Complex-valued Ray-Singer torsion

In the spirit of Ray and Singer we define a complex-valued analytic torsion using non-selfadjoint Laplacians. We establish an anomaly formula which permits to turn this into a topological invariant. Conjecturally this analytically defined invariant computes the complex-valued Reidemeister torsion, including its phase. We establish this conjecture in some non-trivial situations.     

On the Generalized Ray-Singer Conjecture for Covering Spaces

We find a relation between L² -combinatorial torsion and analytic torsion associated with a small spectrum of the Witten-type Laplacian on a covering space. The result is used as one crucial step in a proof of the generalized Ray-Singer conjecture for covering spaces.     

Anomaly formulas for the complex-valued analytic torsion on compact bordisms

We extend the complex-valued analytic torsion, introduced by Burghelea and Haller on closed manifolds, to compact Riemannian bordisms. We do so by considering a flat complex vector bundle over a compact Riemannian manifold, endowed with a fiberwise nondegenerate symmetric bilinear form. The Riemmanian metric and the bilinear form are used to define non-selfadjoint Laplacians acting on vector-valued smooth forms under absolute and relative boundary conditions. In order to define the complex-valued analytic torsion in this situation, we study spectral properties of these generalized Laplacians. Then, as main results, we obtain so-called anomaly formulas for this torsion. Our reasoning takes into account that the coefficients in the heat trace asymptotic expansion associated to the boundary value problem under consideration, are locally computable. The anomaly formulas for the complex-valued Ray–Singer torsion are derived first by using the corresponding ones for the Ray–Singer metric, obtained by Brüning and Ma on manifolds with boundary, and then an argument of analytic continuation. In odd dimensions, our anomaly formulas are in accord with the corresponding results of Su, without requiring the variations of the Riemannian metric and bilinear structures to be supported in the interior of the manifold.     

A Cheeger–Müller theorem for delocalized analytic torsion

We study the delocalized L ²-analytic torsion introduced by John Lott, and define the delocalized L ²-combinatorial torsion. By using the method of Bismut–Zhang, under the conditions of positive Novikov–Shubin invariants and finite conjugacy class, we get the Cheeger–Müller type relation between the delocalized L ²-analytic torsion and the delocalized L ²-combinatorial torsion.      

Twisted analytic torsion

We review the Reidemeister, Ray-Singer’s analytic torsion and the Cheeger-Mller theorem. We describe the analytic torsion of the de Rham complex twisted by a flux form introduced by the current authors and recall its properties. We define a new twisted analytic torsion for the complex of invariant differential forms on the total space of a principal circle bundle twisted by an invariant flux form. We show that when the dimension is even, such a torsion is invariant under certain deformation of the metric and the flux form. Under T-duality which exchanges the topology of the bundle and the flux form and the radius of the circular fiber with its inverse, the twisted torsion of invariant forms are inverse to each other for any dimension.     

Analytic torsion of Hirzebruch surfaces

Using different forms of the arithmetic Riemann-Roch theorem and the computations of Bott-Chern secondary classes, we compute the analytic torsion and the height of Hirzebruch surfaces.     

A Cheeger-M¨uller Theorem for Symmetric Bilinear Torsions

The authors establish a Cheeger-Müller type theorem for the complex valued analytic torsion introduced by Burghelea and Hailer for fiat vector bundles carrying nondegenerate symmetric bilinear forms. As a consequence, they prove the Burghelea-Haller conjecture in full generality, which gives an analytic interpretation of （the square of） the Turaev torsion.     

Analytic torsion and Ruelle zeta functions for hyperbolic manifolds with cusps

In this paper we derive a relationship of the leading coefficient of the Laurent expansion of the Ruelle zeta function at s=0 and the analytic torsion for hyperbolic manifolds with cusps. Here, the analytic torsion is defined by a certain regularized trace following Melrose [R.B. Melrose, The Atiyah-Patodi-Singer Index Theorem, Res. Notes Math., vol. 4, A.K. Peters, Ltd., Wellesley, MA, 1993]. This extends the result of Fried, which was proved for the compact case in [D. Fried, Analytic torsion and closed geodesics on hyperbolic manifolds, Invent. Math. 84 (3) (1986) 523-540], to a noncompact case.     

Ideal version of Egorov's theorem for analytic P-ideals

We introduce the notion of equi-ideal convergence and use it to prove an ideal variant of Egorov's theorem. We also show that this variant usually cannot be strengthen to a direct analogue of Egorov's theorem.     

The gluing formula of the refined analytic torsion for an acyclic Hermitian connection

In the previous study by Huang and Lee (arXiv:1004.1753) we introduced the well-posed boundary conditions ${{\mathcal P}_{-, {\mathcal L}_{0}}}$ and ${{\mathcal P}_{+, {\mathcal L}_{1}}}$ for the odd signature operator to define the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the gluing formula of the refined analytic torsion for an acyclic Hermitian connection with respect to the boundary conditions ${{\mathcal P}_{-, {\mathcal L}_{0}}}$ and ${{\mathcal P}_{+, {\mathcal L}_{1}}}$ . In this case the refined analytic torsion consists of the Ray-Singer analytic torsion, the eta invariant and the values of the zeta functions at zero. We first compare the Ray-Singer analytic torsion and eta invariant subject to the boundary condition ${{\mathcal P}_{-, {\mathcal L}_{0}}}$ or ${{\mathcal P}_{+, {\mathcal L}_{1}}}$ with the Ray-Singer analytic torsion subject to the relative (or absolute) boundary condition and eta invariant subject to the APS boundary condition on a compact manifold with boundary. Using these results together with the well known gluing formula of the Ray-Singer analytic torsion subject to the relative and absolute boundary conditions and eta invariant subject to the APS boundary condition, we obtain the main result.     

紧群作用下的扭化解析挠率

在紧群中的非单位元的作用只有孤立的不动点的情形下, 我们研究了扭化de Rham 复形上的解析挠率的等变情形. 在一些条件下, 我们也研究了Z₂- 分次的椭圆复形上的离域的(delocalized) L²-解析挠率.     

The full basis theorem does not imply analytic wellordering

A finite support product of ${\omega }_1$ clones of Jensen's minimal ${\Pi }_2^1$ singleton forcing is used to define a model in which any non-empty analytically definable set of reals contains an analytically definable real (the full basis theorem), but there is no analytically definable wellordering of the reals.     

A Pochhammer type sampling reconstruction of analytic functions

We give a characterization of the functions f analytic on the half-plane Re(z)>σ that can be expressed in the form for some βn∈C. It will follow that the βn depend on the values of f on the integers so this gives a sampling reconstruction of these functions. Several examples of this representation are given.     

利用行列式求两异面直线间的距离

根据空间两直线方程为一般式方程时共面或异面的充要条件,利用行列式和向量运算的性质,直接给出两异面直线间的距离公式。     

On the gluing formula for the analytic torsion

In this paper, we derive the Cheeger–Müller/Bismut–Zhang theorem for manifolds with boundary and the gluing formula for the analytic torsion of flat vector bundles in full generality, i.e., we do not assume that the Hermitian metric on the flat vector bundle is flat nor that the Riemannian metric has product structure near the boundary.     

On the mean-value theorem for analytic functions

A version of the mean-value theorem (formulas of finite increments) for analytic functions is proved. Volyn University, Lutsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 8, pp. 1143–1147, August, 1997.     

Interpolating sequences for bounded analytic functions

We prove that any sequence in the open ball of a complex Banach space even in that of whose norms are an interpolating sequence for is interpolating for the space of all bounded analytic functions on The construction made yields that the interpolating functions depend linearly on the interpolated values.

     

Additivity for the parametrized topological Euler characteristic and Reidemeister torsion

Dwyer, Weiss, and Williams have recently defined the notions of the parametrized topological Euler characteristic and the parametrized topological Reidemeister torsion which are invariants of bundles of compact topological manifolds. We show that these invariants satisfy additivity formulas paralleling the additive properties of the classical Euler characteristic and the Reidemeister torsion of CW-complexes.     

Isometry theorem for the Segal-Bargmann transform on a noncompact symmetric space of the complex type

We consider the Segal-Bargmann transform on a noncompact symmetric space of the complex type. We establish isometry and surjectivity theorems for the transform, in a form as parallel as possible to the results in the dual compact case. The isometry theorem involves integration over a tube of radius R in the complexification, followed by analytic continuation with respect to R. A cancellation of singularities allows the relevant integral to have a nonsingular extension to large R, even though the function being integrated has singularities.