arXiv research

The paper calculates asymptotic expansions for specific types of oscillatory integrals.

Analytic torsion expansions for symmetric and complex homogeneous spaces.

Study on heat trace expansion for thermoelastic Dirichlet-to-Neumann map.

The article calculates asymptotic expansions for quantum invariants from surgeries on Whitehead link components.

We derive asymptotic expansions for option data to detect infinite variation volatility.

Researchers derive asymptotic expansions for thermoelastic operators on manifolds.

The paper proposes and proves asymptotic expansions for quantum invariants.

This an announcement for the generalized asymptotic expansion of Tian-Yau-Zeldtich.

We establish the existence of the asymptotic expansion of the Bergman kernel associated to the spin-c Dirac operators acting on high tensor powers of line bundles with non-degenerate mixed curvature (negative and positive eigenvalues) by extending the paper " On the asymptotic expansion of Bergman kernel " (math.DG/040…

The paper studies bowl solitons and their asymptotic expansions.

Paper studies the full asymptotic torsion forms of flat bundles.

The paper studies asymptotic expansions of operators related to Bochner-Schrödinger on Riemannian manifolds.

Study improves variance calculation for random zero sets on complex manifolds.

The validity of an approximation formula for European option prices under a general stochastic volatility model is proved in the light of the Edgeworth expansion for ergodic diffusions. The asymptotic expansion is around the Black-Scholes price and is uniform in bounded payoff func- tions. The result provides a validat…

Study verifies asymptotic expansion for Reshetikhin-Turaev invariants of fundamental shadow link pairs.

The study proves an expansion theorem for scalar-flat asymptotically conical Kähler metrics.

We report on recent results of the authors concerning calculations of quantum invariants of Seifert 3-manifolds. These results include a derivation of the Reshetikhin-Turaev invariants of all oriented Seifert manifolds associated with an arbitrary complex finite dimensional simple Lie algebra, and a determination of th…

We derive a detailed asymptotic expansion of the heat trace for the Laplace-Beltrami operator on functions on manifolds with conic singularities, using the Singular Asymptotics Lemma of Jochen Bruening and Robert T. Seeley [BS]. In the subsequent paper we investigate how the terms in the expansion reflect the geometry …

Study oscillatory integrals with degenerate singular points in multivariable phase functions.

Study on quantum invariants of twist knots at specific roots of unity.

We study the asymptotic behavior of Masur-Veech volumes as the genus goes to infinity. We show the existence of a complete asymptotic expansion of these volumes that depends only on the genus and the number of singularities. The computation of the first term of this asymptotics expansion was a long standing problem. Th…

Master thesis proves Bergman kernel asymptotics for positive line bundles.

Authors prove an asymptotic expansion for spectral zeta functions on discrete tori.

Study on quantum invariants of twist knots at specific roots of unity.

Formula derived for torsion of modified Dirac operator.

The paper calculates super Weil-Petersson volumes for large genus.

Given a principal bundle with a connection, we look for an asymptotic expansion of the holonomy of a loop in terms of its length. This length is defined relative to some Riemannian or sub-Riemannian structure. We are able to give an asymptotic formula that is independent of choice of gauge.

Let $H_h = h^2 L +V$ where $L$ is a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold and $V$ is a symmetric endomorphism field. We derive an asymptotic expansion for the heat kernel of $H_h$ as $h \to 0$. As a consequence we get an asymptotic expansion for the …

The paper studies the asymptotic expansion of Gaussian integral operators on Riemannian submanifolds.

In the asymptotic expansion of the hyperbolic specification of the colored Jones polynomial of torus knots, we identify different geometric contributions, in particular Chern--Simons invaraint and Reidemeister torsion.

Study on hyperbolic surfaces' volumes, proving asymptotic expansion for high genus.

The main goal of the paper is to address the issue of the existence of Kempf's distortion function and the Tian-Yau-Zelditch (TYZ) asymptotic expansion for the Kepler manifold - an important example of non compact manfold. Motivated by the recent results for compact manifolds we construct Kempf's distortion function an…

New insights into black hole horizons from asymptotic expansions.

We provide a general method to compute a Taylor expansion in time of implied volatility for stochastic volatility models, using a heat kernel expansion. Beyond the order 0 implied volatility which is already known, we compute the first order correction exactly at all strikes from the scalar coefficient of the heat kern…

We derive formulas for the Reshetikhin-Turaev invariants of all oriented Seifert manifolds associated to an arbitrary complex finite dimensional simple Lie algebra $\mathfrak g$ in terms of the Seifert invariants and standard data for $\mathfrak g$. A main corollary is a determination of the full asymptotic expansions …

Let $X$ be a compact connected strongly pseudoconvex CR manifold of dimension $2n+1, n \ge 1$ with a transversal CR $S^1$ action on $X$. We establish an asymptotic expansion for the $m$-th Fourier component of the Szegő kernel function as $m\rightarrow\infty$, where the expansion involves a contribution in terms of a d…

In this paper, we study the Edgeworth expansion for a pre-averaging estimator of quadratic variation in the framework of continuous diffusion models observed with noise. More specifically, we obtain a second order expansion for the joint density of the estimators of quadratic variation and its asymptotic variance. Our …

Study Szegő kernel on non-compact CR manifolds with specific conditions.

New method improves nonlinear filtering accuracy with reduced computation.

In this paper, we computed the first three coefficients of the asymptotic expansion of Zelditch. We also proved that in general, the $k$-th coefficient is a polynomial of the curvature and its derivative of weight $k$.

Density expansions for hypoelliptic diffusions $(X^1,...,X^d)$ are revisited. In particular, we are interested in density expansions of the projection $(X_T^1,...,X_T^l)$, at time $T>0$, with $l \leq d$. Global conditions are found which replace the well-known "not-in-cutlocus" condition known from heat-kernel asymptot…

Let $φ\in C^\infty(\Complex^n)$ be a given real valued function. We assume that $\pr\ddbarφ$ is non-degenerate of constant signature $(n_-,n_+)$ on $\Complex^n$. When $q=n_-$, it is well-known that the Bergman kernel for $(0,q)$ forms with respect to the $k$-th weight $e^{-2kφ}$, $k>0$, admits a full asymptotic expansi…

We survey recent results about the asymptotic expansion of Toeplitz operators and their kernels, as well as Berezin-Toeplitz quantization. We deal in particular with calculation of the first coefficients of these expansions.

We study the relationship between the geometry and the Laplace spectrum of a Riemannian orbifold O via its heat kernel; as in the manifold case, the time-zero asymptotic expansion of the heat kernel furnishes geometric information about O. In the case of a good Riemannian orbifold (i.e., an orbifold arising as the orbi…

In an abstract Wiener space setting, we constract a rigorous mathematical model of the one-loop approximation of the perturbative Chern-Simons integral, and derive its explicit asymptotic expansion for stochastic Wilson lines.

We investigate the analogy between the large N expansion in normal matrix models and the asymptotic expansion of the determinant of the Hilb map, appearing in the study of critical metrics on complex manifolds via projective embeddings. This analogy helps to understand the geometric meaning of the expansion of matrix m…

The paper derives expansions for Green's operators and resolvents using Hadamard methods.

Paper improves risk estimation for extreme events.