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 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 volume conjecture is extended to all orders for hyperbolic 3-manifolds using complex Chern-Simons theory.
problem Extending the volume conjecture to all orders for hyperbolic 3-manifolds.
method Deriving formulas for the perturbative expansion of the partition function of complex Chern-Simons theory and comparing it to Witten-Reshetikhin-Turaev invariants.
result The conjecture that the perturbative expansion of the partition function of complex Chern-Simons theory matches the Witten-Reshetikhin-Turaev invariants at roots of unity in the limit of infinitely many invariants.
Study superpolynomials and volume conjectures for knot homologies.
problem Understanding superpolynomials and their properties for knot homologies.
method Cyclotomic expansion and volume conjecture for superpolynomials of colored HOMFLY-PT and Kauffman homologies.
result Proved volume conjecture for SU(n) specialized superpolynomial associated to reduced colored HOMFLY homology. 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.
Study volume expansion on convex domains using Blaschke metric.
problem Volume expansion of Blaschke metric on strictly convex domains.
method Expressed logarithmic coefficient L as integrals of affine invariants over the boundary and formulated intrinsic geometry as conformal Codazzi structure.
result L is a global conformal invariant of the boundary.
The paper connects volume conjecture with SU(n) invariants and their limits.
problem Understanding the volume conjecture for SU(n) invariants. method Using symmetry properties and cyclotomic expansions, the authors propose and prove conjectural formulas for SU(n) invariants. result Proof of conjectural formulas for SU(n) invariants, including volume conjecture. Study calculates volume of small sub-Riemannian balls in 3D contact manifolds.
problem Computing the volume of small sub-Riemannian balls in 3D contact manifolds.
method Asymptotic expansion and geometric invariants of the sub-Riemannian structure.
result Expressed first geometric coefficients in terms of sub-Riemannian structure invariants.
Study improves optimal execution model with trading volume considerations.
problem Optimizing trading strategies in models with varying market volumes.
method Introduced a penalization method for an adaptive optimization problem in the Almgren-Chriss model.
result Verified the optimality of the volume-weighted average-price strategy and derived a second-order asymptotic expansion of the optimal strategy.
A singularity theorem based on asymptotic volume growth
problem Proving singularity theorems
method Introducing asymptotic volume-expansion invariants
result Proving an explicit upper bound on the time-separation from a hypersurface to its chronological past
Exact asymptotic value of Weil-Petersson volumes computed for large genus surfaces.
problem Computing the exact asymptotic value of Weil-Petersson volumes for large genus surfaces.
method Analysis of Witten-Kontsevitch intersection numbers and expansion of volumes.
result Exact asymptotic value of volume polynomials computed for hyperbolic surfaces.
Study new volume invariant in Hamiltonian dynamics.
problem Volume behavior in Hamiltonian dynamics.
method Introduce new invariant and study its properties.
result Generalize Riemannian volume expansion to sub-Riemannian manifolds.
The paper analyzes the geometric dynamics of volume expansion.
problem Existence of compact and complete spacelike hypersurfaces in space-time.
method Global geometric analysis of volumetric expansion.
result Insights into the existence of compact and complete spacelike hypersurfaces.
Study on renormalized volume of minimal submanifolds in Poincare-Einstein manifolds.
problem Analyzing minimal submanifolds in Poincare-Einstein manifolds.
method Deriving formulae for first and second variations of renormalized volume, proving asymptotic descriptions, and deriving inner-product relationships.
result Existence of asymptotic description and L2-inner-product relationship for specific cases. Ricci flow preserves renormalized volumes of asymptotically hyperbolic metrics.
problem Volume renormalizability of asymptotically hyperbolic metrics.
method Normalized Ricci flow, Riesz renormalization.
result Renormalized volume is preserved under normalized Ricci flow.
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.
A new Lagrangian formulation of the Raychaudhuri equation in non-Riemannian geometry.
problem Formulating the Raychaudhuri equation in non-Riemannian geometries.
method Established a formal connection between the expansion scalar and the cross-sectional volume of the congruence. Derived a Lagrangian and Hamiltonian formulation.
result The expansion scalar equals the fractional rate of change of volume, weighted by a scalar factor.
Connectedness of small clusters in Riemannian and Finsler manifolds proven.
problem Understanding connectedness of small clusters in Riemannian and Finsler manifolds.
method Proved connectedness and small diameter properties for clusters of small volume in both manifolds.
result Clusters in Riemannian manifolds are connected and have small diameter; in Finsler manifolds, they are at most m connected components of small diameter.
Study shows decay of correlations on specific types of flows.
problem Analyzing decay of correlations in specific flow types.
method Asymptotic expansion of correlation function on Abelian covers.
result Established an expansion in inverse powers of time.
New properties are derived of renormalized volume functionals, which arise as coefficients in the asymptotic expansion of the volume of an asymptotically hyperbolic Einstein (AHE) manifold. A formula is given for the renormalized volume of an even-dimensional AHE manifold in terms of an arbitrary totally geodesic compa…
Develops a new method to calculate volumes of manifolds with boundaries.
problem Calculating volumes of manifolds with boundaries, especially those with singularities.
method Introduces a regulated volume expansion with anomaly terms for singular measures.
result Shows how anomaly terms generate invariant pairs for hypersurfaces with boundaries.
Researchers develop Q-curvature for convex hypersurfaces using ambient metrics.
problem Calculating Q-curvature for convex hypersurfaces in projective manifolds.
method Using the ambient metric, they construct GJMS operators and relate Q-curvature to the logarithmic coefficient in volume expansion.
result Derived first and second variation formulas for strictly convex domains.
Proves volume conjectures for figure-eight knot surgeries.
problem Volume conjectures for hyperbolic 3-manifolds.
method Ohtsuki's method applied to figure-eight knot surgeries.
result Proves Asymptotic Expansion and Volume Conjectures for figure-eight knot surgeries.
In the main theorem of this paper we treat the problem of existence of minimizers of the isoperimetric problem under the assumption of small volumes. Applications of the main theorem to asymptotic expansions of the isoperimetric problem are given.
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.
Sharp Sobolev inequality derived for Riemannian manifolds with bounded Ricci curvature.
problem Deriving a sharp Sobolev inequality for Riemannian manifolds with bounded Ricci curvature.
method Reduction to functions with small volume support, first order uniform asymptotic expansion of isoperimetric profile, local uniform Sobolev inequality.
result Sharp Sobolev inequality for W1,p(M) into Ln−pnp(M) is derived. 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.
The paper modifies asset pricing models using Taylor series expansions and market-based averages.
problem Improving asset pricing models to better reflect market dynamics.
method Derives new pricing equations using Taylor series expansions and market-based averages.
result New expressions for asset prices and volatilities derived from market data.
We will prove that Ruelle L-function for a cuspidal local system on an odd dimensional hyperbolic manifold with finite volume satisfies a functional equation and an analog of the Riemann hypothesis. We will also compute its Laurent expansion at the origin and will prove that the second coefficient coincides with a rati…
New volume functions for random hyperbolic surfaces link to spectral gaps.
problem Analyzing spectral gaps in random hyperbolic surfaces.
method Introduced new volume functions VgT(l), derived their asymptotic expansions, and linked them to spectral gaps. result Coefficients in the asymptotic expansion of VgT(l) are Friedman-Ramanujan functions. 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.
Bayesian optimization tackles unknown search spaces with automatic expansion.
problem Bayesian optimization in unknown search spaces is challenging.
method Proposes a systematic volume expansion strategy to find points close to the objective function maximum without specifying parameters.
result Derives analytic expressions for expansion triggers and sizes, achieving epsilon-accuracy after a finite number of iterations.
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.
We show that for a torus knot the SL(2;C) Chern-Simons invariants and the SL(2;C) twisted Reidemeister torsions appear in an asymptotic expansion of the colored Jones polynomial. This suggests a generalization of the volume conjecture that relates the asymptotic behavior of the colored Jones polynomial of a knot to the…
Paper renormalizes volume of singular Yamabe metrics.
problem Renormalizing volume of singular Yamabe metrics.
method Volume renormalization for singular Yamabe metrics.
result Existence of a conformally invariant energy.
The behavior under conformal change of the renormalized volume coefficients associated to a pseudo-Riemannian metric is investigated. It is shown that they define second order fully nonlinear operators in the conformal factor whose algebraic structure is elucidated via the introduction of "extended obstruction tensors"…
Study calculates spectral invariants for elastic bodies, identifying ball uniquely by spectrum.
problem Identifying elastic bodies from their spectrum.
method Explicitly calculates coefficients of asymptotic expansion of semigroup trace.
result An n-dimensional ball is uniquely determined by its Navier-Lamé spectrum. In the genus expansion of the HOMFLY polynomials their representation dependence is naturally captured by symmetric group characters. This immediately implies that the Ooguri-Vafa partition function (OVPF) is a Hurwitz tau-function. In the planar limit involving factorizable special polynomials, it is actually a trivia…
The paper calculates spectral invariants of magnetic Steklov eigenvalues on Riemannian manifolds.
problem Investigating spectral invariants of magnetic Steklov eigenvalues on Riemannian manifolds.
method Established an effective procedure to calculate all coefficients of the heat trace asymptotic expansion.
result Explicitly provided expressions for the first four coefficients of the heat trace asymptotic expansion.
Study uses renormalized area to determine metric expansion from minimal surfaces.
problem Recovering the expansion of asymptotically hyperbolic metrics from minimal surfaces.
method Uses renormalized area functional on minimal submanifolds to recover metric expansion.
result Proves rigidity for log-analytic metrics and determines obstruction tensor.
To a knot in 3-space, one can associate a sequence of Laurent polynomials, whose nth term is the nth colored Jones polynomial. The Volume Conjecture for small angles states that the value of the n-th colored Jones polynomial at $e^{\a/n}$ is a sequence of complex numbers that grows subexponentially, for a fixed s…
Study shows volumes of knot complements are bounded by linear functions of geodesic periods.
problem Volume calculation of knot complements associated with geodesics on modular surfaces.
method Analyzes geodesics on modular surfaces, their associated knots, and their complements' volumes.
result Volumes of knot complements are bounded linearly by the period of geodesic continued fractions.
3-manifolds with torsion homology expand in all dimensions.
problem Constructing 3-manifolds with good expansion properties.
method Constructing 3-manifolds with specific homological properties and demonstrating their expansion.
result 3-manifolds with torsion homology expand in all dimensions.
The volume weighted average price (VWAP) execution strategy is well known and widely used in practice. In this study, we explicitly introduce a trading volume process into the Almgren-Chriss model, which is a standard model for optimal execution. We then show that the VWAP strategy is the optimal execution strategy for…
This article presents a new definition of Branson's Q-curvature in even-dimensional conformal geometry. We derive the Q-curvature as a coefficient in the asymptotic expansion of the formal solution of a boundary problem at infinity for the Laplacian in the Poincare metric associated to the conformal structure. This giv…
The volume conjecture states that for a hyperbolic knot K in the three-sphere S^3 the asymptotic growth of the colored Jones polynomial of K is governed by the hyperbolic volume of the knot complement S^3\K. The conjecture relates two topological invariants, one combinatorial and one geometric, in a very nonobvious, no…
Study of umbilic points on Willmore surfaces in 3-sphere.
problem Characterizing umbilic points on Willmore surfaces.
method Analysis of conformal Gauss map and Gauss-Bonnet formula.
result Unified expression for Willmore energy in space-forms.
The paper connects Gaussian matrix models to cohomological field theories using topological recursion.
problem Understanding Gaussian matrix model means in all genera.
method Explicit relation between Gaussian means and KPMM, topological recursion.
result Coefficients of Gaussian means in all genera are polynomials in special times weighted by ancestor invariants.