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Singular PoissonKähler geometry of Scorza varieties and their secant varieties
 Differential Geom. Appl
, 2005
"... Abstract. Each Scorza variety and its secant varieties in the ambient projective space are identified, in the realm of singular PoissonKähler geometry, in terms of projectivizations of holomorphic nilpotent orbits in suitable Lie algebras of hermitian type, the holomorphic nilpotent orbits, in turn ..."
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Abstract. Each Scorza variety and its secant varieties in the ambient projective space are identified, in the realm of singular PoissonKähler geometry, in terms of projectivizations of holomorphic nilpotent orbits in suitable Lie algebras of hermitian type, the holomorphic nilpotent orbits, in turn, being affine varieties. The ambient projective space acquires an exotic Kähler structure, the closed stratum being the Scorza variety and the closures of the higher strata its secant varieties. In this fashion, the secant varieties become exotic projective varieties. In the rank 3 case, the four regular Scorza varieties coincide with the four critical Severi varieties. In the standard cases, the Scorza varieties and their secant varieties arise also via Kähler reduction. An interpretation in terms of constrained mechanical systems is included.
LIE GROUP ACTIONS ON MANIFOLDS
"... One of the most successful approaches to geometry is the one suggested by Felix Klein. According to Klein, a geometry is a Gspace M, that is, a set M together with a group G of transformations of M. This approach provides a powerful link between geometry and algebra. Of particular importance is the ..."
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One of the most successful approaches to geometry is the one suggested by Felix Klein. According to Klein, a geometry is a Gspace M, that is, a set M together with a group G of transformations of M. This approach provides a powerful link between geometry and algebra. Of particular importance is the situation when the group G acts transitively on
Severi Varieties and . . .
, 2004
"... Each of the four critical Severi varieties arises from a minimal holomorphic nilpotent orbit in a simple regular rank 3 hermitian Lie algebra and each such variety lies as singular locus in a cubic—the chordal variety—in the corresponding complex projective space; the cubic and projective space are ..."
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Each of the four critical Severi varieties arises from a minimal holomorphic nilpotent orbit in a simple regular rank 3 hermitian Lie algebra and each such variety lies as singular locus in a cubic—the chordal variety—in the corresponding complex projective space; the cubic and projective space are identified in terms of holomorphic nilpotent orbits. The projective space acquires an exotic Kähler structure with three strata, the cubic is an example of an exotic projective variety with two strata, and the corresponding Severi variety is the closed stratum in the exotic variety as well as in the exotic projective space. In the standard cases, these varieties arise also via Kähler reduction. An interpretation in terms of constrained mechanical systems is included.
Arguments for FTheory
, 2005
"... After a brief review of string and MTheory we point out some deficiencies. Partly to cure them, we present several arguments for “FTheory”, enlarging spacetime to (2,10) signature, following the original suggestion of C. Vafa. We introduce a suggestive Supersymmetric 27plet of particles, associat ..."
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After a brief review of string and MTheory we point out some deficiencies. Partly to cure them, we present several arguments for “FTheory”, enlarging spacetime to (2,10) signature, following the original suggestion of C. Vafa. We introduce a suggestive Supersymmetric 27plet of particles, associated to the exceptional symmetric hermitian space E6/Spin c (10). Several possible future directions, including using projective rather than metric geometry, are mentioned. We should emphasize that FTheory is yet just a very provisional attempt, lacking clear dynamical principles. Keywords: Mtheory, Ftheory, Euler multiplets
Coordinating Editors
"... our time, passed away on June 3, 2010, nine days before his seventythird birthday. This article, along with one in the previous issue of the Notices, touches on his outstanding personality and his great contribution to mathematics. ..."
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our time, passed away on June 3, 2010, nine days before his seventythird birthday. This article, along with one in the previous issue of the Notices, touches on his outstanding personality and his great contribution to mathematics.
Problems in Lie Group Theory ∗
, 2003
"... The theory of Lie groups and representations was developed by Lie, Killing, Cartan, Weyl and others to a degree of quasiperfection, in the years 18701930. The main topological features of compact simple Lie groups were elucidated in the 40s by H. Hopf, Pontriagin and others. The exceptional groups ..."
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The theory of Lie groups and representations was developed by Lie, Killing, Cartan, Weyl and others to a degree of quasiperfection, in the years 18701930. The main topological features of compact simple Lie groups were elucidated in the 40s by H. Hopf, Pontriagin and others. The exceptional groups were studied by Chevalley, Borel, Freudenthal etc. in 19491957. Torsion in the exceptional groups was considered by Toda, Adams etc. in the 80s. However, one can still ask some questions for which the answer is either incomplete or absent, at least to this speaker. We would like to raise and discuss some of them in this communication. Submitted to Journ. Opt. B (Wigner issue).
A geometric study of many body systems
, 2008
"... An nbody system is a labelled collection of n point masses in a Euclidean space, and their congruence and internal symmetry properties involve a rich mathematical structure which is investigated in the framework of equivariant Riemannian geometry. Some basic concepts are nconfiguration, configurat ..."
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An nbody system is a labelled collection of n point masses in a Euclidean space, and their congruence and internal symmetry properties involve a rich mathematical structure which is investigated in the framework of equivariant Riemannian geometry. Some basic concepts are nconfiguration, configuration space, internal space, shape space, Jacobi transformation and weighted root system. The latter is a generalization of the root system of SU(n), which provides a bookkeeping for expressing the mutual distances of the point masses in terms of the Jacobi vectors. Moreover, its application to the study of collinear central nconfigurations yields a simple proof of Moulton’s enumeration formula. A major topic is the study of matrix spaces representing the shape space of nbody configurations in Euclidean kspace, the structure of the muniversal shape space and its O(m)equivariant linear model. This also leads to those “orbital fibrations ” where SO(m) or O(m) act on a sphere with a sphere as orbit space. A few of these examples are encountered in the literature, e.g. the special case S 5 /O(2) ≈ S 4 was analyzed independently by Arnold, Kuiper and Massey in the 1970’s. Contents 1