The paper calculates asymptotic expansions for specific types of oscillatory integrals.
problem Analyzing oscillatory integrals with complex phase functions.
method Using asymptotic expansions of simpler phase functions to derive results for more complex cases.
result Explicit computation of coefficients in asymptotic expansions for certain integrals.
Study heat trace expansion on manifolds with conic points.
problem Analyzing heat diffusion on manifolds with sharp corners.
method Using the Singular Asymptotics Lemma to derive an expansion.
result Detailed asymptotic expansion reveals geometric insights.
Analytic torsion expansions for symmetric and complex homogeneous spaces.
problem Calculating the full asymptotic expansion of analytic torsion for various spaces.
method Explicit calculation and comparison with existing results.
result Explicit full asymptotic expansions for symmetric and complex homogeneous spaces.
Study holonomy expansions for loops in principal bundles.
problem Holonomy expansions for loops in principal bundles.
method Asymptotic expansion of holonomy in terms of loop length.
result Independent asymptotic formula for holonomy.
We find the asymptotic expansion of Masur-Veech volumes for large genus.
problem The asymptotic behavior of Masur-Veech volumes as genus increases.
method Combination of combinatorial and algebro-geometric approaches.
result Existence and computation of a complete asymptotic expansion.
Study on heat trace expansion for thermoelastic Dirichlet-to-Neumann map.
problem Asymptotic expansion of heat trace for thermoelastic Dirichlet-to-Neumann map.
method Provided a method to obtain all coefficients of the asymptotic expansion.
result Explicitly gave the first two coefficients involving volume and total mean curvature of the boundary.
The article calculates asymptotic expansions for quantum invariants from surgeries on Whitehead link components.
problem Calculating quantum invariants for 3-manifolds resulting from surgeries on Whitehead link components.
method Asymptotic expansion of relative Reshetikhin-Turaev and Turaev-Viro invariants.
result Asymptotic formulas for both invariants are derived.
We derive asymptotic expansions for option data to detect infinite variation volatility.
problem Detecting infinite variation volatility in high-frequency option data.
method Nonparametric higher-order asymptotic expansions for small-time changes of characteristic functions of Itô semimartingales.
result Evidence of infinite variation volatility in high-frequency option data.
Researchers derive asymptotic expansions for thermoelastic operators on manifolds.
problem Determining precise geometric information from thermoelastic spectra.
method Asymptotic expansions with Dirichlet and Neumann boundary conditions.
result Explicit calculation of first two coefficients for volumes.
The paper proposes and proves asymptotic expansions for quantum invariants.
problem Quantum invariants and their expansions under varying metrics.
method Asymptotic expansion conjectures for relative Reshetikhin-Turaev, Turaev-Viro invariants and quantum 6j-symbols.
result Proved asymptotic expansions for special cases, showing geometric dependence on metrics.
This an announcement for the generalized asymptotic expansion of Tian-Yau-Zeldtich.
Study on CR manifolds with S1 action finds Szegő kernel asymptotics.
problem Analyzing Szegő kernel on CR manifolds with S1 action. method Established asymptotic expansion for the m-th Fourier component of the Szegő kernel function. result Explicit formulas for the first three coefficients of the expansion.
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…
Analyzes the asymptotic expansion of analytic torsion for line bundles and orbifolds.
problem Analyzing the asymptotic expansion of Ray-Singer analytic torsion.
method Proves and calculates coefficients of the asymptotic expansion for specific cases.
result Calculates coefficients for the terms pn−1logp,pn−1 in the Kahler case. The paper studies bowl solitons and their asymptotic expansions.
problem Understanding the behavior of bowl solitons under curvature flows.
method Asymptotic expansion analysis for a large class of fully nonlinear curvature flows.
result Uniqueness of bowl-type solitons in their asymptotic class and construction of wing-like solitons.
Researchers describe the mass of conformal differential operators in terms of their asymptotic expansions.
problem Understanding the mass of conformal differential operators and its invariance under conformal transformations.
method Explicit description of the full asymptotic expansion of the Schwartz kernel of complex powers of m-Laplace type operators. result The mass of conformal differential operators is a conformal invariant in odd dimensions when the kernel is trivial.
Study reproducing kernels on complex Kepler manifolds.
problem Complex geometry of generalized Kepler manifolds.
method Introduce Hilbert spaces of holomorphic functions and find asymptotic expansions of reproducing kernels.
result Complete asymptotic expansions of reproducing kernels found for Kähler potentials.
The paper studies asymptotic expansions of operators related to Bochner-Schrödinger on Riemannian manifolds.
problem Asymptotic expansions of operators related to Bochner-Schrödinger on Riemannian manifolds.
method Analyzes the Bochner-Schrödinger operator Hp and its function φ(Hp) in L2(X,Lp⊗E), providing an asymptotic expansion of its smooth Schwartz kernel. result The trace of the operator φ(Hp) admits a complete asymptotic expansion in powers of p−1/2 as po∞. Paper studies the full asymptotic torsion forms of flat bundles.
problem Analytic torsion forms of flat bundles and their expansions.
method Proves the existence of the full expansion and gives a formula for the sub-leading term.
result Existence and formula for the full asymptotic expansion of torsion forms.
Study improves variance calculation for random zero sets on complex manifolds.
problem Improving the variance calculation for random zero sets on complex manifolds.
method Deriving an asymptotic expansion for the variance of linear statistics of zero divisors of random holomorphic sections.
result Sharpens leading-order asymptotics for the variance of random zero sets.
Study verifies asymptotic expansion for Reshetikhin-Turaev invariants of fundamental shadow link pairs.
problem Verifying the asymptotic expansion conjecture for Reshetikhin-Turaev invariants of fundamental shadow link pairs.
method Using logarithmic holonomies of meridians and hyperbolic cone structures, the study verifies the conjecture for pairs where M∖L is homeomorphic to a fundamental shadow link complement. result The asymptotic expansion conjecture is true for pairs (M,L) with sufficiently small cone angles and M∖L homeomorphic to a fundamental shadow link complement. 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…
The study proves an expansion theorem for scalar-flat asymptotically conical Kähler metrics.
problem Proving an expansion theorem for scalar-flat asymptotically conical Kähler metrics.
method Using weak decay conditions and the standard Kähler metric of the metric cone, the expansion theorem is proven.
result Each scalar-flat AC Kähler metric admits an expansion with a main term given by the standard Kähler metric of the metric cone and a leading error term of O(r^{2-2n}).
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…
Paper solves complex equations near cone points.
problem Solving complex Monge-Ampère equations with conical singularities.
method Proved asymptotic expansion of solutions.
result Found solutions to conical Kähler-Einstein metrics.
Study oscillatory integrals with degenerate singular points in multivariable phase functions.
problem Analyzing oscillatory integrals with degenerate singular points in phase functions.
method Using asymptotic expansions and results from one variable, the study examines multivariable phase functions.
result Asymptotic expansions of oscillatory integrals for multivariable phase functions with degenerate singular points.
Study on quantum invariants of twist knots at specific roots of unity.
problem Asymptotic expansions of quantum invariants for twist knots.
method Saddle point method applied to colored Jones polynomial.
result Asymptotic expansion formula for twist knots at given root of unity.
Master thesis proves Bergman kernel asymptotics for positive line bundles.
problem Proving asymptotic expansion of Bergman kernel for positive line bundles.
method Introduced a semi-classical symbol space and symbolic calculus.
result Established pointwise asymptotic expansion on positive parts of certain semi-positive line bundles.
Authors prove an asymptotic expansion for spectral zeta functions on discrete tori.
problem Proving an asymptotic expansion for spectral zeta functions on discrete tori.
method Inspired by Friedli and Karlsson's work, the authors derive an asymptotic expansion for the spectral zeta function on discrete tori.
result Similar asymptotic expansions hold for m=2 and higher dimensions, equivalent to the Epstein-Riemann conjecture.
Study on quantum invariants of twist knots at specific roots of unity.
problem Asymptotic expansions of quantum invariants for twist knots.
method Asymptotic expansion formula for colored Jones polynomial using twist knots.
result Obtained asymptotic expansion formulas for twist knots at specified roots of unity.
Study on symplectic manifolds finds asymptotic expansions for Bergman kernels.
problem Analyzing asymptotic expansions of Bergman kernels on symplectic manifolds.
method Established asymptotic expansions for renormalized Bochner Laplacians on high tensor powers of positive line bundles.
result Full off-diagonal asymptotic expansion for generalized Bergman kernels.
Study shows asymptotic solutions for scalar waves on long-range Minkowski-like spaces.
problem Analyzing scalar wave behavior on complex spacetimes.
method Proved existence of joint asymptotic expansions at null and timelike infinity.
result Logarithmic term in asymptotic expansions at null infinity is nonzero.
Formula derived for torsion of modified Dirac operator.
problem Analyzing modified Dirac operator's torsion.
method Proved formula for asymptotic expansion.
result Leading term formula for torsion.
The paper calculates super Weil-Petersson volumes for large genus.
problem Calculating super Weil-Petersson volumes for large genus.
method Analyzes super intersection numbers, proves coefficients are polynomials, and provides an algorithm to compute them.
result Proves existence of a complete asymptotic expansion of super Weil-Petersson volumes.
Deep learning method uses asymptotic expansion to solve high-dimensional BSDEs faster.
problem Solving high-dimensional BSDEs efficiently.
method Asymptotic expansion as prior knowledge in deep learning for BSDEs.
result Significantly reduces loss function and accelerates convergence.
Study Edgeworth expansion for quadratic variation estimator with noise.
problem Estimating quadratic variation with noise and error.
method Martingale embedding, Malliavin calculus, stable central limit theorems.
result Obtained a second order expansion for the joint density of estimators.
The paper studies the asymptotic expansion of Gaussian integral operators on Riemannian submanifolds.
problem Analyzing the asymptotic behavior of Gaussian integral operators on Riemannian submanifolds.
method Deriving a full asymptotic expansion of the Gaussian integral operator and computing the first-order correction term.
result Explicit computation of the first-order correction term in terms of mean curvature vector and scalar curvature.
Let Hh=h2L+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 Hh as h→0. As a consequence we get an asymptotic expansion for the …
Study finds wave equation solutions on product cones with new decay rates.
problem Wave equation on product cones with asymptotic expansions near null and future infinities.
method Joint asymptotic expansion, adaptation of Cheeger-Taylor method, new propagation estimates.
result New decay rates at future infinity are resonances of a hyperbolic cone.
Study on hyperbolic surfaces' volumes, proving asymptotic expansion for high genus.
problem Analyzing the volume of moduli spaces of hyperbolic surfaces with varying genus.
method Topological recursion formula by Mirzakhani, asymptotic expansion for high genus.
result Explicit computation of the second term in the asymptotic expansion.
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…
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.
New insights into black hole horizons from asymptotic expansions.
problem Understanding the geometry of black hole horizons.
method Proving the asymptotic expansion of spacetime metrics at non-degenerate Killing horizons.
result The full asymptotic expansion of smooth vacuum metrics at non-degenerate Killing horizons is determined by the horizon geometry.
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 g in terms of the Seifert invariants and standard data for g. A main corollary is a determination of the full asymptotic expansions …
Study finds solutions to Yamabe equation with specific behavior near singular points.
problem Existence of solutions with prescribed asymptotic behavior near singular points of the Yamabe equation.
method Analysis of positive solutions with isolated singularities and asymptotic expansions.
result Existence of solutions with arbitrarily high order of approximation near singular points.
Proves formula for unique Kähler-Einstein metric on quasi-projective manifolds.
problem Finding unique Kähler-Einstein metrics on quasi-projective manifolds.
method Elementary proof using ODE solutions and spectral theory.
result Asymptotic expansion formula for unique complete Kähler-Einstein metric.
Study of Witten-Reshetikhin-Turaev invariants for mapping tori.
problem Asymptotic expansion of Witten-Reshetikhin-Turaev invariants of mapping tori.
method Geometric quantization of moduli spaces of flat connections, Picard-Lefschetz theory for Laplace integrals.
result Full asymptotic expansion provided for pseudo-Anosov mapping classes on a punctured torus.