Karen Vogtmann (born July 13, 1949 in

Pittsburg, California Pittsburg is a city in 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 of the Sacramento–San Joaquin R ...

''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 groups and topological and geometric properties of spaces on which these group ...

. 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''− ...

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 ve ...

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 ...

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 ...

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 ...

.Karen Vogtmann
, 2007 Noether Lecture, 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 Brandeis University is a Private university, private research university in Waltham, Massachusetts. Founded in 1948 as a nonsectarian, non-sectarian, coeducational institution sponsored by the Jews, Jewish community, Brandeis was established on t ...

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 Manha ...

. 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 , mottoeng = Mind moves matter , established = , type = Public research university , endowment = £7.0 million (2021) , budget = £698.2 million (202 ...

. 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 parish in the Warwick District in Warwickshire, England, south-west of Coventry, north of Warwick and north-west of London. It lies on Finham Brook, a tributary of the River Sowe, which joins the ...

. 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'' 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. ...

''.CURRICULUM VITAE - Karen Vogtmann

University of Warwick , mottoeng = Mind moves matter , established = , type = Public research university , endowment = £7.0 million (2021) , budget = £698.2 million (202 ...

. 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 abst ...

'', an editorial board member ''Geometry & Topology Monographs'' book series, and a consulting editor for the '' Proceedings of the Edinburgh Mathematical Society''. 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 o ...

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 AWM may refer to: *Academies of West Memphis, a public high school in West Memphis, Arkansas * Appliance Wiring Material, covered by UL standard 758 *Apostolic Women's Ministries, an organization that serves the women of the Apostolic Church of Pent ...

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

New Orleans New Orleans ( , ,New Orleans

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". On June 21–25, 2010 a 'VOGTMANNFEST' Geometric Group Theory conference in honor of Vogtmann's birthday was held in Luminy, 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 provid ...

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 Rese ...

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 knowledge, including mathematic ...

. 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 Nat ...

(NAS).

Mathematical contributions

Vogtmann's early work concerned homological properties of

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

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 ...

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 ''X_n'', associated to a

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

of a

. 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 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 defo ...

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 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 ...

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 Quotient space may refer to a quotient set when the sets under consideration are considered as spaces. In particular: *Quotient space (topology), in case of topological spaces * Quotient space (linear algebra), in case of vector spaces *Quotient ...

''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 ac ...

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 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 subgrou ...

''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 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 automorphisms 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 a ...

between the cohomology of certain infinite-dimensional

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

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 superv ...

and

Susan P. Holmes Susan P. Holmes is an American statistician and professor at Stanford University. She is noted for her work in applying nonparametric multivariate statistics, bootstrapping methods, and data visualization to biology. She received her PhD in 19 ...

, used the ideas of

and CAT(0) geometry to study the space of phylogenetic trees, 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 ...

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

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