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AN OVERVIEW OF MATHEMATICAL ISSUES ARISING IN THE GEOMETRIC COMPLEXITY THEORY APPROACH TO VP != VNP
"... We discuss the geometry of orbit closures and the asymptotic behavior of Kronecker coefficients in the context of the Geometric Complexity Theory program to prove a variant of Valiant’s algebraic analog of the P ̸ = NP conjecture. We also describe the precise separation of complexity classes that t ..."
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Cited by 40 (13 self)
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We discuss the geometry of orbit closures and the asymptotic behavior of Kronecker coefficients in the context of the Geometric Complexity Theory program to prove a variant of Valiant’s algebraic analog of the P ̸ = NP conjecture. We also describe the precise separation of complexity classes that their program proposes to demonstrate.
Geometry and the complexity of matrix multiplication
, 2007
"... Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, ..."
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Cited by 35 (5 self)
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Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.
The complex Wishart distribution and the symmetric group
"... Let V be the space of (r; r) Hermitian matrices and let\Omega be the cone of the positive definite ones. We say that the random variable S; taking its values has the complex Wishart distribution fl p;oe if IE(exp trace (`S)) = (det(I r \Gamma oe`)) where oe and oe \Gamma ` are ; and where p ..."
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Cited by 35 (2 self)
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Let V be the space of (r; r) Hermitian matrices and let\Omega be the cone of the positive definite ones. We say that the random variable S; taking its values has the complex Wishart distribution fl p;oe if IE(exp trace (`S)) = (det(I r \Gamma oe`)) where oe and oe \Gamma ` are ; and where p = 1; 2; : : : ; r \Gamma 1 or p ? r \Gamma 1. In this paper, we compute all moments of S: More specifically, if h = (h 1 ; : : : ; h n ) is any word of (r; r) complex matrices and if we introduce the complex random variables S(h) = trace (Sh 1 : : : Sh n ) and S (h) = trace (S h 1 \Delta \Delta \Delta S h n ); we are able to compute in a simple way the expressions IE(S(h ) \Delta \Delta \Delta S(h )) for any set of words fh ; \Delta \Delta \Delta ; h g: Similarly we compute IE(S ) \Delta \Delta \Delta S whenever it exists. This provides a general answer to the questions raised by D. Maiwald and D. Kraus (2000) for moments of order 4. Our technique is to use the multilinear forms r (oe)(h 1 ; : : : ; h k ); where belongs to the group S k of permutations of f1; : : : ; kg. For instance, if = (4)(2; 5)(6; 3; 1); the form r (oe) is defined by r (oe)(h 1 ; : : : ; h 6 ) = trace (oeh 4 )trace (oeh 2 oeh 5 )trace (oeh 6 oeh 3 oeh 1 ): Denote by m() the number of cycles of ; and write q for p \Gamma r: Our theorems 2 and 3 can be presented in the following compact way by using the convolution product associated to the group algebra A(S k ): Our proofs of these formulas are elementary. The remainder of the paper is devoted to the inversion of the second formula, i.e. to the computation of )); for any . To this end, we use the irreducible characters of S k
The primary approximation to the cohomology of the moduli space of curves and cocycles for the MumfordMoritaMiller classes
, 2001
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Stable cohomology of the mapping class group with symplectic coefficients and of the universal AbelJacobi map
 J. Alg. Geom
, 1996
"... Abstract. The irreducible representations of the complex symplectic group of genus g are indexed by nonincreasing sequences of integers λ = (λ1 ≥ λ2 ≥ · · ·) with λk = 0 for k> g. A recent result of N.V. Ivanov implies that for a given partition λ, the cohomology group of a given degree of the ..."
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Cited by 26 (1 self)
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Abstract. The irreducible representations of the complex symplectic group of genus g are indexed by nonincreasing sequences of integers λ = (λ1 ≥ λ2 ≥ · · ·) with λk = 0 for k> g. A recent result of N.V. Ivanov implies that for a given partition λ, the cohomology group of a given degree of the mapping class group of genus g with values in the representation associated to λ is independent of g if g is sufficiently large. We prove that this stable cohomology is the tensor product of the stable cohomology of the mapping class group and a finitely generated graded module over Q[c1,..., c λ], where deg(ci) = 2i and λ  = ∑ i λi. We describe this module explicitly. In the same sense we determine the stable rational cohomology of the moduli space of compact Riemann surfaces with s given ordered distinct (resp. not necessarily distinct) points as well as the stable cohomology of the universal Abel–Jacobi map. These results take into account mixed Hodge structures. 1.
BochnerKähler metrics
"... Abstract. A Kähler metric is said to be BochnerKähler if its Bochner curvature vanishes. This is a nontrivial condition when the complex dimension of the underlying manifold is at least 2. In this article it will be shown that, in a certain welldefined sense, the space of BochnerKähler metrics in ..."
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Cited by 25 (1 self)
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Abstract. A Kähler metric is said to be BochnerKähler if its Bochner curvature vanishes. This is a nontrivial condition when the complex dimension of the underlying manifold is at least 2. In this article it will be shown that, in a certain welldefined sense, the space of BochnerKähler metrics in complex dimension n has real dimension n+1 and a recipe for an explicit formula for any BochnerKähler metric will be given. It is shown that any BochnerKähler metric in complex dimension n has local (real) cohomogeneity at most n. The BochnerKähler metrics that can be ‘analytically continued ’ to a complete metric, free of singularities, are identified. In particular, it is shown that the only compact BochnerKähler manifolds are the discrete quotients of the known symmetric examples. However, there are compact BochnerKähler orbifolds that are not locally symmetric. In fact, every weighted projective space carries a BochnerKähler metric. The fundamental technique is to construct a canonical infinitesimal torus action on a BochnerKähler metric whose associated momentum mapping has the orbits of its symmetry pseudogroupoid as fibers.
COHOMOLOGICAL STRUCTURE OF THE MAPPING CLASS GROUP AND BEYOND
, 2005
"... In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various subgroups of them such as the Torelli group, the IA outer au ..."
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Cited by 24 (3 self)
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In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various subgroups of them such as the Torelli group, the IA outer automorphism group of free groups, the symplectomorphism group of surfaces. Based on these, we present several conjectures and problems concerning the cohomology of these groups. We are particularly interested in the possible interplays between these cohomology groups rather than merely the structures of individual groups. It turns out that, we have to include, in our considerations, two other groups which contain the mapping class group as their core subgroups and whose structures seem to be deeply related to that of the mapping class group. They are the arithmetic mapping class group and the group of homology cobordism classes of homology cylinders. 1
The periodic cyclic homology of IwahoriHecke algebras
, 2008
"... We determine the periodic cyclic homology of the IwahoriHecke algebras Hq, for q ∈ C∗ not a “proper root of unity.” (In this paper, by a proper root of unity we shall mean a root of unity other than 1.) Our method is based on a general result on periodic cyclic homology, which states that a “weakl ..."
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Cited by 22 (7 self)
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We determine the periodic cyclic homology of the IwahoriHecke algebras Hq, for q ∈ C∗ not a “proper root of unity.” (In this paper, by a proper root of unity we shall mean a root of unity other than 1.) Our method is based on a general result on periodic cyclic homology, which states that a “weakly spectrum preserving” morphism of finite type algebras induces an isomorphism in periodic cyclic homology. The concept of a weakly spectrum preserving morphism is defined in this paper, and most of our work is devoted to understanding this class of morphisms. Results of Kazhdan–Lusztig and Lusztig show that, for the indicated values of q, there exists a weakly spectrum preserving morphism φq: Hq → J, to a fixed finite type algebra J. This proves that φq induces an isomorphism in periodic cyclic homology and, in particular, that all algebras Hq have the same periodic cyclic homology, for the indicated values of q. The periodic cyclic homology groups of the algebra H1 can then be determined directly, using results of Karoubi and Burghelea, because it is the group algebra of an extended affine Weyl group.
Frontiers of reality in Schubert calculus
 Bulletin of the AMS
"... Abstract. The theorem of Mukhin, Tarasov, and Varchenko (formerly the Shapiro conjecture for Grassmannians) asserts that all (a priori complex) solutions to certain geometric problems in the Schubert calculus are actually real. Their proof is quite remarkable, using ideas from integrable systems, Fu ..."
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Cited by 21 (9 self)
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Abstract. The theorem of Mukhin, Tarasov, and Varchenko (formerly the Shapiro conjecture for Grassmannians) asserts that all (a priori complex) solutions to certain geometric problems in the Schubert calculus are actually real. Their proof is quite remarkable, using ideas from integrable systems, Fuchsian differential equations, and representation theory. There is now a second proof of this result, and it has ramifications in other areas of mathematics, from curves to control theory to combinatorics. Despite this work, the original Shapiro conjecture is not yet settled. While it is false as stated, it has several interesting and not quite understood modifications and generalizations that are likely true, and the strongest and most subtle version of the Shapiro conjecture for Grassmannians remains open.