Zlil Sela
Encyclopedia
Zlil Sela is an Israel
i mathematician
working in the area of geometric group theory
.
He is a Professor of Mathematics at the Hebrew University of Jerusalem
. Sela is known for the solution of the isomorphism problem for torsion-free word-hyperbolic groups and for the solution of the Tarski conjecture about equivalence of first order theories of finitely generated non-abelian
free group
s.
, where his doctoral advisor was Eliyahu Rips
.
Prior to his current appointment at the Hebrew University, he held an Associate Professor position at Columbia University
in New York. While at Columbia, Sela won the Sloan Fellowship
from the Sloan Foundation.
Sela gave an Invited Address at the 2002 International Congress of Mathematicians
in Beijing. Sela gave a plenary talk at the 2002 annual meeting of the Association for Symbolic Logic
.
He delivered an AMS Invited Address at the October 2003 meeting of the American Mathematical Society
and the 2005 Tarski Lectures at the University of California at Berkeley.
He was also awarded the 2003 Erdős Prize
from the Israel Mathematical Union.
Sela also received the 2008 Carol Karp Prize from the Association for Symbolic Logic
for his work on the Tarski conjecture and on discovering and developing new connections between model theory
and geometric group theory
.
for torsion-free word-hyperbolic groups. The machinery of group actions on real tree
s, developed by Eliyahu Rips
, played a key role in Sela's approach. The solution of the isomorphism problem also relied on the notion of canonical representatives for elements of hyperbolic groups, introduced by Rips and Sela in a joint 1995 paper. The machinery of the canonical representatives allowed Rips and Sela to prove algorithmic solvability of finite systems of equations in torsion-free hyperbolic groups, by reducing the problem to solving equations in free group
s where the Makanin-Razborov algorithm can be applied. The technique of canonical representatives was later generalized by Dahmani to the case of relatively hyperbolic group
s and played a key role in the solution of the isomorphism problem for toral relatively hyprbolic groups.
In his work on the isomorphism problem Sela also introduced and developed the notion of a JSJ-decomposition for word-hyperbolic groups, motivated by the notion of a JSJ decomposition
for 3-manifold
s. A JSJ-decomposition is a representation of a word-hyperbolic group as the fundamental group of a graph of groups which encodes in a canonical way all possible splittings over infinite cyclic subgroup
s. The idea of JSJ-decomposition was later extended by Rips and Sela to torsion-free finitely presented groups and this work gave rise a systematic development of the JSJ-decomposition theory with many further extensions and generalizations by other mathematicians. Sela applied a combination of his JSJ-decomposition and real tree
techniques to prove that torsion-free word-hyperbolic groups are Hopfian
. This result and Sela's approach were later generalized by others to finitely generated subgroup
s of hyperbolic groups and to the setting of relatively hyperbolic groups.
Sela's most important work came in early 2000s when he produced a solution to a famous Tarski conjecture. Namely, in a long series of papers, he proved that any two non-abelian finitely generated free group
s have the same first-order theory. Sela's work relied on applying his earlier JSJ-decomposition and real tree
techniques as well as developing new ideas and machinery of "algebraic geometry" over free group
s.
Sela pushed this work further to study first-order theory of arbitrary torsion-free word-hyperbolic groups and to characterize all groups that are elementarily equivalent to (that is, have the same first order theory as) a given torsion-free word-hyperbolic group. In particular, his work implies that if a finitely generated group G is elementarily equivalent to a word-hyperbolic group then G is word-hyperbolic as well.
Sela also proved that the first order theory of a finitely generated free group
is stable
in the model-theoretic sense, providing a brand-new and qualitatively different source of examples for the stability theory
.
An alternative solution for the Tarski conjecture has been presented by Olga Kharlampovich
and Alexei Myasnikov.
The work of Sela on first-order theory of free and word-hyperbolic groups substantially influenced the development of geometric group theory
, in particular by stimulating the development and the study of the notion of limit groups and of relatively hyperbolic group
s.
| volume=120 |issue=3| pages=489–512 }} | year=1995 | journal=Annals of Mathematics
(2) | volume=141 | issue=2 | pages=217–283 | jstor=2118520}} | year=1997 | journal=Geometric and Functional Analysis | volume=7|issue=3 | pages=561–593 | url=http://www.springerlink.com/content/qn2dqfw0xquukbw0/?p=7370f12b59a04dc7bfb4bcaba1999688&pi=6}} | year=1997 | journal=Annals of Mathematics
(2) | volume=146 |issue=1| pages=53–109 | jstor=2951832 }} | year=2001 | journal=Publications Mathématiques de l'IHÉS
| volume=93 | issue=1 | pages=31–105}} | year=2003 | journal=Israel Journal of Mathematics
| volume=134 | issue=1 | pages=173–254 | url=http://www.springerlink.com/content/v27x52636w4l2981/}} | year=2006 | journal=Geometric and Functional Analysis| volume=16|issue=3 | pages=707–730 | url=http://www.springerlink.com/content/r53x383mj14206v1/?p=2d33d0052c1749aeb7b127871d580675&pi=2}}
Israel
The State of Israel is a parliamentary republic located in the Middle East, along the eastern shore of the Mediterranean Sea...
i mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
working in the area 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...
.
He is a Professor of Mathematics at the Hebrew University of Jerusalem
Hebrew University of Jerusalem
The Hebrew University of Jerusalem ; ; abbreviated HUJI) is Israel's second-oldest university, after the Technion – Israel Institute of Technology. The Hebrew University has three campuses in Jerusalem and one in Rehovot. The world's largest Jewish studies library is located on its Edmond J...
. Sela is known for the solution of the isomorphism problem for torsion-free word-hyperbolic groups and for the solution of the Tarski conjecture about equivalence of first order theories of finitely generated non-abelian
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
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.
Biographical data
Sela received his Ph.D. in 1991 from the Hebrew University of JerusalemHebrew University of Jerusalem
The Hebrew University of Jerusalem ; ; abbreviated HUJI) is Israel's second-oldest university, after the Technion – Israel Institute of Technology. The Hebrew University has three campuses in Jerusalem and one in Rehovot. The world's largest Jewish studies library is located on its Edmond J...
, where his doctoral advisor was Eliyahu Rips
Eliyahu Rips
Eliyahu Rips, also Ilya Rips is a Latvian-born Israeli mathematician known for his research in geometric group theory. He became known to the general public following his coauthoring a paper on the Torah Code....
.
Prior to his current appointment at the Hebrew University, he held an Associate Professor position at 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...
in New York. While at Columbia, Sela won the Sloan Fellowship
Sloan Fellowship
The Sloan Research Fellowships are awarded annually by the Alfred P. Sloan Foundation since 1955 to "provide support and recognition to early-career scientists and scholars". This is distinct from the Sloan Fellows in business....
from the Sloan Foundation.
Sela gave an Invited Address at the 2002 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 Beijing. Sela gave a plenary talk at the 2002 annual meeting of the Association for Symbolic Logic
Association for Symbolic Logic
The Association for Symbolic Logic is an international organization of specialists in mathematical logic and philosophical logic—the largest such organization in the world. The ASL was founded in 1936, a crucial year in the development of modern logic, and its first president was Alonzo Church...
.
He delivered an AMS Invited Address at the October 2003 meeting 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...
and the 2005 Tarski Lectures at the University of California at Berkeley.
He was also awarded the 2003 Erdős Prize
Erdős Prize
- References :*...
from the Israel Mathematical Union.
Sela also received the 2008 Carol Karp Prize from the Association for Symbolic Logic
Association for Symbolic Logic
The Association for Symbolic Logic is an international organization of specialists in mathematical logic and philosophical logic—the largest such organization in the world. The ASL was founded in 1936, a crucial year in the development of modern logic, and its first president was Alonzo Church...
for his work on the Tarski conjecture and on discovering and developing new connections between model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
and 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...
.
Mathematical contributions
Sela's early important work was his solution in mid-1990s of the isomorphism problemGroup isomorphism problem
In abstract algebra, the group isomorphism problem is the decision problem of determining whether two group presentations present isomorphic groups....
for torsion-free word-hyperbolic groups. The machinery of group actions 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, developed by Eliyahu Rips
Eliyahu Rips
Eliyahu Rips, also Ilya Rips is a Latvian-born Israeli mathematician known for his research in geometric group theory. He became known to the general public following his coauthoring a paper on the Torah Code....
, played a key role in Sela's approach. The solution of the isomorphism problem also relied on the notion of canonical representatives for elements of hyperbolic groups, introduced by Rips and Sela in a joint 1995 paper. The machinery of the canonical representatives allowed Rips and Sela to prove algorithmic solvability of finite systems of equations in torsion-free hyperbolic groups, by reducing the problem to solving equations in 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 where the Makanin-Razborov algorithm can be applied. The technique of canonical representatives was later generalized by Dahmani to the case of relatively hyperbolic group
Relatively hyperbolic group
In mathematics, the concept of a relatively hyperbolic group is an important generalization of the geometric group theory concept of a hyperbolic group...
s and played a key role in the solution of the isomorphism problem for toral relatively hyprbolic groups.
In his work on the isomorphism problem Sela also introduced and developed the notion of a JSJ-decomposition for word-hyperbolic groups, motivated by the notion of a JSJ decomposition
JSJ decomposition
In mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem:The acronym JSJ is for William Jaco, Peter Shalen, and Klaus Johannson...
for 3-manifold
3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
s. A JSJ-decomposition is a representation of a word-hyperbolic group as the fundamental group of a graph of groups which encodes in a canonical way all possible splittings over infinite cyclic 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...
s. The idea of JSJ-decomposition was later extended by Rips and Sela to torsion-free finitely presented groups and this work gave rise a systematic development of the JSJ-decomposition theory with many further extensions and generalizations by other mathematicians. Sela applied a combination of his JSJ-decomposition and 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...
techniques to prove that torsion-free word-hyperbolic groups are Hopfian
Hopfian group
In mathematics, a Hopfian group is a group G for which every epimorphismis an isomorphism. Equivalently, a group is Hopfian if and only if it is not isomorphic to any of its proper quotients. A group G is co-Hopfian if everymonomorphism...
. This result and Sela's approach were later generalized by others to finitely generated 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...
s of hyperbolic groups and to the setting of relatively hyperbolic groups.
Sela's most important work came in early 2000s when he produced a solution to a famous Tarski conjecture. Namely, in a long series of papers, he proved that any two non-abelian finitely generated 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 have the same first-order theory. Sela's work relied on applying his earlier JSJ-decomposition and 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...
techniques as well as developing new ideas and machinery of "algebraic geometry" over 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.
Sela pushed this work further to study first-order theory of arbitrary torsion-free word-hyperbolic groups and to characterize all groups that are elementarily equivalent to (that is, have the same first order theory as) a given torsion-free word-hyperbolic group. In particular, his work implies that if a finitely generated group G is elementarily equivalent to a word-hyperbolic group then G is word-hyperbolic as well.
Sela also proved that the first order theory of a finitely generated 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...
is stable
Stable theory
In model theory, a complete theory is called stable if it does not have too many types. One goal of classification theory is to divide all complete theories into those whose models can be classified and those whose models are too complicated to classify, and to classify all models in the cases...
in the model-theoretic sense, providing a brand-new and qualitatively different source of examples for the stability theory
Stable theory
In model theory, a complete theory is called stable if it does not have too many types. One goal of classification theory is to divide all complete theories into those whose models can be classified and those whose models are too complicated to classify, and to classify all models in the cases...
.
An alternative solution for the Tarski conjecture has been presented by Olga Kharlampovich
Olga Kharlampovich
Olga Kharlampovich is a Canadian mathematician working in the area of group theory. She is a Professor of Mathematics at McGill University, Montreal, Canada, where she has been working since 1990. She is mostly known for her example of a finitely presented 3-step solvable group with...
and Alexei Myasnikov.
The work of Sela on first-order theory of free and word-hyperbolic groups substantially influenced the development 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...
, in particular by stimulating the development and the study of the notion of limit groups and of relatively hyperbolic group
Relatively hyperbolic group
In mathematics, the concept of a relatively hyperbolic group is an important generalization of the geometric group theory concept of a hyperbolic group...
s.
Published work
| year=1995 | journal=Inventiones MathematicaeInventiones Mathematicae
Inventiones Mathematicae, often just referred to as Inventiones, is a mathematical journal published monthly by Springer Verlag. It was founded in 1966.-Editors:...
| volume=120 |issue=3| pages=489–512 }} | year=1995 | journal=Annals of Mathematics
Annals of Mathematics
The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study. It ranks amongst the most prestigious mathematics journals in the world by criteria such as impact factor.-History:The journal began as The Analyst in 1874 and was...
(2) | volume=141 | issue=2 | pages=217–283 | jstor=2118520}} | year=1997 | journal=Geometric and Functional Analysis | volume=7|issue=3 | pages=561–593 | url=http://www.springerlink.com/content/qn2dqfw0xquukbw0/?p=7370f12b59a04dc7bfb4bcaba1999688&pi=6}} | year=1997 | journal=Annals of Mathematics
Annals of Mathematics
The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study. It ranks amongst the most prestigious mathematics journals in the world by criteria such as impact factor.-History:The journal began as The Analyst in 1874 and was...
(2) | volume=146 |issue=1| pages=53–109 | jstor=2951832 }} | year=2001 | journal=Publications Mathématiques de l'IHÉS
Publications Mathématiques de l'IHÉS
Publications Mathématiques de l'IHÉS is a mathematical journal. It is published by Institut des Hautes Études Scientifiques, with the help of the Centre National de la Recherche Scientifique....
| volume=93 | issue=1 | pages=31–105}} | year=2003 | journal=Israel Journal of Mathematics
Israel Journal of Mathematics
Israel Journal of Mathematics is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem .Founded in 1963, as a continuation of the Bulletin of the Research Council of Israel , the journal publishes articles on all areas of mathematics.The journal is indexed by...
| volume=134 | issue=1 | pages=173–254 | url=http://www.springerlink.com/content/v27x52636w4l2981/}} | year=2006 | journal=Geometric and Functional Analysis| volume=16|issue=3 | pages=707–730 | url=http://www.springerlink.com/content/r53x383mj14206v1/?p=2d33d0052c1749aeb7b127871d580675&pi=2}}
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...
- Stable theoryStable theoryIn model theory, a complete theory is called stable if it does not have too many types. One goal of classification theory is to divide all complete theories into those whose models can be classified and those whose models are too complicated to classify, and to classify all models in the cases...
- Free groupFree groupIn 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...
- Word-hyperbolic group
- Group isomorphism problemGroup isomorphism problemIn abstract algebra, the group isomorphism problem is the decision problem of determining whether two group presentations present isomorphic groups....
- Real treeReal treeA 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 - JSJ decompositionJSJ decompositionIn mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem:The acronym JSJ is for William Jaco, Peter Shalen, and Klaus Johannson...