Georgia Benkart
Georgia Benkart in 2010
Born
Georgia McClure Benkart

(1947-12-30)December 30, 1947
Youngstown, Ohio, U.S.
DiedApril 29, 2022(2022-04-29) (aged 74)
Madison, Wisconsin, U.S.
NationalityAmerican
Education
Known forClassification of simple modular Lie algebras
Awards
Scientific career
FieldsLie algebras, Representation Theory, Combinatorics
InstitutionsUniversity of Wisconsin–Madison
Doctoral advisorNathan Jacobson

Georgia McClure Benkart (December 30, 1947 – April 29, 2022)[1][2][3] was an American mathematician who was known for her work in the structure and representation theory of Lie algebras and related algebraic structures. She published over 130 journal articles[4][5] and co-authored three American Mathematical Society memoirs[6][7][8] in four broad categories: modular Lie algebras; combinatorics of Lie algebra representations; graded algebras and superalgebras; and quantum groups and related structures.

Education and career

Benkart received her BS degree summa cum laude from the Ohio State University[4] in 1970 and an MPhil in mathematics from Yale University in 1973. She completed her doctoral work at Yale under Nathan Jacobson and wrote a dissertation entitled Inner Ideals and the Structure of Lie Algebras. She was awarded a PhD in mathematics from the Yale University in 1974.[9]

Upon completing her doctoral degree, Benkart began her long career at the University of Wisconsin–Madison, first as a MacDuffee Instructor and eventually as a E. B. Van Vleck Professor of Mathematics until she retired from teaching in 2006.[10]

She held visiting positions at the Mathematical Sciences Research Institute in Berkeley, California, the Institute for Advanced Study in Princeton, New Jersey, the Aspen Center for Physics, and the University of Virginia.[5] During her career, Benkart delivered over 350 invited talks including 3 plenary lectures at the Joint Mathematics Meetings and the Emmy Noether Lecture at the International Congress of Mathematicians in Seoul, South Korea in 2014.[4]

Personal life

Benkart was born on December 30, 1947, in Youngstown, Ohio,[3] to George Benkart II and Florence K. Benkart.[2] Her father served in the Army Corps of Engineers and her mother was a teacher in Youngstown's "ethnically rich south side."[11]

Benkart died on April 29, 2022, aged 74, from undisclosed causes in Madison, Wisconsin. Her survivors included her sister, Paula Kaye Benkart.[1][11]

Research

Benkart made a contribution to the classification of simple modular Lie algebras. Her work with J. Marshall Osborn on toroidal rank-one Lie algebras[12] became one of the building blocks of the classification.[13] The complete description of Hamiltonian Lie Algebras (with Gregory, Osborn, Strade, Wilson) can stand alone, and also has applications in the theory of pro-p groups.[13]

In 2009, she published, jointly with Thomas Gregory and Alexander Premet,[8] the first complete proof of the recognition theorem for graded Lie algebras in characteristics at least 5.[14]

In the early 1990s, Benkart and Efim Zelmanov started to work on classification of root-graded Lie algebras and intersection matrix algebras. The latter were introduced by Peter Slodowy in his work on singularities. Berman and Moody recognized that these algebras (generalizations of affine Kac–Moody algebras) are universal root graded Lie algebras and classified them for simply laced root systems. Benkart and Zelmanov tackled the remaining cases involving the Freudenthal magic square and extended this square to exceptional Lie superalgebras.[13]

Later Benkart extended these results in two directions. In a series of papers with Alberto Elduque she developed the theory of root graded Lie superalgebras.[15] In a second series of works with Bruce Allison, Arturo Pianzola, Erhard Neher, et al. she determined the universal central covers of these algebras.[16]

One of the pillars of the representation theory of quantum groups (and applications to combinatorics) is Masaki Kashiwara's theory of crystal bases. These are highly invariant bases which are well suited for decompositions of tensor products. In a paper with Seok-Jin Kang and Kashiwara, Benkart extended the theory of crystal bases to quantum superalgebras.[13][17]

Benkart's work on noncommutative algebras related to algebraic combinatorics became a basic tool in the construction of tensor categories.[13]

Service to the profession

Benkart served on several editorial boards including the boards of the American Mathematical Society for Surveys and Monographs and Abstracts,[18] Communications in Algebra,[19] and the Journal of Algebra.[20] She served as the associate secretary of the American Mathematical Society for the Central Section from 2010 to 2020,[21] and was a member of the governing council in 1995 and from 2010 to 2021.[22] Benkart was active in the Association for Women in Mathematics(AWM) over many years, served as chair of the membership portfolio and the nominating committee, and helped plan the first AWM Research Symposium, which was also a celebration of the 40th anniversary of AWM.[23]

Awards and honors

Benkart received a Woodrow Wilson Fellowship from the Woodrow Wilson National Fellowship Foundation. Her work at Wisconsin was recognized by a Romnes Fellowship in 1985, a Distinguished Teaching Award in 1987, and a WARF Mid-Career Faculty Research Award in 1996.[24] In 2008 the University of California Lie Groups and Lie Algebras meeting was held in Benkart's honor. Her talks and lectures include two invited lectures at the Joint Mathematics Meetings[25] and a plenary lecture at a meeting of the Canadian Mathematical Society.[26]

In 2000–2002 Benkart was named a Polya Lecturer[27] by the Mathematical Association of America. She was elected a Fellow of the American Mathematical Society (AMS)[28] in the inaugural class of 2013.

She was elected and served as president of the Association for Women in Mathematics from 2009 to 2011.[29] In 2014 she was selected to deliver the AWM-AMS Noether Lecture.[30] The title of her talk was Walking on Graphs the Representation Theory Way.[31]

In 2014 at the International Congress of Mathematicians held in Seoul, she delivered the ICM Emmy Noether Lecture.[32]

She was selected as a fellow of the Association for Women in Mathematics in the inaugural 2018 class.[33]

In 2022, a tribute to her contributions to her field, "Gems from the Work of Georgia Benkart", appeared in the Notices of the American Mathematical Society.[13]

The American Mathematical Society[34] and the University of Wisconsin–Madison mathematics department[35] posted remembrances of Benkart.

Selected publications

  • with Daniel Britten, Frank Lemire: Stability in Modules for Classical Lie Algebras: A Constructive Approach. Memoirs of the American Mathematical Society. Vol. 85. Providence, R.I.: American Mathematical Society. 1990. ISBN 9780821861530. MR 1010997.
  • with Seok-Jin Kang, Masaki Kashiwara: Benkart, Georgia; Kang, Seok-Jin; Kashiwara, Masaki (2000). "Crystal bases for the quantum superalgebra Uq(𝔤𝔩(m,n))". J. Amer. Math. Soc. 13 (2): 295–331. doi:10.1090/S0894-0347-00-00321-0. MR 1694051.
  • with Bruce Allison, Yun Gao: Lie algebras graded by the root systems BCr, r ≥ 2. Memoirs of the American Mathematical Society. Vol. 158. American Mathematical Society. 2002. ISBN 9780821864739. MR 1902499.
  • with Thomas Gregory, Alexander Premet: Benkart, Georgia; Gregory, Thomas; Premet, Alexander (2009). The recognition theorem for graded Lie algebras in prime characteristic. Memoirs of the American Mathematical Society. Vol. 197. American Mathematical Society. arXiv:math/0508373. ISBN 9780821842263. MR 2488391.

References

  1. 1 2 "In Memoriam: Georgia Benkart". Department of Mathematics. April 30, 2022. Retrieved May 2, 2022.
  2. 1 2 "1950 Census". United States Archives. 1950. Retrieved May 2, 2022.
  3. 1 2 "Georgia Benkart". OfficialUSA.com Records. December 30, 1947. Retrieved May 2, 2022.
  4. 1 2 3 "Georgia Benkart". Ohio State University Department of Mathematics. Retrieved May 5, 2022.
  5. 1 2 "Personal Profile of Dr. Georgia M. Benkart". Mathematical Sciences Research Institute. Retrieved May 5, 2022.
  6. ↑ Allison, Bruce; Benkart, Georgia; Gao, Yun (2002). Lie algebras graded by the root systems BCr, r ≥ 2. Memoirs of the American Mathematical Society. Vol. 158. American Mathematical Society. ISBN 978-0-8218-2811-3. MR 1902499.
  7. ↑ Benkart, Georgia M.; Britten, Daniel; Lemire, Frank (May 1990). Stability in Modules for Classical Lie Algebras: A Constructive Approach. Memoirs of the American Mathematical Society. Vol. 85. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-2492-4. MR 2488391. Retrieved June 10, 2014.
  8. 1 2 Benkart, Georgia; Gregory, Thomas; Premet, Alexander (2009). The recognition theorem for graded Lie algebras in prime characteristic. Memoirs of the American Mathematical Society. Vol. 197. American Mathematical Society. ISBN 978-0-8218-4226-3.
  9. ↑ "Georgia Benkart". The Mathematics Genealogy Project. Retrieved April 30, 2022.
  10. ↑ "A Brief History of Women Mathematicians at Wisconsin". University of Wisconsin Math Department. Archived from the original on May 29, 2014. Retrieved May 21, 2014.
  11. 1 2 "Georgia Bankart, Youngstown, Ohio". MyValleyTributes. May 4, 2022. Retrieved May 5, 2022.
  12. ↑ Benkart, Georgia; J. Marshall Osborn (1988). "Toral rank one Lie algebras". Journal of Algebra. 115 (1): 238–250. doi:10.1016/0021-8693(88)90293-1.
  13. 1 2 3 4 5 6 "Gems from the Work of Georgia Benkart" (PDF). Notices of the American Mathematical Society. 69 (3): 375–384. March 2022. Retrieved April 30, 2022.
  14. ↑ Review of The recognition theorem for graded Lie algebras in prime characteristic by Murray R. Bremner (2009); MR2488391
  15. ↑ Martínez, Consuelo; Zelmanov, Efim I. (June 27, 2003). "Lie superalgebras graded by P ( n ) and Q ( n )". Proceedings of the National Academy of Sciences. 100 (14): 8130–8137. Bibcode:2003PNAS..100.8130M. doi:10.1073/pnas.0932706100. ISSN 0027-8424. PMC 166194. PMID 12832611.
  16. ↑ Benkart, Georgia; Neher, Erhard (2006). "The centroid of extended affine and root graded Lie algebras". J. Pure Appl. Algebra. 205 (1): 117–145. arXiv:math/0502561. doi:10.1016/j.jpaa.2005.06.007. MR 2193194. S2CID 14256793.
  17. ↑ Benkart, Georgia; Kang, Seok-Jin; Kashiwara, Masaki (2000). "Crystal bases for the quantum superalgebra Uq(𝔤𝔩(m,n))". J. Amer. Math. Soc. 13 (2): 295–331. doi:10.1090/S0894-0347-00-00321-0. MR 1694051.
  18. ↑ "Mathematical Surveys and Monographs Editorial Committee Past Members". American Mathematical Society. Retrieved May 6, 2022.
  19. ↑ "Editorial board". Communications in Algebra. 15 (6): ebi. January 1, 1987. doi:10.1080/00927878708823457. ISSN 0092-7872. Retrieved May 6, 2022.
  20. ↑ "Georgia Benkart". Mathematical Association of America. Retrieved May 6, 2022.
  21. ↑ "American Mathematical Society Council | February 1, 2017 - January 31, 2018". Archived from the original (web.archive.org) on February 4, 2017. Retrieved May 6, 2022. Professor Georgia Benkart Associate Secretary (Central Section) Term: February 1, 2010 - January 31, 2020 University of Wisconsin, Madison
  22. ↑ "Council Past Members". American Mathematical Society. Retrieved May 6, 2022.
  23. ↑ "Committees". Association for Women in Mathematics. Retrieved May 6, 2022.
  24. ↑ "University of Wisconsin Honors and Awards". University of Wisconsin. Archived from the original on January 21, 2020. Retrieved May 23, 2014.
  25. ↑ "Joint Mathematics Meeting AMS-MAA Invited Address". American Mathematical Society. Retrieved May 23, 2014.
  26. ↑ "2001 CMS Summer Meeting". Photos from the 2001 CMS Summer Meeting. Canadian Mathematical Society. Retrieved June 10, 2014.
  27. ↑ "Pólya Lectures". Mathematical Association of America. Retrieved May 21, 2014.
  28. ↑ "List of Fellows of the American Mathematical Society". American Mathematical Society. Retrieved May 21, 2014.
  29. ↑ "President's Report". AWM Newsletter. 39 (2): 1–5. March–April 2009. Retrieved May 27, 2014.
  30. ↑ "List of Noether Lecturers". Retrieved January 26, 2019.
  31. ↑ "Noether Lecture Profile". Association for Women in Mathematics. Retrieved January 26, 2019.
  32. ↑ "ICM Emmy Noether Lecture". the organizing committee of Seoul ICM 2014. Archived from the original on October 6, 2014. Retrieved October 2, 2014.
  33. ↑ "2018 Inaugural Class of AWM Fellows". Association for Women in Mathematics. Retrieved November 23, 2020.
  34. ↑ "Remembering Georgia Benkart (1947 - 2022)". American Mathematical Society. Retrieved May 4, 2022.
  35. ↑ "Georgia Benkart Remembrances". University of Wisconsin – Madison Mathematics. Retrieved May 4, 2022.
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