The Experts below are selected from a list of 156 Experts worldwide ranked by ideXlab platform
Xiaojun Chen - One of the best experts on this subject based on the ideXlab platform.
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Quantization of the Lie Bialgebra of String Topology
Communications in Mathematical Physics, 2011Co-Authors: Xiaojun Chen, Farkhod EshmatovAbstract:Let M be a smooth, simply-connected, closed Oriented Manifold, and LM the free loop space of M . Using a Poincaré duality model for M , we show that the reduced equivariant homology of LM has the structure of a Lie bialgebra, and we construct a Hopf algebra which quantizes the Lie bialgebra.
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Quantization of the Lie Bialgebra of String Topology
Communications in Mathematical Physics, 2010Co-Authors: Xiaojun Chen, Farkhod EshmatovAbstract:Let M be a smooth, simply-connected, closed Oriented Manifold, and LM the free loop space of M. Using a Poincare duality model for M, we show that the reduced equivariant homology of LM has the structure of a Lie bialgebra, and we construct a Hopf algebra which quantizes the Lie bialgebra.
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Batalin–Vilkovisky coalgebra of string topology
Pacific Journal of Mathematics, 2010Co-Authors: Xiaojun ChenAbstract:We prove that the reduced Hochschild homology of a commutative DG Frobenius algebra has the natural structure of a Batalin-Vilkovisky coalgebra, and the reduced cyclic homology has the natural structure of a gravity coalgebra. As an application, this gives an algebraic model for a Batalin-Vilkovisky coalgebra structure on the reduced homology of the free loop space of a simply connected closed Oriented Manifold, and a gravity coalgebra structure on the reduced equivariant homology.
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Batalin-Vilkovisky coalgebra of string topology
arXiv: Quantum Algebra, 2008Co-Authors: Xiaojun ChenAbstract:We show that the reduced Hochschild homology of a DG open Frobenius algebra has the natural structure of a Batalin-Vilkovisky coalgebra, and the reduced cyclic homology has the natural structure of a gravity coalgebra. This gives an algebraic model for a Batalin-Vilkovisky coalgebra structure on the reduced homology of the free loop space of a simply connected closed Oriented Manifold, and a gravity coalgebra structure on the reduced equivariant homology.
Farkhod Eshmatov - One of the best experts on this subject based on the ideXlab platform.
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Quantization of the Lie Bialgebra of String Topology
Communications in Mathematical Physics, 2011Co-Authors: Xiaojun Chen, Farkhod EshmatovAbstract:Let M be a smooth, simply-connected, closed Oriented Manifold, and LM the free loop space of M . Using a Poincaré duality model for M , we show that the reduced equivariant homology of LM has the structure of a Lie bialgebra, and we construct a Hopf algebra which quantizes the Lie bialgebra.
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Quantization of the Lie Bialgebra of String Topology
Communications in Mathematical Physics, 2010Co-Authors: Xiaojun Chen, Farkhod EshmatovAbstract:Let M be a smooth, simply-connected, closed Oriented Manifold, and LM the free loop space of M. Using a Poincare duality model for M, we show that the reduced equivariant homology of LM has the structure of a Lie bialgebra, and we construct a Hopf algebra which quantizes the Lie bialgebra.
B. Levit - One of the best experts on this subject based on the ideXlab platform.
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A Bayesian approach to the estimation of maps between Riemannian Manifolds. II: Examples
Mathematical Methods of Statistics, 2009Co-Authors: L. T. Butler, B. LevitAbstract:Let gJ be a smooth compact Oriented Manifold without boundary, imbedded in a Euclidean space E ^ s , and let γ be a smooth map of gJ into a Riemannian Manifold Λ. An unknown state θ ∈ gJ is observed via X = θ + ɛ ξ, where ɛ > 0 is a small parameter and ξ is a white Gaussian noise. For a given smooth prior λ on gJ and smooth estimators g ( X ) of the map γ we have derived a second-order asymptotic expansion for the related Bayesian risk [3]. In this paper, we apply this technique to a variety of examples. The second part examines the first-order conditions for equality-constrained regression problems. The geometric tools that are utilized in [3] are naturally applicable to these regression problems.
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A Bayesian approach to the estimation of maps between Riemannian Manifolds
Mathematical Methods of Statistics, 2007Co-Authors: L. T. Butler, B. LevitAbstract:Let Θ be a smooth compact Oriented Manifold without boundary, imbedded in a Euclidean space E ^s, and let γ be a smooth map of Θ into a Riemannian Manifold Λ. An unknown state θ ∈ Θ is observed via X = θ + εξ , where ε > 0 is a small parameter and ξ is a white Gaussian noise. For a given smooth prior λ on Θ and smooth estimators g ( X ) of the map γ we derive a second-order asymptotic expansion for the related Bayesian risk. The calculation involves the geometry of the underlying spaces Θ and Λ, in particular, the integration-by-parts formula. Using this result, a second-order minimax estimator of γ is found based on the modern theory of harmonic maps and hypo-elliptic differential operators.
Eric Leichtnam - One of the best experts on this subject based on the ideXlab platform.
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ON THE HOMOTOPY INVARIANCE OF HIGHER SIGNATURES FOR ManifoldS WITH BOUNDARY
Journal of Differential Geometry, 2020Co-Authors: Eric Leichtnam, John Lott, Paolo PiazzaAbstract:If M is a compact Oriented Manifold-with-boundary whose fundamental group is virtually nilpotent or Gromov-hyperbolic, we show that the higher signatures of M are Oriented-homotopy invariants. We give applications to the question of when higher signatures of closed Manifolds are cut-and-paste invariant.
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on the cut and paste property of higher signatures of a closed Oriented Manifold
Topology, 2002Co-Authors: Eric Leichtnam, Wolfgang Luck, Matthias KreckAbstract:Abstract We extend the notion of the symmetric signature σ( M ,r)∈L n (R) for a compact n-dimensional Manifold M without boundary, a reference map r : M→BG and a homomorphism of rings with involutions β : Z G→R to the case with boundary ∂M, where ( M , ∂M )→(M, ∂M) is the G-covering associated to r. We need the assumption that C ∗ ( ∂M ) ⊗ Z G R is R-chain homotopy equivalent to a R-chain complex D ∗ with trivial mth differential for n=2m resp. n=2m+1. We prove a glueing formula, homotopy invariance and additivity for this new notion. Let Z be a closed Oriented Manifold with reference map Z→BG. Let F⊂Z be a cutting codimension one subManifold F⊂Z and let F →F be the associated G-covering. Denote by α m ( F ) the mth Novikov–Shubin invariant and by b m (2) ( F ) the mth L2-Betti number. If for the discrete group G the Baum–Connes assembly map is rationally injective, then we use σ( M ,r) to prove the additivity (or cut and paste property) of the higher signatures of Z, if we have α m ( F )=∞ + in the case n=2m and, in the case n=2m+1, if we have α m ( F )=∞ + and b m (2) ( F )=0 . This additivity result had been proved (by a different method) in (On the Homotopy Invariance of Higher Signatures for Mainfolds with Boundary, preprint, 1999, Corollary 0.4) when G is Gromov hyperbolic or virtually nilpotent. We give new examples, where these conditions are not satisfied and additivity fails. We explain at the end of the introduction why our paper is greatly motivated by and partially extends some of the work of Leichtnam et al. (On the Homotopy Invariance of Higher Signatures for Mainfolds with Boundary, preprint, 1999), Lott (Math. Ann., 1999) and Weinberger (Contemporary Mathematics, 1999, p. 231).
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on the cut and paste property of higher signatures of a closed Oriented Manifold
arXiv: Geometric Topology, 2000Co-Authors: Eric Leichtnam, Wolfgang LueckAbstract:We extend the notion of the symmetric signature $\sigma(\bar{M},r)$ in L^n(R) for a compact n-dimensional Manifold M without boundary, a reference map r from M to BG and a homomorphism of rings with involutions from ZG to R to the case with boundary $\partial M$, where $(\bar{M},\bar{\partial M}) \to (M,\partial M)$ is the G-covering associated to r. We need the assumption that $C_*(\bar{\partial M}) \otimes_{\zz G} R$ isR-chain homotopy equivalent to a R-chain complex D_* with trivial m-th differential for n = 2m resp. n = 2m+1. Let Z be a closed Oriented Manifold with reference map BG. Let F be a cutting codimension one subManifold in Z and let $\bar{F} \to F$ be the associated $G$-covering. Denote by $\alpha_m(\bar{F})$ the m-th Novikov-Shubin invariant and by $b_m^{(2)}(\bar{F})$ the m-th L^2-Betti number. We use $\sigma(\bar{M},r)$ to prove the additivity (or cut and paste property) of the higher signatures of Z if we have $\alpha_m(\bar{F}) = \infty^+$ in the case n = 2m and, in the case n = 2m+1, if we have $\alpha_m(\bar{F}) = \infty^+$ and $b_m^{(2)}(\bar{F}) = 0$. We give examples, where these conditions are not satisfied and additivity fails. Our work is motivated by the one of Leichtnam-Lott-Piazza, Lott and Weinberger.
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On the homotopy invariance of higher signatures for Manifolds with boundary
arXiv: Differential Geometry, 1999Co-Authors: Eric Leichtnam, John Lott, Paolo PiazzaAbstract:If M is a compact Oriented Manifold-with-boundary whose fundamental group is virtually nilpotent or Gromov-hyperbolic, we show that the higher signatures of M are Oriented-homotopy invariants.
John Lott - One of the best experts on this subject based on the ideXlab platform.
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ON THE HOMOTOPY INVARIANCE OF HIGHER SIGNATURES FOR ManifoldS WITH BOUNDARY
Journal of Differential Geometry, 2020Co-Authors: Eric Leichtnam, John Lott, Paolo PiazzaAbstract:If M is a compact Oriented Manifold-with-boundary whose fundamental group is virtually nilpotent or Gromov-hyperbolic, we show that the higher signatures of M are Oriented-homotopy invariants. We give applications to the question of when higher signatures of closed Manifolds are cut-and-paste invariant.
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Signatures and higher signatures of S^1-quotients
Mathematische Annalen, 2000Co-Authors: John LottAbstract:We define and study the signature, \(\widehat{A}\)-genus and higher signatures of the quotient space of an \(S^1\)-action on a closed Oriented Manifold. We give applications to questions of positive scalar curvature and to an Equivariant Novikov Conjecture.
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On the homotopy invariance of higher signatures for Manifolds with boundary
arXiv: Differential Geometry, 1999Co-Authors: Eric Leichtnam, John Lott, Paolo PiazzaAbstract:If M is a compact Oriented Manifold-with-boundary whose fundamental group is virtually nilpotent or Gromov-hyperbolic, we show that the higher signatures of M are Oriented-homotopy invariants.
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Signatures and Higher Signatures of $S^1$-Quotients
arXiv: Differential Geometry, 1998Co-Authors: John LottAbstract:We define and study the signature, A-hat genus and higher signatures of the quotient space of an $S^1$-action on a closed Oriented Manifold. We give applications to questions of positive scalar curvature and to an Equivariant Novikov Conjecture.
