# Ranee Brylinski

Ranee Kathryn Brylinski (née Gupta, born January 28, 1957)[1] is an American mathematician known for her research in representation theory and quantum logic gates. Formerly a professor of mathematics at Pennsylvania State University, she left academia in 2003 to found the mathematical consulting company Brylinski Research with her husband, Jean-Luc Brylinski.

## Education and career

Brylinski was born in Detroit, Michigan.[1] She graduated from Princeton University in 1977,[1][2] and completed her Ph.D. at the Massachusetts Institute of Technology (MIT) in 1981. Her dissertation, Abelian Algebras and Adjoint Orbits, was supervised by Steven Kleiman.[1][2][3]

After a year as an NSF Mathematical Sciences Postdoctoral Fellow at MIT, she joined the faculty at Brown University as Tamarkin Assistant Professor of Mathematics in 1982. She moved from Brown to Pennsylvania State University in 1991. At Pennsylvania State, she was co-director of the Center for Geometry and Mathematical Physics.[1][2]

## Contributions

The Brylinski–Kostant filtration of weight spaces is named after her. She originally developed this filtration in 1989, motivated by earlier work of Bertram Kostant.[4][LWS] She is also known for Brylinski's theorem, a theorem from her dissertation on the closures of orbits of algebraic groups.[5]

Another result, also called "Brylinski's theorem", comes from a paper written jointly by Brylinski and her husband, characterizing universal quantum logic gates.[6][UQG]

## Recognition

Brylinski won a Sloan Research Fellowship in 1990. In 1998, she won the G. de B. Robinson Award [de] of the Canadian Mathematical Society[1][2] for her work on quantization of algebraic groups.[Q4D]

## Selected publications

With Goong Chen, Brylinski is the editor of the book Mathematics of Quantum Computation (Chapman & Hall/CRC, 2002). She is also one of the editors of Lie Theory and Geometry: In Honor of Bertram Kostant (Springer, 1994) and Advances in Geometry (Springer, 1999).

Her research papers include:

 LWS. Brylinski, Ranee Kathryn (1989), "Limits of weight spaces, Lusztig's ${\displaystyle q}$-analogs, and fiberings of adjoint orbits", Journal of the American Mathematical Society, 2 (3): 517–533, doi:10.2307/1990941, MR 0984511.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"\"""\"""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
 Q4D. Brylinski, Ranee (1997), "Quantization of the 4-dimensional nilpotent orbit of ${\displaystyle \mathrm {SL} (3,\mathbb {R} )}$", Canadian Journal of Mathematics, 49 (5): 916–943, doi:10.4153/CJM-1997-048-0, MR 1604118
 UQG. Brylinski, Jean-Luc; Brylinski, Ranee (2002), "Universal quantum gates", Mathematics of Quantum Computation, Comput. Math. Ser., Boca Raton, FL: Chapman & Hall/CRC, pp. 101–116, MR 2007944

## References

1. Curriculum vitae (PDF), August 29, 2001, archived from the original (PDF) on 2004-08-28, retrieved 2018-10-06
2. ^ a b c d The 3rd G de B Robinson Award: Biographical Information (PDF), Canadian Mathematical Society, 1998, retrieved 2018-10-06
3. ^
4. ^ Hague, Chuck (2009), "Cohomology of flag varieties and the Brylinski-Kostant filtration", Journal of Algebra, 321 (12): 3790–3815, doi:10.1016/j.jalgebra.2009.04.001, MR 2517813
5. ^ Kostant, Bertram; Wallach, Nolan (2009), "On a theorem of Ranee Brylinski", Symmetry in Mathematics and Physics, Contemporary Mathematics, 490, Providence, RI: American Mathematical Society, pp. 105–132, doi:10.1090/conm/490/09590, MR 2555973
6. ^ Zhang, Yong; Kauffman, Louis H.; Ge, Mo-Lin (2005), "Yang-Baxterizations, universal quantum gates and Hamiltonians", Quantum Information Processing, 4 (3): 159–197, doi:10.1007/s11128-005-7655-7, MR 2187341