Polyhomogeneous expansions for Calabi-Yau metrics near singularities.
problem Analyzing metrics near conical singularities of Calabi-Yau conifolds.
method Weighted Melrose-type blow-ups, gluing, and solving complex Monge-Ampère equations.
result Polyhomogeneous expansions of smooth Calabi-Yau metrics on resolutions and smoothings.
Study revisits Bondi mass and discusses memory effect in polyhomogeneous spacetimes.
problem Analyzing the asymptotic behavior and memory effect in polyhomogeneous spacetimes.
method Revisits Bondi mass using Iyer-Wald formalism and discusses memory effect in vacuum polyhomogeneous spacetimes.
result The balance law remains unchanged in polyhomogeneous spacetimes with logarithmic terms.
Study of Nahm pole solutions over 3-manifolds, proving smoothness at boundary.
problem Analyzing Nahm pole solutions over 3-manifolds.
method Examining polyhomogeneous solutions and their expansions.
result Smoothness of sub-leading terms at boundary for Einstein 3-manifolds.
Characterizes polyhomogeneous symbols and applies to Heisenberg calculus.
problem Understanding polyhomogeneous symbols and their applications.
method Simple characterisation and generalization of A.~Connes' tangent groupoid.
result Heisenberg calculus on contact manifolds coincides with groupoid calculus.
We establish multiparameter resolvent trace expansions for elliptic boundary value problems, polyhomogeneous both in the resolvent and the auxiliary parameter. The present analysis is rooted in the joint project with Matthias Lesch on multiparameter resolvent trace expansions on revolution surfaces with applications to…
Study of conformally compact metrics and Lovelock tensors in even dimensions.
problem Understanding conformally compact metrics satisfying Lovelock equations.
method Polyhomogeneous expansions and formal solutions to singular Yamabe-(2q) problem.
result Identification of a boundary obstruction in even dimensions that generalizes the ambient obstruction tensor.
Sharp eigenvalue estimates on degenerating hyperbolic surfaces.
problem Estimating the first non-zero eigenvalue of Laplacian on hyperbolic surfaces as a collar degenerates.
method Using the relationship between the eigenvalue and Fenchel-Nielsen length coordinate, proving estimates with optimal error rates.
result Improved estimates and new information on leading order terms of eigenvalue expansion.
We present a new multiparameter resolvent trace expansion for elliptic operators, polyhomogeneous in both the resolvent and auxiliary variables. For elliptic operators on closed manifolds the expansion is a simple consequence of the parameter dependent pseudodifferential calculus. As an additional nontrivial toy exampl…
Study on stability of Minkowski spacetime using advanced mathematical techniques.
problem Nonlinear stability of Minkowski spacetime in Einstein vacuum equation.
method Iteration scheme, linearized equation, generalized harmonic gauge, constraint damping, energy and vector field methods, Melrose's b-analysis.
result Polyhomogeneous spacetime metric produced from polyhomogeneous initial data.
We consider the Riemann moduli space Mγ of conformal structures on a compact surface of genus γ>1 together with its Weil-Petersson metric gWP. Our main result is that gWP admits a complete polyhomogeneous expansion in powers of the lengths of the short geodesics up to the singu…
This article considers the existence and regularity of Kahler-Einstein metrics on a compact Kahler manifold M with edge singularities with cone angle 2πβ along a smooth divisor D. We prove existence of such metrics with negative, zero and some positive cases for all cone angles 2πβ≤2π. The results in the po…
Study polyhomogeneity of metrics with Lie structure along Ricci flow.
problem Polyhomogeneity of metrics with Lie structure along Ricci flow.
method Analyzing polyhomogeneity of complete Riemannian metrics with Lie structure fibered at infinity under Ricci flow.
result Polyhomogeneity of metrics compatible with a Lie structure fibered at infinity is locally preserved by the Ricci-DeTurck flow.
Paper proves nonpositive boundary integral for Liouville's equation, zero only for discs.
problem Properties of Liouville's equation and boundary integrals.
method Polyhomogeneous expansions and rigidity/gap theorems.
result Boundary integral is nonpositive and zero only for discs.
The article studies mapping properties of Radon transform and backprojection on a unit ball.
problem Polyhomogeneous mapping properties of Radon transform and backprojection operator on the unit ball.
method Constructs a double b-fibration to desingularize the point-hyperplane relation, provides formulas and sharper estimates.
result Sharper estimates on polyhomogeneous mapping properties of Radon transform and backprojection compared to classic estimates.
We prove that the deformation theory of compactifiable asymptotically cylindrical Calabi-Yau manifolds is unobstructed. This relies on a detailed study of the Dolbeault-Hodge theory and its description in terms of the cohomology of the compactification. We also show that these Calabi-Yau metrics admit a polyhomogeneous…
The study of Einstein metrics with corners and boundaries.
problem Formal study of Einstein metrics with corners and boundaries.
method Generalization and extension of previous work; formal expansion and existence demonstration.
result Existence of Einstein metrics up to a certain order in a cornered asymptotically hyperbolic normal form.
Study linear differential operators on special manifolds.
problem Analyzing elliptic differential operators on specific types of manifolds.
method Examining a linear elliptic differential operator of the form Δ + V - λ on quasi-asymptotically conical manifolds.
result Established an isomorphism theorem for these operators.
We study various aspects of the noncommutative residue for an algebra of pseudodifferential operators whose symbols have an expansion a∼∑j=0∞am−j,am−j(x,ξ)=∑l=0kam−j,l(x,ξ)logl∣ξ∣, where am−j,l is homogeneous in ξ of degree m−j. We will explain why this algebra of pseudo…
We prove a local well-posedness theorem for the (n+1)-dimensional Einstein equations in Lorentzian signature, with initial data (g~,K) whose asymptotic geometry at infinity is similar to that anti-de Sitter (AdS) space, and compatible boundary data g^ prescribed at the time-like conformal boundary of spa…
Let X be a quasiprojective manifold given by the complement of a divisor $\bD$ with normal crossings in a smooth projective manifold $\bX$. Using a natural compactification of X by a manifold with corners $\tX$, we describe the full asymptotic behavior at infinity of certain complete Kahler metrics of finite volume o…
We find a resonance free region polynomially close to the critical line on Conformally compact manifolds with polyhomogeneous metric.
We prove local polyhomogeneity of asymptotically real or complex hyperbolic Einstein metrics, with application to unique continuation problems.
We show that C^2 conformally compact Riemannian Einstein metrics have conformal compactifications that are smooth up to the boundary in dimension 3 and all even dimensions, and polyhomogeneous in odd dimensions greater than 3.
This work removes logarithmic singularities from hyperboloidal initial data without creating new ones.
problem Logarithmic singularities in hyperboloidal initial data sets.
method Evolutionary framework of the constraint equations and generalization of Beyer and Ritchie's result.
result Generic solutions of the constraint equations are free of logarithmic singularities.
We show that any polyhomogeneous asymptotically hyperbolic constant-mean-curvature solution to the vacuum Einstein constraint equations can be approximated, arbitrarily closely in Hölder norms determined by the physical metric, by shear-free smoothly conformally compact vacuum initial data.
New calculus solves boundary value problems for elliptic operators.
problem Boundary value problems for 0-elliptic operators.
method Developed a new calculus called symbolic 0-calculus to handle boundary value problems.
result Construct left and right parametrices for 0-elliptic operators with boundary conditions.
Study X-ray transform on manifolds, desingularize, and improve mapping properties.
problem Characterize mapping properties of X-ray transform and its adjoint on manifolds with strictly convex boundary.
method Use b-fibrations, desingularize, and apply Melrose's Pushforward Theorem to analyze polyhomogeneous functions.
result Improved mapping properties of X-ray transform and its adjoint, recovering sharp results.
Study analyzes low-energy behavior of Schrödinger operators with Coulomb potentials.
problem Analyzing the limiting resolvent of Schrödinger operators at low energies.
method Using Vasy's second microlocal approach (Lagrangian approach), uniformly analyzing the resolvent from E=0. result Obtained oscillatory asymptotics for the resolvent output at low energy, differing from short-range cases.
New boundary conditions for knot polynomials solved via elliptic KW equations.
problem Computing Jones polynomial coefficients from knot solutions.
method Formulated generalized Nahm pole boundary conditions for KW equations, proved ellipticity, analyzed polyhomogeneity.
result KW equations with generalized Nahm pole conditions are elliptic and solutions are polyhomogeneous.
We show that the polyhomogeneity at infinity of an asymptotically complex hyperbolic metric is preserved along the Ricci-DeTurck flow. Moreover, if the initial metric is `smooth up to the boundary', this will be preserved by the Ricci-DeTurck flow and the normalized Ricci flow. When the initial metric is Kähler, sharpe…
New equations simplify gauge-theoretic Khovanov homology solutions.
problem Solving the Haydys-Witten equations for Khovanov homology.
method Introduced decoupled version of Haydys-Witten equations; investigated asymptotic behavior.
result Decoupled equations simplify analysis of full equations on manifolds with ends and boundaries.
We consider the family of constant curvature fiber metrics for a Lefschetz fibration with regular fibers of genus greater than one. A result of Obitsu and Wolpert is refined by showing that on an appropriate resolution of the total space, constructed by iterated blow-up, this family is log-smooth, i.e. polyhomogeneous …
Study heat kernel on manifolds with fibred boundary metrics.
problem Analyzing spectral problems in manifolds with fibred boundary metrics.
method Construct heat kernel as polyhomogeneous conormal distribution.
result Fundamental step towards analysis of Ray-Singer torsion, eta-invariants and index theorems.
Our main result is that if a generic convex domain in Rn collapses to a domain in Rn−1, then the difference between the first two Dirichlet eigenvalues of the Euclidean Laplacian, known as the fundamental gap, diverges. The boundary of the domain need not be smooth, merely Lipschitz continuous. To motivate th…
Rescaling expansiveness proven for k*-expansive vector fields.
problem Proving rescaling expansiveness for k*-expansive vector fields.
method Introducing and exploring singular-expansive flows.
result Rescaling expansiveness established for k*-expansive vector fields.
The paper derives expansions for Green's operators and resolvents using Hadamard methods.
problem Analyzing normally hyperbolic operators and their Green's functions.
method Hadamard expansions for powers of Green's operators and resolvents.
result Derives expansions involving Hadamard coefficients for advanced/retarded Green's operators.
Develops a martingale expansion for stochastic volatility models.
problem Approximating marginal distributions of stochastic volatility models.
method Martingale expansion framework for continuous stochastic volatility models.
result First-order perturbation expansions for small volatility-of-volatility and fast mean-reversion models.
Taylor expansions improve reinforcement learning policies.
problem Improving reinforcement learning policy optimization.
method Taylor expansion policy optimization.
result Taylor expansions enhance performance of distributed algorithms.
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.
Proved cyclotomic expansion for double twist knots' HOMFLY-PT invariants.
problem Proving cyclotomic expansion for colored HOMFLY-PT invariants of double twist knots.
method Cyclotomic expansion formula for double twist knots.
result Confirmed cyclotomic expansion conjecture for SU(N)-invariants.
A new hypergraph expansion method treats vertices and hyperedges equally, improving node classification.
problem Information loss in hypergraph expansions on either vertex or hyperedge level.
method Proposes a new hypergraph formulation named line expansion (LE) that treats vertices and hyperedges symmetrically.
result The proposed line expansion method outperforms state-of-the-art baselines on five hypergraph datasets.
Study evaluates methods for expanding communities in hypergraphs using random walks.
problem Expanding communities in hypergraphs using random walks.
method Clique-expansion and tensor methods evaluated; hybrid method proposed.
result Parameter regimes identified where methods outperform each other.
Study of hypersurfaces with specific expansion properties.
problem Existence and properties of hypersurfaces with prescribed null expansion.
method Adapted Eichmair's Perron approach for existence.
result Topology theorem for hypersurfaces with prescribed null expansion.
Asymmetric expansion preserves convexity in hyperbolic geometry.
problem Maintaining convexity in hyperbolic geometry under asymmetric expansions.
method Generalizing earlier results on radial expansion to asymmetric expansion.
result Asymmetric expansion of hyperbolic convex sets remains convex.
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.
Sequence learning improves query expansion in information retrieval.
problem Improving query expansion in information retrieval systems.
method Used sequence to sequence algorithms to extract keywords from sentence embeddings and trained a neural network on open datasets.
result Sequence to sequence models can capture complex query expansion relations in word embeddings.
Researchers exhaust curve graph using rigid expansions on surfaces.
problem Exhausting the curve graph of surfaces with genus ≥ 3.
method Constructing a finite set of curves and using iterated rigid expansions.
result The constructed set exhausts the curve graph via rigid expansions.
Dropout increases the generalization of neural networks by expanding the weight space.
problem Understanding and improving the generalization of neural networks.
method Introducing weight expansion and showing that dropout leads to it.
result Dropout increases the generalization of neural networks by expanding the weight space.