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ALGEBRAIC GEOMETRY
"... Algebraic geometry is the mathematical study of geometric objects by means of algebra. Its origins go back to the coordinate geometry introduced by Descartes. A classic example is the circle of radius 1 in the plane, which is ..."
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Cited by 523 (6 self)
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Algebraic geometry is the mathematical study of geometric objects by means of algebra. Its origins go back to the coordinate geometry introduced by Descartes. A classic example is the circle of radius 1 in the plane, which is
On the mathematical foundations of learning
 Bulletin of the American Mathematical Society
, 2002
"... The problem of learning is arguably at the very core of the problem of intelligence, both biological and arti cial. T. Poggio and C.R. Shelton ..."
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Cited by 329 (12 self)
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The problem of learning is arguably at the very core of the problem of intelligence, both biological and arti cial. T. Poggio and C.R. Shelton
Split States, Entropy Enigmas, Holes and Halos
, 2007
"... We investigate degeneracies of BPS states of Dbranes on compact CalabiYau manifolds. We develop a factorization formula for BPS indices using attractor flow trees associated to multicentered black hole bound states. This enables us to study background dependence of the BPS spectrum, to compute e ..."
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Cited by 243 (21 self)
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We investigate degeneracies of BPS states of Dbranes on compact CalabiYau manifolds. We develop a factorization formula for BPS indices using attractor flow trees associated to multicentered black hole bound states. This enables us to study background dependence of the BPS spectrum, to compute explicitly exact indices of various nontrivial Dbrane systems, and to clarify the subtle relation of DonaldsonThomas invariants to BPS indices of stable D6D2D0 states, realized in supergravity as “hole halos. ” We introduce a convergent generating function for D4 indices in the large CY volume limit, and prove it can be written as a modular average of its polar part, generalizing the fareytail expansion of the elliptic genus. We show polar states are “split ” D6antiD6 bound states, and that the partition function factorizes accordingly, leading to a refined version of the OSV conjecture. This differs from the original conjecture in several aspects. In particular we obtain a nontrivial measure factor g −2 top e−K and find factorization requires a cutoff. We show that the main factor determining the cutoff and therefore the error is the existence of “swing states ” — D6 states which exist at large radius but do not form stable D6antiD6 bound states. We point out a likely breakdown of the OSV conjecture at small gtop (in the large background CY volume limit), due to the surprising phenomenon that for sufficiently large background Kähler moduli, a charge ΛΓ supporting single centered black holes of entropy ∼ Λ2S(Γ) also admits twocentered BPS black hole realizations whose entropy grows like Λ3 when Λ → ∞.
Special transverse slices and their enveloping algebras
 Adv. Math
"... Abstract. Let G be a simple, simply connected algebraic group over C, g = LieG, N (g) the nilpotent cone in g, and (E,H,F) an sl2triple in g. Let S = E+Ker adF, the special transverse slice to the adjoint orbit Ω of E, and S0 = S ∩ N (g). The coordinate ring C[S0] is naturally graded (see [35]). Le ..."
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Cited by 84 (8 self)
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Abstract. Let G be a simple, simply connected algebraic group over C, g = LieG, N (g) the nilpotent cone in g, and (E,H,F) an sl2triple in g. Let S = E+Ker adF, the special transverse slice to the adjoint orbit Ω of E, and S0 = S ∩ N (g). The coordinate ring C[S0] is naturally graded (see [35]). Let Z(g) be the centre of the enveloping algebra U(g) and η: Z(g) → C an algebra homomorphism. Identify g with g ∗ via a Killing isomorphism and let χ denote the linear function on g corresponding to E. Following [32] we attach to χ a nilpotent subalgebra mχ ⊂ g of dimension (dim Ω)/2 and a 1dimensional mχmodule Cχ. Let H̃χ denote the algebra opposite to Endg(U(g) ⊗U(mχ) Cχ) and H̃χ,η = H̃χ ⊗Z(g) Cη. It is proved in the paper that the algebra H̃χ,η has a natural filtration such that gr(H̃χ,η), the associated graded algebra, is isomorphic to C[S0]. This construction yields natural noncommutative deformations of all singularities associated with the adjoint quotient map of g. 1.
CRITICAL SETS OF SOLUTIONS TO ELLIPTIC EQUATIONS
"... P bjDju = 0 that the critical set jruj,1 f0g has locally Let u 6 const satisfy an elliptic equation L0u P aijDiju + with smooth coefficients in a domain in R n. It is shown finite n, 2 dimensional Hausdorff measure. This implies in particular that for a solution u 6 0 of (L 0 + c)u =0,with c 2 C 1 ..."
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Cited by 69 (2 self)
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P bjDju = 0 that the critical set jruj,1 f0g has locally Let u 6 const satisfy an elliptic equation L0u P aijDiju + with smooth coefficients in a domain in R n. It is shown finite n, 2 dimensional Hausdorff measure. This implies in particular that for a solution u 6 0 of (L 0 + c)u =0,with c 2 C 1, the singular set u,1 f0g \jruj,1 f0g has locally finite n  2 dimensional Hausdorff measure.
Isoparametric hypersurfaces with four principal curvatures
"... Abstract. The classication of isoparametric hypersurfaces with four principal curvatures in spheres in [2] hinges on a crucial characterization, in terms of four sets of equations of the 2nd fundamental form tensors of a focal submanifold, of an isoparametric hypersurface of the type constructed b ..."
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Cited by 42 (4 self)
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Abstract. The classication of isoparametric hypersurfaces with four principal curvatures in spheres in [2] hinges on a crucial characterization, in terms of four sets of equations of the 2nd fundamental form tensors of a focal submanifold, of an isoparametric hypersurface of the type constructed by Ferus, Karcher and Munzner. The proof of the characterization in [2] is an extremely long calculation by exterior derivatives with remarkable cancellations, which is motivated by the idea that an isoparametric hypersurface is dened by an overdetermined system of partial dierential equations. Therefore, exterior dierentiating suciently many times should gather us enough information for the conclusion. In spite of its elementary nature, the magnitude of the calculation and the surprisingly pleasant cancellations make it desirable to understand the underlying geometric principles. In this paper, we give a conceptual, and considerably shorter, proof of the characterization based on Ozeki and Takeuchi's expansion formula for the CartanMunzner polynomial. Along the way the geometric meaning of these four sets of equations also becomes clear. 1.
On the characteristic and deformation varieties of a knot
 in: Proceedings of the CassonFest
, 2004
"... Abstract. The colored Jones function of a knot is a sequence of Laurent polynomials in one variable, whose nth term is the Jones polynomial of the knot colored with the ndimensional irreducible representation of sl2. It was recently shown by TTQ Le and the author that the colored Jones function of ..."
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Cited by 40 (11 self)
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Abstract. The colored Jones function of a knot is a sequence of Laurent polynomials in one variable, whose nth term is the Jones polynomial of the knot colored with the ndimensional irreducible representation of sl2. It was recently shown by TTQ Le and the author that the colored Jones function of a knot is qholonomic, i.e., that it satisfies a nontrivial linear recursion relation with appropriate coefficients. Using holonomicity, we introduce a geometric invariant of a knot: the characteristic variety, an affine 1dimensional variety in 2. We then compare it with the character variety of SL2 ( ) representations, viewed from the boundary. The comparison is stated as a conjecture which we verify (by a direct computation) in the case of the trefoil and figure eight knots. We also propose a geometric relation between the peripheral subgroup of the knot group, and basic operators that act on the colored Jones function. We also define a noncommutative version (the socalled noncommutative Apolynomial) of the characteristic variety of a knot. Holonomicity works well for higher rank groups and goes beyond hyperbolic geometry, as we explain in the last chapter. Contents
Duality Relating Spaces of Algebraic Cocycles and Cycles
"... In this paper a fundamental duality is established between algebraic cycles and algebraic cocycles on a smooth projective variety. The proof makes use of a new Chow moving lemma for families. If X is a smooth projective variety of dimension n, our duality map induces isomorphisms L s H k (X) → Ln−s ..."
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Cited by 38 (14 self)
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In this paper a fundamental duality is established between algebraic cycles and algebraic cocycles on a smooth projective variety. The proof makes use of a new Chow moving lemma for families. If X is a smooth projective variety of dimension n, our duality map induces isomorphisms L s H k (X) → Ln−sH2n−k(X) for 2s ≤ k which carry over via natural transformations to the Poincaré duality isomorphism H k (X; Z) → H2n−k(X; Z). More generally, for smooth projective varieties X and Y the natural graphing homomorphism sending algebraic cocycles on X with values in Y to algebraic cycles on the product X ×Y is a weak homotopy equivalence. Among applications presented are the determination of the homotopy type of certain algebraic mapping complexes and a computation of the group of algebraic scocycles modulo algebraic equivalence on a smooth projective variety.