Georgia Benkart

Georgia Benkart
Born	1949 Youngstown, Ohio
Nationality	United States
Fields	Lie algebras, Representation Theory, Combinatorics
Institutions	University of Wisconsin–Madison
Alma mater	B.S., Ohio State University, 1970 Ph.D., Yale University, 1974
Doctoral advisor	Nathan Jacobson
Known for	Classification of simple modular Lie algebras

Georgia McClure Benkart is an American mathematician who is known for her work in the structure and representation theory of Lie algebras and related algebraic structures. She has published over 100 journal articles^[1] and co-authored 3 American Mathematical Society Memoirs^[2][3][4] in four broad categories: modular Lie algebras; combinatorics of Lie algebra representations; graded algebras and superalgebras; and quantum groups and related structures. Benkart's role as a teacher has led to her work in mentoring 22 doctoral students.

Research

Benkart made a contribution to the classification of simple modular Lie algebras. Her work with J. M. Osborn on toroidal rank-one Lie algebras^[5] became one of the building blocks of the classification. 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.

In 2009 she published, jointly with T. Gregory and A. Premet,^[4] the first complete proof of the recognition theorem for graded Lie algebras in characteristics at least 5.^[6]

In the early 90s Benkart and Efim Zelmanov started to work on classification of root-graded Lie algebras and intersection matrix algebras. The latter were introduced by P. 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 so-called Magic Freudenthal–Tits “Square” and extended this square to exceptional Lie superalgebras.

Later Benkart extended these results in two directions. In a series of papers with A. Elduque she developed the theory of root graded Lie superalgebras. In a second series of works with B. Allison, A. Pianzola, E. Neher, et al. she determined the universal central covers of these algebras.

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

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

Education and career

Benkart received her B.S. from the Ohio State University in 1970 and an M. Phil. 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 Ph.D. in Mathematics from the Yale University in 1974.

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.^[7] 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.

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.^[8] In 2008 the University of California Lie Groups and Lie Algebras meeting was held in Benkart's honor. She has given numerous talks and series of lectures throughout the U.S., Canada, France, Germany, Hong Kong, Korea, Mexico, and Spain, including two invited lectures at the Joint Mathematics Meetings^[9] and a plenary lecture at a meeting of the Canadian Mathematical Society.^[10]

In 2000–2002 Benkart was named a Polya Lecturer^[11] by the Mathematical Association of America. She was elected a Fellow of the American Mathematical Society (AMS)^[12] in the inaugural class of 2013.

She has served on the editorial boards of the American Mathematical Society for Surveys and Monographs and Abstracts,^[13] the Journal of Algebra,^[14] the Korean Mathematical Colloquium, the Nova Journal of Algebra and Geometry, Communications in Algebra, and Algebras, Groups, and Geometries. She served as the Associate Secretary of the American Mathematical Society for the Central Section from 2010–2016.^[15]

Benkart has been active in the Association for Women in Mathematics (AWM). She was elected and served as President of the AWM from 2009–2011.^[16] In 2014 she was selected to deliver the AWM-AMS Noether Lecture.^[17] The title of her talk was Walking on Graphs the Representation Theory Way.^[18]

In 2014 at the International Congress of Mathematicians held in Seoul, she delivered the ICM Emmy Noether Lecture.^[19]

Selected publications

• with Daniel Britten, Frank Lemire: Stability in Modules for Classical Lie Algebras: A Constructive Approach. Memoirs of the American Mathematical Society. 85. Providence, R.I.: American Mathematical Society. 1990. 
• with Bruce Allison, Yun Gao: Lie algebras graded by the root systems BC_r, r ≥ 2. Memoirs of the American Mathematical Society. 158. American Mathematical Society. 2002. 
• with Thomas Gregory, Alexander Premet: The recognition theorem for graded Lie algebras in prime characteristic. Memoirs of the American Mathematical Society. 197. American Mathematical Society. 2009. 

References