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Hang Wang

Publications and source records attributed to Hang Wang.

108 records · Page 6Linked to original sources

An equivariant orbifold index for proper actions

For a proper, cocompact action by a locally compact group of the form $H \times G$, with $H$ compact, we define an $H \times G$-equivariant index of $H$-transversally elliptic operators, which takes values in $KK_*(C^*H, C^*G)$. This simultaneously generalises the Baum--Connes analytic assembly map, Atiyah's index of transversally elliptic operators, and Kawasaki's orbifold index. This index also generalises the assembly map to elliptic operators on orbifolds. In the special case where the manifold in question is a real semisimple Lie group, $G$ is a cocompact lattice and $H$ is a maximal compact subgroup, we realise the Dirac induction map from the Connes--Kasparov conjecture as a Kasparov product and obtain an index theorem for Spin-Dirac operators on compact locally symmetric spaces.

math.KT↗

A K-theoretic Selberg trace formula

Let G be a semisimple Lie group and H a uniform lattice in G. The Selberg trace formula is an equality arising from computing in two different ways the traces of convolution operators on the Hilbert space L^2(G/H) associated to test functions. In this paper we present a cohomological interpretation of the trace formula involving the K-theory of the maximal group C*-algebras of G and H. As an application, we exploit the role of group C*-algebras as recipients of higher indices of elliptic differential operators and we obtain the index theoretic version of the Selberg trace formula developed by Barbasch and Moscovici from ours.

math.NT↗

Bi-stream Pose Guided Region Ensemble Network for Fingertip Localization from Stereo Images

In human-computer interaction, it is important to accurately estimate the hand pose especially fingertips. However, traditional approaches for fingertip localization mainly rely on depth images and thus suffer considerably from the noise and missing values. Instead of depth images, stereo images can also provide 3D information of hands and promote 3D hand pose estimation. There are nevertheless limitations on the dataset size, global viewpoints, hand articulations and hand shapes in the publicly available stereo-based hand pose datasets. To mitigate these limitations and promote further research on hand pose estimation from stereo images, we propose a new large-scale binocular hand pose dataset called THU-Bi-Hand, offering a new perspective for fingertip localization. In the THU-Bi-Hand dataset, there are 447k pairs of stereo images of different hand shapes from 10 subjects with accurate 3D location annotations of the wrist and five fingertips. Captured with minimal restriction on the range of hand motion, the dataset covers large global viewpoint space and hand articulation space. To better present the performance of fingertip localization on THU-Bi-Hand, we propose a novel scheme termed Bi-stream Pose Guided Region Ensemble Network (Bi-Pose-REN). It extracts more representative feature regions around joint points in the feature maps under the guidance of the previously estimated pose. The feature regions are integrated hierarchically according to the topology of hand joints to regress the refined hand pose. Bi-Pose-REN and several existing methods are evaluated on THU-Bi-Hand so that benchmarks are provided for further research. Experimental results show that our new method has achieved the best performance on THU-Bi-Hand.

cs.CV↗

Higher Nahm transform in non commutative geometry

Anti-self-dual (ASD) connections for a compact smooth four manifold arise as critical values for the Yang-Mills action functional. Nahm transform is a nice correspondence between a vector bundle with ASD connections and a vector bundle with ASD connections over Picard torus associated to X. In this talk we propose a noncommutative geometric version of the Nahm transform that generalises the Connes-Yang-Mills action functional formulated using Dixmier trace.

math.OA↗

Positive Scalar Curvature and Poincare Duality for Proper Actions

For G an almost-connected Lie group, we study G-equivariant index theory for proper co-compact actions with various applications, including obstructions to and existence of G-invariant Riemannian metrics of positive scalar curvature. We prove a rigidity result for almost-complex manifolds, generalising Hattori's results, and an analogue of Petrie's conjecture. When G is an almost-connected Lie group or a discrete group, we establish Poincare duality between G-equivariant K-homology and K-theory, observing that Poincare duality does not necessarily hold for general G.

math.KT↗

Orbital integrals and $K$-theory classes

Let $G$ be a semisimple Lie group with discrete series. We use maps $K_0(C^*_rG)\to \mathbb{C}$ defined by orbital integrals to recover group theoretic information about $G$, including information contained in $K$-theory classes not associated to the discrete series. An important tool is a fixed point formula for equivariant indices obtained by the authors in an earlier paper. Applications include a tool to distinguish classes in $K_0(C^*_rG)$, the (known) injectivity of Dirac induction, versions of Selberg's principle in $K$-theory and for matrix coefficients of the discrete series, a Tannaka-type duality, and a way to extract characters of representations from $K$-theory. Finally, we obtain a continuity property near the identity element of $G$ of families of maps $K_0(C^*_rG)\to \mathbb{C}$, parametrised by semisimple elements of $G$, defined by stable orbital integrals. This implies a continuity property for $L$-packets of discrete series characters, which in turn can be used to deduce a (well-known) expression for formal degrees of discrete series representations from Harish-Chandra's character formula.

math.KT↗

Shelstad's character identity from the point of view of index theory

Shelstad's character identity is an equality between sums of characters of tempered representations in corresponding $L$-packets of two real, semisimple, linear, algebraic groups that are inner forms to each other. We reconstruct this character identity in the case of the discrete series, using index theory of elliptic operators in the framework of $K$-theory. Our geometric proof of the character identity is evidence that index theory can play a role in the classification of group representations via the Langlands program.

math.RT↗

Noncommutative Geometry and Conformal Geometry. I. Local Index Formula and Conformal Invariants

This paper is part of a series of articles on noncommutative geometry and conformal geometry. In this paper, we reformulate the local index formula in conformal geometry in such a way to take into account of the action of conformal diffeomorphisms. We also construct and compute a whole new family of geometric conformal invariants associated with conformal diffeomorphisms. This includes conformal invariants associated with equivariant characteristic classes. The approach of this paper involves using various tools from noncommutative geometry, such as twisted spectral triples and cyclic theory. An important step is to establish the conformal invariance of the Connes-Chern character of the conformal Dirac spectral triple of Connes-Moscovici. Ultimately, however, the main results of the paper are stated in a purely differential-geometric fashion.

math.DG↗

A fixed point formula and Harish-Chandra's character formula

The main result in this paper is a fixed point formula for equivariant indices of elliptic differential operators, for proper actions by connected semisimple Lie groups on possibly noncompact manifolds, with compact quotients. For compact groups and manifolds, this reduces to the Atiyah-Segal-Singer fixed point formula. Other special cases include an index theorem by Connes and Moscovici for homogeneous spaces, and an earlier index theorem by the second author, both in cases where the group acting is connected and semisimple. As an application of this fixed point formula, we give a new proof of Harish-Chandra's character formula for discrete series representations.

math.DG↗

A fixed point theorem on noncompact manifolds

We generalise the Atiyah-Segal-Singer fixed point theorem to noncompact manifolds. Using $KK$-theory, we extend the equivariant index to the noncompact setting, and obtain a fixed point formula for it. The fixed point formula is the explicit cohomological expression from Atiyah-Segal-Singer's result. In the noncompact case, however, we show in examples that this expression yields characters of infinite-dimensional representations. In one example, we realise characters of discrete series representations on the regular elements of a maximal torus, in terms of the index we define. Further results are a fixed point formula for the index pairing between equivariant $K$-theory and $K$-homology, and a non-localised expression for the index we use, in terms of deformations of principal symbols. The latter result is one of several links we find to indices of deformed symbols and operators studied by various authors.

math.KT↗

Monoid and group of pseudo braids

In this paper we define a monoid of pseudo braids and prove that this monoid is isomorphic to a singular braid monoid. We also prove an analogue of Markov's theorem for pseudo braids.

math.GT↗

Noncommutative Geometry and Conformal Geometry. II. Connes-Chern character and the local equivariant index theorem

This paper is the second part of a series of papers on noncommutative geometry and conformal geometry. In this paper, we compute explicitly the Connes-Chern character of an equivariant Dirac spectral triple. The formula that we obtain for which was used in the first paper of the series. The computation has two main steps. The first step is the justification that the CM cocycle represents the Connes-Chern character. The second step is the computation of the CM cocycle as a byproduct of a new proof of the local equivariant index theorem of Donnelly-Patodi, Gilkey and Kawasaki. The proof combines the rescaling method of Getzler with an equivariant version of the Greiner-Hadamard approach to the heat kernel asymptotics. Finally, as a further application of this approach, we computate the short-time limit of the JLO cocycle of an equivariant Dirac spectral triple.

math.DG↗

Noncommutative Geometry and Conformal Geometry. III. Vafa-Witten Inequality and Poincaré Duality

This paper is the the third part of a series of paper whose aim is to use of the framework of \emph{twisted spectral triples} to study conformal geometry from a noncommutive geometric viewpoint. In this paper we reformulate the inequality of Vafa-Witten \cite{VW:CMP84} in the setting of twisted spectral triples. This involves a notion of Poincaré duality for twisted spectral triples. Our main results have various consequences. In particular, we obtain a version in conformal geometry of the original inequality of Vafa-Witten, in the sense of an explicit control of the Vafa-Witten bound under conformal changes of metric. This result has several noncommutative manifestations for conformal deformations of ordinary spectral triples, spectral triples associated to conformal weights on noncommutative tori, and spectral triples associated to duals of torsion-free discrete cocompact subgroups satisfying the Baum-Connes conjecture.

math.DG↗

Index map, $σ$-connections, and Connes-Chern character in the setting of twisted spectral triples

Twisted spectral triples are a twisting of the notion of spectral triple aiming at dealing with some type III geometric situations. In the first part of the paper, we give a geometric construction of the index map of a twisted spectral triple in terms of $σ$-connections on finitely generated projective modules. This makes it more transparent the analogy with the indices of Dirac operators with coefficients in vector bundles. In the second part, we give a direct construction of the Connes-Chern character of a twisted spectral, both in the invertible and non-invertible cases. Combining these two parts we obtain an analogue the Atiyah-Singer index formula for twisted spectral triples.

math.OA↗

Noncommutative geometry, conformal geometry, and the local equivariant index theorem

We prove a local index formula in conformal geometry by computing the Connes-Chern character for the conformal Dirac (twisted) spectral triple recently constructed by Connes-Moscovici. Following an observation of Moscovici, the computation reduces to the computation of the CM cocycle of an equivariant Dirac (ordinary) spectral triple. This computation is obtained as a straightforward consequence of a new proof of the local equivariant index theorem of Patodi, Donelly-Patodi and Gilkey. This proof is obtained by combining Getzler's rescaling with an equivariant version of Greiner's approach to the heat kernel asymptotic. It is believed that this approach should hold in various other geometric settings. On the way we give a geometric description of the index map of a twisted spectral in terms of (twisted) connections on finitely generated projective modules.

math.OA↗

Localized Index and $L^2$-Lefschetz fixed point formula for orbifolds

We study a class of localized indices for the Dirac type operators on a complete Riemannian orbifold, where a discrete group acts properly, co-compactly and isometrically. These localized indices, generalizing the $L^2$-index of Atiyah, are obtained by taking certain traces of the higher index for the Dirac type operators along conjugacy classes of the discrete group. Applying the local index technique, we also obtain an $L^2$-version of the Lefschetz fixed point formula for orbifolds. These cohomological formulae for the localized indices give rise to a class of refined topological invariants for the quotient orbifold.

math.DG↗

$L^2$-index formula for proper cocompact group actions

We study the index of the $G$-invariant elliptic pseudo-differential operator acting on a complete Riemannian manifold, where a unimodular, locally compact group $G$ acts properly and cocompactly. An $L^2$-index formula was obtained using the heat kernel method.

math.AT↗

Markov Theorem For Free Links

The notion of free link is a generalized notion of virtual link. In the present paper we define the group of free braids, prove the Alexander theorem that all free links can be obtained as closures of free braids and prove a Markov theorem, which gives necessary and sufficient conditions for two free braids to have the same free link closure. Our result is expected to be useful in study the topology invariants for free knots and links.

math.GT↗