George Mostow

George Daniel Mostow (born July 4, 1923 - † April 4, 2017 ) was an American mathematician who mainly deals with differential geometry , algebraic groups and Lie groups .

Life

Mostow studied at Harvard University , where he received his doctorate under Garrett Birkhoff in 1948 ( The Extensibility of Local Lie Groups of Transformations and Groups on Surfaces ). From 1952 he was at Johns Hopkins University and from 1961 professor at Yale University , where he retired in 1999.

In 1957 Mostow was a Guggenheim Fellow . In 1970 he was invited speaker at the International Congress of Mathematicians in Nice ( The rigidity of locally symmetric spaces ). He has been a member of the National Academy of Sciences since 1974 and of the American Academy of Arts and Sciences since 1978 . In 1987/8 he was President of the American Mathematical Society . In 1993 he received the Leroy P. Steele Prize for research achievements and his book Strong rigidity in locally symmetric spaces from 1973. In 2013 he received the Wolf Prize for Mathematics.

plant

He discovered and investigated the rigidity properties of co-compact lattices in semi-simple Lie groups (without compact factor groups and a center ), i.e. discrete subsets of semi-simple Lie groups, so that the quotient spaces of the Lie group are compact modulo the discrete subgroup . His 1972 Rigidity Theorem states that isomorphisms of the lattices in these Lie groups can be extended to analytic isomorphisms of the Lie groups, except for . Applied to hyperbolic spaces , it means that hyperbolic manifolds of finite volume in more than two dimensions are uniquely determined by their fundamental group . Mostow's work revived the investigation of locally symmetrical spaces ( William Thurston's classification of three-dimensional manifolds ) and were a model for similar rigidity theorems, for example by Grigori Alexandrowitsch Margulis , who, based on Mostow, in 1974 proved the arithmetic of grids in semi-simple Lie groups with rank > 1. ${\displaystyle SL(2,R)/\pm 1}$

Fonts

• with Pierre Deligne : Commensurabilities among lattices in . ${\displaystyle PU(1,n)}$Annals of Mathematical Studies, Princeton University Press 1993.
• with Deligne: Monodromy of hypergeometric functions and nonlattice integral monodromy. Inst. Hautes Études Sci. Publ. Math. No. 63: 5-89 (1986).
• Generalized Picard lattices arising from half-integral conditions. Inst. Hautes Études Sci. Publ. Math. No. 63: 91-106 (1986).
• with Yum-Tong Siu : A compact Kähler surface of negative curvature not covered by the ball. Ann. of Math. (2) 112 (1980) no. 2, 321-360.
• On a remarkable class of polyhedra in complex hyperbolic space. Pacific J. Math. 86 (1980) no. 1, 171-276.
• Strong rigidity of locally symmetric spaces. Annals of Mathematical Studies, Princeton 1973.
• Quasi-conformal mappings in n -space and the rigidity of hyperbolic space forms. Inst. Hautes Études Sci. Publ. Math. No. 34 1968 53-104.
• Cohomology of topological groups and solvmanifolds. Ann. of Math. (2) 73 1961 20-48.
• Equivariant embeddings in Euclidean space. Ann. of Math. (2) 65: 432-446 (1957).
• Fully reducible subgroups of algebraic groups. Amer. J. Math. 78: 200-221 (1956).
• Some new decomposition theorems for semi-simple groups. Mem. Amer. Math. Soc. 1955, (1955). no. 14, 31-54.
• Factor spaces of solvable groups. Ann. of Math. (2) 60, (1954). 1-27.

literature

• Roger Howe (Ed.): Discrete groups in geometry and analysis- Papers in Honor of GDMostow on his 60th Birthday. (Yale University Conference 1986), Birkhäuser

Web links

• Mostow: A brief survey of symmetry in mathematics. 1993
• Enlightenment at a red traffic light: Wolf Prize laureate Prof. George Daniel Mostow made his greatest scientific breakthrough while driving. Biographical article in Haaretz

Remarks

1. ↑ In Memoriam: George Daniel Mostow 1923-2017. (No longer available online.) In: Yale University. Archived from the original on April 22, 2017 ; accessed on April 23, 2017 (English). Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.  
2. ↑ Special case of the rigidity theorem for lattices for , proven with quasi-conformal mappings in -dimensional space. Proved by Mostow for compact hyperbolic spaces, extended to finite volumes by Gopal Prasad . Mostow: Quasiconformal mappings in -space and rigidity of hyperbolic space-forms. IHES 1968, Prasad Inventiones Mathematicae Vol. 21, 1973, p. 255 ${\displaystyle SO(n,1)}$ ${\displaystyle n}$ ${\displaystyle n}$
3. ↑ in two dimensions (compact Riemann surfaces ) it does not apply. There are a number of hyperbolic structures there, parameterized by the Teichmüller rooms
4. ↑ as a consequence of the law of rigidity, the metric invariants of hyperbolic manifolds of finite volume in more than two dimensions are topological invariants