Vladimir Turaev Explained

Vladimir Georgievich Turaev (Владимир Георгиевич Тураев, born in 1954) is a Russian mathematician, specializing in topology.

Turaev received in 1979 from the Steklov Institute of Mathematics his Candidate of Sciences degree (PhD) under Oleg Viro. Turaev was a professor at the University of Strasbourg and then became a professor at Indiana University. In 2016, he was elected a Fellow of the American Mathematical Society.

Turaev's research deals with low-dimensional topology, quantum topology, and knot theory and their interconnections with quantum field theory. In 1991, Reshetikhin and Turaev published a mathematical construction of new topological invariants of compact oriented 3-manifolds and framed links in these manifolds, corresponding to a mathematical implementation of ideas in quantum field theory published by Witten;^[1] the invariants are now called Witten-Reshetikhin-Turaev (or Reshetikhin-Turaev) invariants. In 1992, Turaev and Viro introduced a new family of invariants for 3-manifolds by using state sums computed on triangulations of manifolds;^[2] these invariants are now called Turaev-Viro invariants.

In 1990, Turaev was an Invited Speaker with talk State sum models in low dimensional topology at the ICM in Kyōto.^[3] In 2016, he shared, with Alexis Virelizier, the Ferran Sunyer i Balaguer Prize for their monograph Monoidal categories and topological field theory.

Selected publications

Articles

• The Yang-Baxter equation and invariants of links. Inventiones Mathematicae. 92. 3. 1988. 527–553. 10.1007/BF01393746. Turaev. V. G.. 1988InMat..92..527T. 122703873.
• with Nicolai Reshetikhin: 10.1007/BF02096491. Ribbon graphs and their invariants derived from quantum groups. Communications in Mathematical Physics. 127. 1. 1990. 1–26. Reshetikhin. N. Y.. Turaev. V. G.. 1990CMaPh.127....1R. 122572484.
• Skein quantization of Poisson algebras of loops on surfaces. Annales Scientifiques de l'École Normale Supérieure. 24. 6. 635–704. 1991. 10.24033/asens.1639. Turaev. Vladimir G.. free.

Books

• Quantum invariants of Knots and 3-Manifolds, de Gruyter 1994;^[4] Book: 2nd edition. 2010. ;^[5] Book: 3rd edition. 2016. 9783110435221. Turaev. Vladimir G.. Walter de Gruyter GmbH & Co KG .
• with Christian Kassel and Marc Rosso: Quantum groups and knot invariants, SMF (Panoramas et Synthèses) 1997
• as editor with Anatoly Vershik: Topology, ergodic theory, real algebraic geometry - Rokhlin´s memorial, American Mathematical Society 2001
• Introduction to combinatorial torsions, Birkhäuser 2001
• Torsions of 3-dimensional manifolds, Birkhäuser 2002
• with Christian Kassel: Braid Groups, Springer 2008, ^{[6] [7]}
• Homotopy quantum field theory, European Mathematical Society 2010^[8]
• with Alexis Virelizier: Monoidal Categories and Topological Field Theory, Birkhäuser 2015

See also

• Reshetikhin–Turaev invariant

External links

• Vladimir Touraev, Mathematics Department, Indiana University
• mathnet.ru

Notes and References

1. Reshetikhin, N.. Turaev, V. G.. Invariants of 3-manifolds via link polynomials and quantum groups. Inventiones Mathematicae. 103. 3. 1991. 547–597. 10.1007/BF01239527. 1991InMat.103..547R. 123376541.
2. Turaev, V. G.. Viro, O. Y.. State sum invariants of 3-manifolds and quantum 6j-symbols. Topology. 31. 4. 1992. 865–902. 10.1016/0040-9383(92)90015-A.
3. Book: Turaev, Vladimir G.. State sum models in low-dimensional topology. In: Proc. ICM Kyoto. 1. 689–698. 1990. https://www.researchgate.net/publication/265451900.
4. Review of Quantum invariants of knots and 3-manifolds by V. G. Turaev. Kuperberg, Greg. Greg Kuperberg. Bull. Amer. Math. Soc.. 33. 1996. 107–110. 10.1090/S0273-0979-96-00621-0. free.
5. Web site: Taylor, Scott. 10 June 2011. Review of Quantum Invariants of Knots and 3-Manifolds by Vladimir G. Turaev. MAA Reviews, Mathematical Association of America.
6. Review of 2 books: Braid groups by Christian Kassel and Vladimir Turaev (with the graphical assistance of Olivier Douane); Ordering braids by Patrick Dehornoy. Birman, Joan S.. Joan Birman. Bull. Amer. Math. Soc.. 48. 2011. 137–146. 10.1090/S0273-0979-2010-01305-7. free.
7. Web site: Review of Braid Groups by Christian Kassel and Vladimir Turaev. Taylor, Scott. 13 October 2011. MAA Reviews, Mathematical Association of America.
8. Review of Homotopy quantum field theory by Vladimir Turaev. Porter, Timothy. Bull. Amer. Math. Soc.. 49. 2. 2012. 337–345. 10.1090/S0273-0979-2011-01351-9. free.