arXiv research

Derives Selberg trace formula on Riemann surfaces and generalizes to other spaces.

In this article, we will discuss the smooth $(X_{M}+\sqrt{-1}Y_{M})$-invariant forms on $M$ and to establish a localization formulas. As an application, we get a localization formulas for characteristic numbers.

We prove a localization formula for group-valued equivariant de Rham cohomology of a compact G-manifold. This formula is a non-trivial generalization of the localization formula of Berline-Vergne and Atiyah-Bott for the usual equivariant de Rham cohomology. As an application, we obtain a version of the Duistermaat-Heck…

We formulate a conjectural Lefschetz formula for locally symmetric spaces of finite volume. The formula can be verified in the compact case and for Riemann surfaces.

Short proof given for a complex formula in spin manifold geometry.

We provide explicit formulas for integrating multiplicative forms on local Lie groupoids in terms of infinitesimal data. Combined with our previous work [8], which constructs the local Lie groupoid of a Lie algebroid, these formulas produce concrete integrations of several geometric stuctures defined infinitesimally. I…

Paper provides an explicit formula for local volatility in Cheyette models.

New principle for supersymmetric localization on Lie groups.

We prove a localization formula for a "holomorphic equivariant cohomology" attached to the Atiyah algebroid of an equivariant holomorphic vector bundle. This generalizes Feng-Ma, Carrell-Liebermann, Baum-Bott and K. Liu's localization formulas.

In this article we give a totally new proof of the integral localization formula for equivariantly closed differential forms (Theorem 7.11 in [BGV]). We restate it here as Theorem 2. This localization formula is very well known, but the author hopes to adapt this proof to obtain a more general result in the future.

We give an explicit geometric formula for the twisted orbital integrals using the method of the hypoelliptic Laplacian developed by Bismut. Combining with the twisted trace formula, we can evaluate the equivariant trace of the heat operators of the Laplacians on a compact locally symmetric space. In particular, we revi…

In this paper, we consider three typical problems on a locally finite connected graph. The first one is to study the Bochner formula for the Laplacian operator on a locally finite connected graph. We use the Bochner formula to derive the Bernstein type estimate of the heat equation. The second is to derive the Reilly t…

For an $S^{1}$-manifold with boundary, we prove a localization formula applying to any equivariant cohomology theory satisfying a certain algebraic condition. We show how the localization result of Kalkman and a case of the quantization commutes with reduction theorem follow easily from the localization formula.

Paper derives local Plücker formulas for special orthogonal groups.

We prove a holomorphic residue localization formula for odd holomorphic vector fields on compact complex supermanifolds whose fermionic and bosonic dimensions coincide. Under isolated non-degeneracy hypotheses on the reduced zero set, we give an explicit local residue formula.

Derives localization formulas in Batalin-Vilkovisky formalism.

New parametrizations for minimal timelike surfaces discovered.

We survey recent results in hermitian integral geometry, i.e. integral geometry on complex vector spaces and complex space forms. We study valuations and curvature measures on complex space forms and describe how the global and local kinematic formulas on such spaces were recently obtained. While the local and global k…

We give a direct geometric proof of a Danilov-type formula for toric origami manifolds by using the localization of Riemann-Roch number.

In this article, we will discuss a localization formulas of equivariant cohomology about two Killing vector fields on the set of zero points ${\rm{Zero}}(X_{M}-\sqrt{-1}Y_{M})=\{x\in M \mid |Y_{M}(x)|=|X_{M}(x)|=0 \}.$ As application, we use it to get formulas about characteristic numbers and to get a Duistermaat-Heckm…

In this paper, we obtain a localization formula in differential K-theory for $S^1$-action. Then by combining an extension of Goette's result on the comparison of two types of equivariant $η$-invariants, we establish a version of localization formula for equivariant $η$-invariants. An important step of our approach is t…

Formula for Lefschetz numbers on locally symmetric spaces.

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 operat…

In this article we establish a local parabolic almost monotonicity formula for two phase free boundary problems on Riemannian manifolds, which is an extension of a work of Edquist-Petrosyan.

The paper is devoted to the problem of finding explicit combinatorial formulae for the Pontryagin classes. We discuss two formulae, the classical Gabrielov-Gelfand-Losik formula based on investigation of configuration spaces and the local combinatorial formula obtained by the author in 2004. The latter formula is based…

The paper calculates the full asymptotics of analytic torsions for compact orbifolds.

Surveying locally homogeneous almost-Hermitian spaces with formulas for curvature.

We extend Dupire's formula for stochastic interest rates and local volatility.

We derive Mok-Siu-Yeung type formulas for horizontal maps from compact contact locally sub-symmetric spaces into strictly pseudoconvex CR manifolds and we obtain some rigidity theorems for the horizontal pseudoharmonic maps.

A general Lefschetz formula for the geodesic action on locally symmetric spaces is proven.

The paper derives a local formula for the Euler number of circle bundles.

We use localization formulas in the theory of equivariant cohomology to rederive the wall crossing formulas of Li-Liu and Okonek-Teleman for Seiberg-Witten invariants.

We develop a Chern-Weil theory for compact Lie group action whose generic stabilizers are finite in the framework of equivariant cohomology. This provides a method of changing an equivariant closed form within its cohomological class to a form more suitable to yield localization results. This work is motivated by our w…

We derive identities for general flows of Riemannian metrics that may be regarded as local mean-value, monotonicity, or Lyapunov formulae. These generalize previous work of the first author for mean curvature flow and other nonlinear diffusions. Our results apply in particular to Ricci flow, where they yield a local mo…

This note is an addendum to our earlier work \cite{humi}. In \cite{humi}, we studied a Hamiltonian action for a generalized Calabi-Yau manifold and showed that the Duistermaat-Heckman theorem holds. The purpose of this note is to show that the density function of the Duistermaa-Heckman measure is a piecewise polynomial…

We develop two types of integral formulas for the perimeter of a convex body K in planar geometries. We derive Cauchy-type formulas for perimeter in planar Hilbert geometries. Specializing to H^2 we get a formula that appears to be new. We show that it implies the standard Cauchy-Santalo formula involving a central ang…

In this paper, we show a local Blaschke-Petkantschin formula for a Riemannian manifold. Namely, we compute the Jacobian determinant of the parametrization of $(n+1)$-tuples of the manifold by the center and the radius of their common circumscribed sphere as well as the $(n+1)$ directions characterizing the positions of…

Formula proves invariant matches for smooth and orbifold test configurations.

We prove that higher moment maps on area measures of a euclidean vector space are injective, while the kernel of the centroid map equals the image of the first variation map. Based on this, we introduce the space of smooth dual area measures on a finite-dimensional euclidean vector space and prove that it admits a natu…

New formula for dual knots using involutions.

Using heat kernel methods developed by Vaillant, a local index formula is obtained for families of d-bar operators on the Teichmuller universal curve of Riemann surfaces of genus g with n punctures. The formula also holds on the moduli space M{g,n} in the sense of orbifolds where it can be written in terms of Mumford-M…

We prove an analogue of the Atiyah-Bott-Berline-Vergne localization formula in the setting of equivariant basic cohomology of $K$-contact manifolds. As a consequence, we deduce analogues of Witten's nonabelian localization and the Jeffrey-Kirwan residue formula, which relate equivariant basic integrals on a contact man…

The LS-category of a topological space is a numerical homotopy invariant, introduced originally in a course on the global calculus of variations by Lyusternik and Schnirelmann, to estimate the number of critical points of a smooth function. When the topological space is a smooth manifold equipped with a proper action o…

Goussarov, Polyak, and Viro proved that finite type invariants of knots are ``finitely multi-local'', meaning that on a knot diagram, sums of quantities, defined by local information, determine the value of the knot invariant. The result implies the existence of Gauss diagram combinatorial formulas for finite type inva…

We give an explicit local formula for any formal deformation quantization, with separation of variables, on a Kähler manifold. The formula is given in terms of differential operators, parametrized by acyclic combinatorial graphs.

The paper extends a formula linking surface curvature to projection invariants.

Formula for sections on complex manifolds with non-isolated components.

Following a hedging based approach to model free financial mathematics, we prove that it should be possible to make an arbitrarily large profit by investing in those one-dimensional paths which do not possess local times. The local time is constructed from discrete approximations, and it is shown that it is $α$-Hölder …