Nolan Russell Wallach (born August 3, 1940) is a mathematician known for work in the representation theory of reductive algebraic groups. He is the author of the 2-volume treatise Real Reductive Groups.[1]
Education and career
Wallach did his undergraduate studies at the University of Maryland, graduating in 1962.[2] He earned his Ph.D. from Washington University in St. Louis in 1966, under the supervision of Jun-Ichi Hano.[2][3]
He became an instructor and then lecturer at the University of California, Berkeley. At Rutgers University he became in 1969 an assistant professor, in 1970 an associate professor, in 1972 a full professor, and in 1986 the Hermann Weyl Professor of Mathematics. In 1989 he became a professor at the University of California, San Diego, where he is now a professor emeritus. From 1997 to 2003 he was an associate editor of the Annals of Mathematics and from 1996 to 1998 an associate editor of the Bulletin of the American Mathematical Society.
Wallach was a Sloan Fellow from 1972 to 1974. In 1978 he was an Invited Speaker with talk The spectrum of compact quotients of semisimple Lie groups[4] at the International Congress of Mathematicians in Helsinki. He was elected in 2004 a Fellow of the American Academy of Arts and Sciences and in 2012 a Fellow of the American Mathematical Society.[2][5] His doctoral students include AMS Fellow Alvany Rocha.[3] He has supervised more than 18 Ph.D. theses.[3] Besides representation theory, Wallach has also published more than 150 papers in the fields of algebraic geometry, combinatorics, differential equations, harmonic analysis, number theory, quantum information theory, Riemannian geometry, and ring theory.[6]
Selected publications
Articles
with Michel Cahen: Lorentzian symmetric spaces, Bull. Amer. Math. Soc., vol. 76, no. 3, 1970, pp. 585–591. MR0267500 doi:10.1090/S0002-9904-1970-12448-X
with M. do Carmo: Minimal immersions of spheres into spheres, Annals of Mathematics, vol. 93, 1971, pp. 43–62. JSTOR 1970752
Compact homogeneous Riemannian manifolds with strictly positive curvature, Annals of Mathematics, vol. 96, 1972, pp. 277–295. doi:10.2307/1970789
with S. Aloff: An infinite number of distinct 7-manifolds admitting positively curved Riemannian structures, Bull. Amer. Math. Soc., vol. 81, 1975, pp. 93–97 MR0370624 doi:10.1090/S0002-9904-1975-13649-4
with D. DeGeorge: Limit formulas for multiplicities in L2(Γ \G), Annals of Mathematics, vol. 107, 1978, pp. 133–150. doi:10.2307/1971140
with Roe Goodman: Classical and quantum mechanical systems of Toda lattice type, 3 Parts, Comm. Math. Phys., Part I, vol. 83, 1982, pp. 355–386, MR0649809 doi:10.1007/BF01213608; Part II, vol. 94, 1984, pp. 177–217, MR0761793 doi:10.1007/BF01209301; Part III, vol. 105, 1986, pp. 473–509, MR0848652 doi:10.1007/BF01205939
with A. Rocha-Caridi: Characters of irreducible representations of the Lie algebra of vector fields on the circle, Invent. Math., vol. 72, 1983, pp. 57–75 doi:10.1007/BF01389129
with A. Rocha-Caridi: Highest weight modules over graded Lie algebras: resolutions, filtrations and character formulas, Transactions of the American Mathematical Society, vol. 277, 1983, pp. 133–162 MR690045 doi:10.1090/S0002-9947-1983-0690045-3
with T. Enright, R. Howe: A classification of unitary highest weight modules, in: Representation theory of reductive groups (Park City, Utah 1982), Progress in Mathematics 40, Birkhäuser 1983, pp. 97–143 doi:10.1007/978-1-4684-6730-7_7
with A. Rocha-Caridi: Characters of irreducible representations of the Virasoro-Algebra, Mathematische Zeitschrift, vol. 185, 1984, pp. 1–21
Invariant differential operators on a reductive Lie algebra and Weyl group representations, J. Amer. Math. Soc., vol. 6, no. 4, 1993, pp. 779–816. doi:10.2307/2152740
Quantum computing and entanglement for mathematicians, in: Representation theory and complex analysis, pp. 345–376, Lecture Notes in Math. No. 1931, Springer 2008 doi:10.1007/978-3-540-76892-0_6
with G. Gour: Classification of multipartite entanglement of all finite dimensionality, Phys. Rev. Lett., vol. 111, 2013, 060502 doi:10.1103/PhysRevLett.111.060502
Books
Harmonic analysis on homogeneous spaces, New York: Marcel Dekker 1973
Symplectic geometry and Fourier analysis, Brookline: Math. Science Press 1977[7]
Real Reductive Groups, 2 vols., Academic Press 1988,[8] 1992[9]
with Roe Goodman: Representations and invariants of the classical groups, Cambridge University Press 1998;[10] 1st pbk edition, 1999; reprint with corrections, 2003
with Armand Borel: Continuous cohomology, discrete subgroups and representations of reductive groups, Annals of Mathematical Studies 94, 1980, 2nd edition, American Mathematical Society 2013
with Roe Goodman: Symmetry, representations, and invariants, Graduate Texts in Mathematics, Springer 2009[11]
Geometric Invariant Theory: Over the Real and Complex Numbers, Universitext, Springer 2017
References
Collingwood, David H. (1990), "Review: Nolan R. Wallach, Real reductive groups. I", Bulletin of the American Mathematical Society, New Series, 22 (1): 183–198, doi:10.1090/s0273-0979-1990-15876-8.
UCSD Mathematics Profile: Nolan Wallach Archived 2013-09-27 at the Wayback Machine, retrieved 2013-09-01.
Nolan Wallach at the Mathematics Genealogy Project
Wallach, Nolan R. "The spectrum of compact quotients of semisimple Lie groups." In Proceedings of the International Congress of Mathematicians (Helsinki, 1978 (Helsinki), pp. 715–719 (Acad. Sci. Fennica, 1980), MR562677 (81f: 22021). 1980.
List of Fellows of the American Mathematical Society, retrieved 2013-09-01.
Howe, Roger; Hunziker, Markus; Willenbring, Jeb F., eds. (2014). Symmetry: Representation theory and its applications. In Honor of Nolan R. Wallach. Birkhäuser.
Marsden, Jerrold E.; Weinstein, Alan (1979). "Review of Geometric asymptotics by Victor Guillemin and Shlomo Sternberg and Symplectic geometry and Fourier analysis by Nolan R. Wallach" (PDF). Bull. Amer. Math. Soc. New Series. 1 (3): 545–553. doi:10.1090/s0273-0979-1979-14617-2.
Collingwood, David H. (1990). "Review: Real reductive groups I by Nolan R. Wallach" (PDF). Bull. Amer. Math. Soc. New Series. 22 (1): 183–188. doi:10.1090/S0273-0979-1990-15876-8.
Trombi, Peter (1994). "Review: Real reductive groups II by Nolan R. Wallach" (PDF). Bull. Amer. Math. Soc. New Series. 30 (1): 157–158. doi:10.1090/s0273-0979-1994-00454-9.
Towber, Jacob (1999). "Review: Representations and invariants of the classical groups by Roe Goodman and Nolan R. Wallach" (PDF). Bull. Amer. Math. Soc. New Series. 36 (4): 533–538. doi:10.1090/s0273-0979-99-00795-8.
Berg, Michael (21 August 2009). "Review: Symmetry, Representations, and Invariants by Roe Goodman and Nolan R. Wallach". MAA Reviews, Mathematical Association of America.
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