Karen Vogtmann (born July 13, 1949 in

Pittsburg, California Pittsburg is a city in Contra Costa County, California, Contra Costa County, California, United States. It is an industrial suburb located on the southern shore of the Suisun Bay in the East Bay region of the San Francisco Bay Area, and is part ...

''Biographies of Candidates 2002.''

Notices of the American Mathematical Society ''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue. The first volume appeared in 1953. Each issue of the magazine since ...

. September 2002, Volume 49, Issue 8, pp. 970–981) is an American mathematician working primarily in the area of

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 group (mathematics), groups and topology, topological and geometry, geometric pro ...

. She is known for having introduced, in a 1986 paper with

Marc Culler Marc Edward Culler (born November 22, 1953) 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 ...

, an object now known as the Culler–Vogtmann Outer space. The Outer space is a

free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...

analog of the

Teichmüller space In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmülle ...

of a

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

and is particularly useful in the study of the

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

of outer automorphisms of the free group on ''n'' generators, Out(''F''_''n''). Vogtmann is a professor of mathematics at

Cornell University Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to teach an ...

and the University of Warwick.

Biographical data

Vogtmann was inspired to pursue mathematics by a

National Science Foundation The National Science Foundation (NSF) is an independent agency of the United States government that supports fundamental research and education in all the non-medical fields of science and engineering. Its medical counterpart is the National I ...

summer program for high school students at the

University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...

. She received a B.A. from the

in 1971. Vogtmann then obtained a PhD in mathematics, also from the

in 1977.''Biographies of Candidates 2007.''

. September 2007, Volume 54, Issue 8, pp. 1043–1057 Her PhD advisor was John Wagoner and her doctoral thesis was on

algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense o ...

.Karen Vogtmann
, 2007

Noether Lecture The Noether Lecture is a distinguished lecture series that honors women "who have made fundamental and sustained contributions to the mathematical sciences". The Association for Women in Mathematics (AWM) established the annual lectures in 1980 as t ...

, Profiles of Women in Mathematics. The Emmy Noether Lectures.

Association for Women in Mathematics The Association for Women in Mathematics (AWM) is a professional society whose mission is to encourage women and girls to study and to have active careers in the mathematical sciences, and to promote equal opportunity for and the equal treatment o ...

. Accessed November 28, 2008 She then held positions at

University of Michigan , mottoeng = "Arts, Knowledge, Truth" , former_names = Catholepistemiad, or University of Michigania (1817–1821) , budget = $10.3 billion (2021) , endowment = $17 billion (2021)As o ...

Brandeis University , mottoeng = "Truth even unto its innermost parts" , established = , type = Private research university , accreditation = NECHE , president = Ronald D. Liebowitz , pro ...

and

Columbia University Columbia University (also known as Columbia, and officially as Columbia University in the City of New York) is a private research university in New York City. Established in 1754 as King's College on the grounds of Trinity Church in Manhatt ...

. Vogtmann has been a faculty member at

since 1984, and she became a full professor at Cornell in 1994. In September 2013, she also joined the

University of Warwick The University of Warwick ( ; abbreviated as ''Warw.'' in post-nominal letters) is a public research university on the outskirts of Coventry between the West Midlands (county), West Midlands and Warwickshire, England. The university was founded i ...

. She is married to the mathematician John Smillie. The couple moved in 2013 to England and settled in

Kenilworth Kenilworth ( ) is a market town and Civil parishes in England, civil parish in the Warwick (district), Warwick District in Warwickshire, England, south-west of Coventry, north of Warwick and north-west of London. It lies on Finham Brook, a ...

. She is currently a professor of mathematics at Warwick, and a Goldwin Smith Professor of Mathematics Emeritus at Cornell. Vogtmann has been the vice-president of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

(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 2018. Vogtmann is a former editorial board member (2006–2016) of the journal ''

Algebraic and Geometric Topology '' Algebraic & Geometric Topology'' is a peer-reviewed mathematics journal published quarterly by Mathematical Sciences Publishers. Established in 2001, the journal publishes articles on topology. Its 2018 MCQ was 0.82, and its 2018 impact factor ...

'' and a former associate editor of ''

Bulletin of the American Mathematical Society The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. I ...

''.CURRICULUM VITAE - Karen Vogtmann

. Accessed September 14, 2017 She is currently an associate editor of the ''

Journal of the American Mathematical Society The ''Journal of the American Mathematical Society'' (''JAMS''), is a quarterly peer-reviewed mathematical journal published by the American Mathematical Society. It was established in January 1988. Abstracting and indexing This journal is abstr ...

'', an editorial board member ''Geometry & Topology Monographs'' book series, and a consulting editor for the ''

Proceedings of the Edinburgh Mathematical Society In academia and librarianship, conference proceedings is a collection of academic papers published in the context of an academic conference or workshop. Conference proceedings typically contain the contributions made by researchers at the conferen ...

''. She is also a member of the

ArXiv arXiv (pronounced "archive"—the X represents the Greek letter chi ⟨χ⟩) is an open-access repository of electronic preprints and postprints (known as e-prints) approved for posting after moderation, but not peer review. It consists of ...

advisory board. Since 1986 Vogtmann has been a co-organizer of the annual conference called the ''Cornell Topology Festival'' that usually takes places at

each May.

Awards, honors and other recognition

Vogtmann gave an invited lecture at the International Congress of Mathematicians in

Madrid, Spain Madrid ( , ) is the capital and most populous city of Spain. The city has almost 3.4 million inhabitants and a metropolitan area population of approximately 6.7 million. It is the second-largest city in the European Union (EU), and ...

, in August 2006. She gave the 2007 annual AWM

titled "Automorphisms of Free Groups, Outer Space and Beyond" at the annual meeting of

New Orleans New Orleans ( , ,New Orleans
Merriam-Webster. ; french: La Nouvelle-Orléans , es, Nuev ...

in January 2007. Vogtmann was selected to deliver the

for "her fundamental contributions to geometric group theory; in particular, to the study of the automorphism group of a free group". On June 21–25, 2010 a 'VOGTMANNFEST' Geometric Group Theory conference in honor of Vogtmann's birthday was held in

Luminy Aix-Marseille University Faculty of Sciences is one of the faculties of Aix-Marseille University. The faculty is divided into seven departments across six campuses. With 8000 students and 1600 staff and faculty it is one of the largest science facu ...

, France. In 2012 she became a fellow of the

. She became a member of the

Academia Europaea The Academia Europaea is a pan-European Academy of Humanities, Letters, Law, and Sciences. The Academia was founded in 1988 as a functioning Europe-wide Academy that encompasses all fields of scholarly inquiry. It acts as co-ordinator of Europea ...

in 2020. Vogtmann received the

Royal Society Wolfson Research Merit Award The Royal Society Wolfson Research Merit Award was an award made by the Royal Society from 2000 to 2020. It was administered by the Royal Society and jointly funded by the Wolfson Foundation and the UK Office of Science and Technology, to provide ...

in 2014. She also received the

Humboldt Research Award The Humboldt Prize, the Humboldt-Forschungspreis in German, also known as the Humboldt Research Award, is an award given by the Alexander von Humboldt Foundation of Germany to internationally renowned scientists and scholars who work outside of G ...

from the

Humboldt Foundation The Alexander von Humboldt Foundation (german: Alexander von Humboldt-Stiftung) is a foundation established by the government of the Federal Republic of Germany and funded by the Federal Foreign Office, the Federal Ministry of Education and Resear ...

in 2014. She was named MSRI Clay Senior Scholar in 2016 and Simons Professor for 2016-2017. Vogtmann gave a plenary talk at the 2016

European Congress of Mathematics The European Congress of Mathematics (ECM) is the second largest international conference of the mathematics community, after the International Congresses of Mathematicians (ICM). The ECM are held every four years and are timed precisely betwee ...

in Berlin. In 2018 she won the Pólya Prize of the

London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical S ...

"for her profound and pioneering work in geometric group theory, particularly the study of automorphism groups of free groups". In May 2021 she was elected a

Fellow of the Royal Society Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the judges of the Royal Society of London to individuals who have made a "substantial contribution to the improvement of natural science, natural knowledge, incl ...

. In 2022 she was elected to the

National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nati ...

(NAS).

Mathematical contributions

Vogtmann's early work concerned

homological Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor *Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chromo ...

properties of

orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...

s associated to

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...

s over various

fields Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 *Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...

. 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 ''X_n'', associated to a

''F''_''n'', is a free group analog of the

of a

. Instead of marked

conformal structure In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two d ...

s (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 In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...

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_n'' can also be thought of as free and discrete minimal isometric actions ''F''_''n'' on

real tree In mathematics, real trees (also called \mathbb R-trees) are a class of metric spaces generalising simplicial trees. They arise naturally in many mathematical contexts, in particular geometric group theory and probability theory. They are also the s ...

s where the quotient graph has volume one. By construction the Outer space ''X_n'' is a finite-dimensional

simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set ...

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, obtained via Morse-theoretic methods, was that the Outer space ''X_n'' is contractible. Thus the quotient space ''X_n'' /Out(''F''_''n'') is "almost" a

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'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free acti ...

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 In mathematics, especially in the area of abstract algebra that studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index. Given a property P, the group ''G'' is said to b ...

torsion-free, so for any torsion-free

subgroup In group theory, a branch of mathematics, 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 ...

''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

and

cohomological In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...

information about Out(''F''_''n''). In particular, Culler and Vogtmann proved 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, the term ''Outer space'' for the complex ''X_n'' was later coined by

Peter Shalen Peter B. Shalen (born c. 1946) 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 Coll ...

. 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

, and studying the action of Out(''F''_''n'') on this compactification yields interesting information about dynamical properties of

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 ...

s of

s.Gilbert Levitt and Martin Lustig, ''Irreducible automorphisms of F_n have north-south dynamics on compactified Outer space.'' Journal of the Institute of Mathematics of Jussieu, vol. 2 (2003), no. 1, 59–72 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 obtained a number of homological stability results for Out(''F''_''n'') and Aut(''F''_''n''). In her papers with Conant, Vogtmann explored the connection found by

Maxim Kontsevich Maxim Lvovich Kontsevich (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques an ...

between the cohomology of certain infinite-dimensional

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

s and the homology of Out(''F''_''n''). A 2001 paper of Vogtmann, joint with

Louis Billera Louis Joseph Billera is a Professor of Mathematics at Cornell University. Career Billera completed his B.S. at the Rensselaer Polytechnic Institute in 1964. He earned his Ph.D. from the City University of New York in 1968, under the joint supervi ...

and Susan P. Holmes, used the ideas of

and CAT(0) geometry to study the space of

phylogenetic tree A phylogenetic tree (also phylogeny or evolutionary tree Felsenstein J. (2004). ''Inferring Phylogenies'' Sinauer Associates: Sunderland, MA.) is a branching diagram or a tree showing the evolutionary relationships among various biological spec ...

s, that is trees showing possible evolutionary relationships between different species. Identifying precise evolutionary trees is an important basic problem in

mathematical biology Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical models and abstractions of the living organisms to investigate the principles that govern the structure, development a ...

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.Julie Rehmeyer
''A Grove of Evolutionary Trees''.

Science News ''Science News (SN)'' is an American bi-weekly magazine devoted to articles about new scientific and technical developments, typically gleaned from recent scientific and technical journals. History ''Science News'' has been published since 1 ...

. May 10, 2007. Accessed November 28, 2008 The fact that the space of

s has "non-positively curved geometry", particularly the uniqueness of shortest paths or ''geodesics'' in

CAT(0) space In mathematics, a \mathbf(k) space, where k is a real number, is a specific type of metric space. Intuitively, triangles in a \operatorname(k) space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature k. I ...

s, 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.

Selected works

* * * * *

References

External links

Karen Vogtmann's webpage
at