric group theory.-Outer space:...
and it can be thought of as a classifying space over Q
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
. Moreover, Out(Fn) is known to be virtually
Virtually
In mathematics, especially in the area of abstract algebra which studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index...
torsion-free, so for any torsion-free subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
H of Out(Fn) the action of H on Xn is discrete and free, so that Xn/H is a classifying space for H. For these reasons the Outer space is a particularly useful object in obtaining homological
Homology (mathematics)
In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...
and cohomological
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
information about Out(Fn). In particular, Culler and Vogtmann proved that Out(Fn) has virtual cohomological dimension 2n − 3.
In their 1986 paper Culler and Vogtmann do not assign Xn a specific name. According to Vogtmann, the term Outer space for the complex Xn was later coined by Peter Shalen
Peter Shalen
Peter B. Shalen is an American mathematician, working primarily in low-dimensional topology. He is the "S" in JSJ decomposition.-Life:He graduated from Stuyvesant High School in 1962, and went on to earn a B.A. from Harvard College in 1966 and his Ph.D. from Harvard University in 1972...
. In subsequent years the Outer space became a central object in the study of Out(Fn)
Out(Fn)
In mathematics, Out is the outer automorphism group of a free group on n generators. These groups play an important role in geometric group theory.-Outer space:...
. In particular, the Outer space has a natural compactification, similar to Thurston
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
's compactification of the Teichmüller space
Teichmüller space
In mathematics, the Teichmüller space TX of a topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism...
, and studying the action of Out(Fn) on this compactification yields interesting information about dynamical properties of automorphism
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
s of free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
s.
Much of Vogtmann's subsequent work concerned the study of the Outer space Xn, particularly its homotopy, homological and cohomological properties, and related questions for Out(Fn). For example, Hatcher and Vogtmann obtained a number of homological stability results for Out(Fn) and Aut(Fn).
In her papers with Conant, Vogtmann explored the connection found by Maxim Kontsevich
Maxim Kontsevich
Maxim Lvovich Kontsevich is a Russian mathematician. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished professor at the University of Miami...
between the cohomology of certain infinite-dimensional Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s and the homology of Out(Fn).
A 2001 paper of Vogtmann, joint with Billera and Holmes, used the ideas of geometric group theory
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 groups and topological and geometric properties of spaces on which these groups act .Another important...
and CAT(0) geometry to study the space of phylogenetic tree
Phylogenetic tree
A phylogenetic tree or evolutionary tree is a branching diagram or "tree" showing the inferred evolutionary relationships among various biological species or other entities based upon similarities and differences in their physical and/or genetic characteristics...
s, that is trees showing possible evolutionary relationships between different species. Identifying precise evolutionary trees is an important basic problem in mathematical biology
Mathematical biology
Mathematical and theoretical biology is an interdisciplinary scientific research field with a range of applications in biology, medicine and biotechnology...
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. The fact that the space of phylogenetic tree
Phylogenetic tree
A phylogenetic tree or evolutionary tree is a branching diagram or "tree" showing the inferred evolutionary relationships among various biological species or other entities based upon similarities and differences in their physical and/or genetic characteristics...
s 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.
See also
- Geometric group theoryGeometric group theoryGeometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
- Teichmüller spaceTeichmüller spaceIn mathematics, the Teichmüller space TX of a topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism...
- Mapping class groupMapping class groupIn mathematics, in the sub-field of geometric topology, the mapping class groupis an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space.-Motivation:...
- Train track mapTrain track mapIn 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 iterations...
External links
- Karen Vogtmann's webpage at Cornell UniversityCornell UniversityCornell University is an Ivy League university located in Ithaca, New York, United States. It is a private land-grant university, receiving annual funding from the State of New York for certain educational missions...
- Karen Vogtmann at the Mathematics Genealogy Project
- Cornell Topology Festival