Zlil Sela

Zlil Sela is an Isareli 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 groups.

Biographical data

Sela received his PhD in 1991 from the Hebrew University of Jerusalem, 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. [http://www.columbia.edu/cu/record/archives/vol21/vol21_iss27/record2127.18.html Faculty Members Win Fellowships] Columbia University Record, May 15, 1996, Vol. 21, No. 27.] While at Columbia, Sela won the Sloan Fellowship from the Sloan Foundation [ [http://www.ams.org/notices/199607/people.pdf Sloan Fellowships Awarded] Notices of the American Mathematical Society, vol. 43 (1996), no. 7, pp. 781–782] .

Sela gave an Invited Address at the 2002 International Congress of Mathematicians in Beijing.Z. Sela. "Diophantine geometry over groups and the elementary theory of free and hyperbolic groups." Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pp. 87 92, Higher Ed. Press, Beijing, 2002. ISBN: 7-04-008690-5] He also gave a plenary talk at the 2002 annual meeting of the Association for Symbolic Logic [ [http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdffirstpage_1&handle=euclid.bsl/1046288726 The 2002 annual meeting of the Association for Symbolic Logic.] Bulletin of Symbolic Logic, vol. 9 (2003), pp. 51–70] .

Sela delivered an AMS Invited Address at the October 2003 meeting of the American Mathematical Society [ [http://www.ams.org/notices/200309/mtgs.pdf AMS Meeting at Binghamton, New York.] Notices of the American Mathematical Society, vol. 50 (2003), no. 9, p. 1174] and the 2005 Tarski Lectures at the University of California at Berkeley [ [http://math.berkeley.edu/index.php?module=announce&ANN_user_op=view&ANN_id=37 2005 Tarski Lectures.] Department of Mathematics, University of California at Berkeley. Accessed September 14, 2008.] .

Sela was awarded the 2003 Erdős Prize from the Israel Mathematical Union. [ [http://imu.org.il/erdos_prize.html#english Erdős Prize.] Israel Mathematical Union. Accessed September 14, 2008] . Sela 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 [ [http://www.aslonline.org/Karp_recipients.html Karp Prize Recipients.] Association for Symbolic Logic. Accessed September 13, 2008] .

Mathematical contributions

Sela's early important work was his solutionZ. Sela. "The isomorphism problem for hyperbolic groups. I."
Annals of Mathematics (2), vol. 141 (1995), no. 2, pp. 217–283.] in mid-1990s of the isomorphism problem for torsion-free word-hyperbolic groups. The machinery of group actions on real trees, developed by Eliyahu Rips, played a key role in Sela's approach. In his work on the isomorphism problem Sela also introduced and developed the notion of a JSJ-decomposition for word-hyperbolic groups [Z. Sela. Structure "and rigidity in (Gromov) hyperbolic groups and discrete groups in rank $1$ Lie groups. II." Geometric and Functional Analysis, vol. 7 (1997), no. 3, pp. 561–593 ] , motivated by the notion of a JSJ decomposition for 3-manifolds. 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 subgroups. The idea of JSJ-decomposition was later extended by Rips and Sela to torsion-free finitely presented groups [E. Rips, and Z. Sela. "Cyclic splittings of finitely presented groups and the canonical JSJ decomposition." Annals of Mathematics (2), vol. 146 (1997), no. 1, pp. 53–109 ] and this work gave rise a systematic development of the JSJ-decomposition theory with many further extensions and generalizations by other mathematicians [M. J. Dunwoody, and M. E. Sageev. "JSJ-splittings for finitely presented groups over slender groups." Inventiones Mathematicae, vol. 135 (1999), no. 1, pp. 25 44] [P. Scott and G. A. Swarup. "Regular neighbourhoods and canonical decompositions for groups." Electronic Research Announcements of the American Mathematical Society, vol. 8 (2002), pp. 20–28] [B. H. Bowditch. "Cut points and canonical splittings of hyperbolic groups." Acta Mathematica, vol. 180 (1998), no. 2, pp. 145–186] [K. Fujiwara, and P. Papasoglu, "JSJ-decompositions of finitely presented groups and complexes of groups." Geometric and Functional Analysis, vol. 16 (2006), no. 1, pp. 70–125] .

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 [Z. Sela. "Diophantine geometry over groups. I. Makanin-Razborov diagrams." Publications Mathématiques. Institut de Hautes Études Scientifiques, vol. 93 (2001), pp. 31–105] [Z. Sela. "Diophantine geometry over groups. II. Completions, closures and formal solutions." Israel Journal of Mathematics, vol. 134 (2003), pp. 173–254] [Z. Sela. Diophantine "geometry over groups. III. Rigid and solid solutions." Israel Journal of Mathematics, vol. 147 (2005), pp. 1–73 ] [Z. Sela. "Diophantine geometry over groups. IV. An iterative procedure for validation of a sentence." Israel Journal of Mathematics, vol. 143 (2004), pp. 1–130] [Z. Sela. "Diophantine geometry over groups. V₁. Quantifier elimination. I." Israel Journal of Mathematics, vol. 150 (2005), pp. 1–197 ] [Z. Sela. "Diophantine geometry over groups. V₂. Quantifier elimination. II." Geometric and Functional Analysis, vol. 16 (2006), no. 3, pp. 537–706 ] [Z. Sela. "Diophantine geometry over groups. VI. The elementary theory of a free group." Geometric and Functional Analysis, vol. 16 (2006), no. 3, pp. 707–730] , he proved that any two non-abelian finitely generated free groups 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 groups.

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 [O. Kharlampovich, and A. Myasnikov. "Tarski's problem about the elementary theory of free groups has a positive solution." Electronic Research Announcements of the American Mathematical Society, vol. 4 (1998), pp. 101–108 ] [O. Kharlampovich, and A. Myasnikov. "Implicit function theorem over free groups." Journal of Algebra, vol. 290 (2005), no. 1, pp. 1–203 ] [O. Kharlampovich, and A. Myasnikov. "Algebraic geometry over free groups: lifting solutions into generic points." Groups, languages, algorithms, pp. 213–318, Contemporary Mathematics, vol. 378, American Mathematical Society, Providence, RI, 2005 ] [O. Kharlampovich, and A. Myasnikov. "Elementary theory of free non-abelian groups." Journal of Algebra, vol. 302 (2006), no. 2, pp. 451–552] , although Sela's solution remains better understood. 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 groups [Frédéric Paulin."Sur la théorie élémentaire des groupes libres (d'après Sela)." Astérisque No. 294 (2004), pp. 63–402 ] .

elected publications

ee also

*Geometric group theory
*Stable theory
*Free group
*Word-hyperbolic group
*Group isomorphism problem

References

External links

* [http://www.ma.huji.ac.il/~zlil/ Zlil Sela's webpage at the Hebrew University]
* [http://www.genealogy.math.ndsu.nodak.edu/id.php?id=118496 Zlil Sela at the Mathematics Genealogy Project]