We find G2-manifolds with specific asymptotic properties.
problem Existence and structure of G2-manifolds with ALC asymptotics.
method Robust Fredholm theory for ALC spaces, proving existence and rigidity results.
result Existence of a G2-analogue of the Atiyah-Hitchin metric and good moduli theory for ALC G2-holonomy metrics.
Survey of spectral theory and dynamics for infinite volume hyperbolic manifolds.
problem Understanding infinite volume asymptotically hyperbolic manifolds.
method Survey of geometry, spectral theory, dynamics, and quantum/classical mechanics.
result Recent results, ideas, and conjectures discussed.
New self-expander found between two given asymptotic ones.
problem Finding new self-expanders between given asymptotic ones.
method Developed a min-max theory for asymptotically conical self-expanders of mean curvature flow.
result Existence of a new asymptotically conical self-expander trapped between two given ones.
Study instantons on asymptotically conical Spin(7)-manifolds, identifying deformation spaces.
problem Deformation theory of instantons on specific Spin(7)-manifolds.
method Relating deformation complex to spinors, identifying kernel of twisted negative Dirac operator.
result Virtual dimension of moduli space calculated using index theorem and Dirac operator spectrum.
The asymptotic dimension theory was founded by Gromov in the early 90s. In this paper we give a survey of its recent history where we emphasize two of its features: an analogy with the dimension theory of compact metric spaces and applications to the theory of discrete groups.
New insights into manifold properties using Seiberg-Witten and L2 harmonic theories.
problem Characterizing properties of 4-manifolds with specific geometric conditions.
method Combining Seiberg-Witten theory on compact manifolds and L2 harmonic theory on non-compact manifolds, with a new argument for asymptotic properties. result Found a pair of homeomorphic 4-manifolds with distinct geometric properties under Riemannian metrics.
The paper develops AMP theory for sparse and robust regression with polynomial iterations.
problem Challenges in high-dimensional statistical estimation due to asymptotic theory breakdown.
method Non-asymptotic distributional theory of AMP for sparse and robust regression.
result First finite-sample non-asymptotic distributional theory of AMP for polynomial iterations.
New degree theory proves existence of solitons on 4D manifolds.
problem Existence of gradient expanding solitons on 4D manifolds.
method Developed new degree theory for 4D, asymptotically conical gradient expanding solitons.
result Existence of solitons asymptotic to any cone over S^3 with non-negative scalar curvature.
New theory for eigenvectors of generalized Laplacian matrices, addressing dependency issues.
problem Dependency in random matrix theory hinders eigenvector analysis for latent embeddings.
method Introduces generalized Laplacian matrices and a new asymptotic theory framework.
result Established asymptotic normalities for spiked eigenvectors and eigenvalues.
Establishes scattering theory for de Sitter vacuum solutions in even dimensions.
problem Quantitative nonlinear scattering theory for asymptotically de Sitter vacuum solutions in even spatial dimensions.
method Geometric Littlewood-Paley decomposition of the solution, constructing the scattering map.
result Existence and uniqueness of scattering states, asymptotic completeness, and invertible scattering map with quantitative control.
A tutorial on non-asymptotic system identification methods.
problem Identifying system parameters in linear models.
method Covering technique, Hanson-Wright Inequality, method of self-normalized martingales.
result Streamlined proofs of least-squares based estimator performance.
Estimates growth of reciprocal classes in Hecke groups.
problem Estimating the growth of reciprocal conjugacy classes in Hecke groups.
method Using free product structure and word lengths of reciprocal elements, with tools from basic probability theory.
result Estimates the asymptotic growth of reciprocal conjugacy classes in Hecke groups.
We clarify and refine the relation between the asymptotic behavior of the colored Jones polynomial and Chern-Simons gauge theory with complex gauge group SL(2,C). The precise comparison requires a careful understanding of some delicate issues, such as normalization of the colored Jones polynomial and the choice of pola…
The asymptotic lattices and their transformations are studied within the line geometry approach. It is shown that the discrete asymptotic nets are represented by isotropic congruences in the Plucker quadric. On the basis of the Lelieuvre-type representation of asymptotic lattices and of the discrete analog of the Mouta…
We introduce dynamic asymptotic dimension, a notion of dimension for actions of discrete groups on locally compact spaces, and more generally for locally compact étale groupoids. We study our notion for minimal actions of the integer group, its relation with conditions used by Bartels, Lück, and Reich in the context of…
Paper examines risk measure expansions under FGM dependence, improving accuracy at extreme levels.
problem Capturing higher-order tail behavior and dependence effects in risk measures.
method Second-order asymptotic expansions using extreme value theory and regular variation theory.
result Second-order approximations reduce approximation errors, especially at extreme confidence levels.
We review the spectral analysis and the time-dependent approach of scattering theory for manifolds with asymptotically cylindrical ends. For the spectral analysis, higher order resolvent estimates are obtained via Mourre theory for both short-range and long-range behaviors of the metric and the perturbation at infinity…
Study asymptotic properties of generalized shortfall risk measures for heavy-tailed risks.
problem Understanding risk measures for heavy-tailed risks.
method Derive asymptotic expansions for generalized shortfall risk measures.
result Unified theory for risk measures including distortion and utility-based measures.
Extends Arnold's linking theory to higher dimensions and submanifolds.
problem Volume-preserving actions in higher dimensions and submanifolds.
method Generalization of V. Arnold's theory to Rk and Rℓ. result Extension of asymptotic linking to higher dimensions and submanifolds.
Develops a new asymptotic efficiency theory for non-Euclidean parameter spaces.
problem Lack of a unified efficiency theory for non-Euclidean parameter spaces.
method Introduces a new theory for Riemannian manifolds with regularity conditions.
result Establishes efficiency bounds for non-Euclidean parameter spaces.
The paper proves constant mean curvature surfaces in specific manifold types.
problem Existence of surfaces with constant mean curvature in asymptotically flat and hyperbolic manifolds.
method Combines min-max theory with inverse mean curvature flow.
result Existence of compact surfaces with constant mean curvature in asymptotically flat and hyperbolic manifolds.
New proof of Penrose inequality using potential theory.
problem Proving the Riemannian Penrose inequality for black holes.
method Establishing a monotonicity formula for the p-capacitary potential.
result A new proof of the Penrose inequality for black holes.
The paper develops asymptotic theory for QRF variable importance, revealing a bias-variance trade-off.
problem Challenges in statistical inference for QRF variable importance due to non-smoothness and bias-variance trade-off.
method Developed asymptotic theory using pinball loss and Knight's identity, uncovered phase transition phenomenon, derived asymptotic bias.
result Theoretical foundation for understanding QRF inference limitations in high-dimensional settings.
The article develops deformation theory for ACyl associative submanifolds in ACyl G2-manifolds.
problem Deformation theory of ACyl associative submanifolds in ACyl G2-manifolds.
method Study of moduli spaces with fixed and varying asymptotic data, computing virtual dimensions.
result The moduli space of ACyl associative submanifolds embeds as a Lagrangian submanifold in the moduli space of holomorphic curves.
The paper proves asymptotic normality for multinomial logistic regression on null covariates.
problem Classical asymptotic normality results fail in high-dimensional multinomial logistic models.
method Developed asymptotic normality and chi-square results for multinomial logistic MLE on null covariates.
result Validated new methodology to test feature significance in high-dimensional classification problems.
Extends Floquet-Bloch theory to nilpotent groups for geometric applications.
problem Asymptotic problems on nilpotent covers of negatively curved manifolds.
method Generalized Floquet-Bloch theory using Malcev completions.
result Branching formula relating finite and infinite-dimensional representations.
New theory for clustering in geometric and adaptive settings.
problem Clustering in non-Euclidean spaces and adaptive parameters.
method Asymptotic theory for k-means and related methods. result Strong consistency and asymptotic limit theorems for various clustering procedures.
Study geometric characterization of asymptotic pseudodifferential calculus on spinor bundles.
problem Geometric characterization of asymptotic pseudodifferential calculus on spinor bundles.
method Groupoid approach to pseudodifferential calculus, rescaled bundle.
result Rescaled bundle provides geometric characterization to asymptotic pseudodifferential calculus on spinor bundles.
Extreme value theory enhances statistical learning extrapolation for rare events.
problem Challenges in traditional machine learning methods for extreme data.
method Asymptotic theory and statistical tools for tail behavior.
result Effective extrapolation methods for extreme quantiles and anomalies.
Paper improves risk estimation for extreme events.
problem Estimating extreme risks accurately.
method Modified Bayes risk for expectiles, asymptotic expansions, efficient estimators.
result Asymptotic normality of estimators proved.
We study gluings of asymptotically cylindrical special Lagrangian submanifolds in asymptotically cylindrical Calabi--Yau manifolds. We prove both that there is a well-defined gluing map, and, after reviewing the deformation theory for special Lagrangians, prove that this gluing map defines a local diffeomorphism from m…
The study connects norms and filtrations on section rings of projective manifolds.
problem Understanding norms and filtrations on section rings of polarized projective manifolds.
method Analyzes submultiplicative norms and their equivalence to sup-norms, discusses applications to spectral theory and holomorphic extension.
result Injective and projective tensor norms on symmetric algebras are asymptotically equivalent.
For an asymptotically hyperbolic metric on the interior of a compact manifold with boundary, we prove that the resolvent and scattering operators are continuous functions of the metric in the appropriate topologies.
We introduce a new geometric evolution equation for hypersurfaces in asymptotically flat spacetime initial data sets, that unites the theory of marginally outer trapped surfaces (MOTS) with the study of inverse mean curvature flow in asymptotically flat Riemannian manifolds. A theory of weak solutions is developed usin…
We study the spectral theory of asymptotically hyperbolic manifolds with ends of warped product type. Our main result is an upper bound on the resonance counting function with a geometric constant expressed in terms of the respective Weyl constants for the core of the manifold and the base manifold defining the ends.
We develop a local theory for the construction of singular spacetimes in all spacetime dimensions which become asymptotically self-similar as the singularity is approached. The techniques developed also allow us to construct and classify exact self-similar solutions which correspond to the formal asymptotic expansions …
Applying standard techniques from Toeplitz operator theory, we analyze the asymptotics of the Hilbert-Smith norms of the TQFT operators coming from isotopy classes of one dimensional oriented submanifolds on a closed oriented surface. We thereby obtain a Toeplitz operator interpretation and generalization of the asympt…
In this paper we introduce a notion of scattering theory for the Laplace-Beltrami operator on non-compact, connected and complete Riemannian manifolds. A principal condition is given by a certain positive lower bound of the second fundamental form of angular submanifolds at infinity. Another condition is certain bounds…
Study of height jumps in Ceresa cycle using asymptotic Hodge theory.
problem Understanding height jumps in the Ceresa cycle.
method Analysis of asymptotic behavior of Hain-Reed beta-invariant in degenerating families of curves.
result Height jump of Ceresa cycle is equal to the slope of the dual graph of the curve.
We extend asymptotic formulas for saddle connections on translation surfaces.
problem Counting saddle connections on translation surfaces with large genus.
method Recursive formulas and asymptotic analysis for all strata and multiplicities.
result Asymptotics for all saddle connections on translation surfaces of growing genus.
Develops adiabatic theory for ACW flow on surfaces.
problem Evolution of large closed surfaces under area-constrained Willmore flow.
method Constructs a map on a four-dimensional manifold of barycenters to characterize ACW flow dynamics.
result Explicit four-dimensional effective dynamics of barycenters serves as an asymptotic approximation for ACW flow.
Study on high-codimensional minimal surfaces in hyperbolic space.
problem Understanding high-codimensional minimal surfaces in hyperbolic space.
method Investigating asymptotic behavior and boundary regularity of area-minimizing currents.
result Established boundary regularity results for high-codimensional minimal surfaces near their asymptotic boundaries.
The abstract proposes a neural network theory using quantum field theory.
problem Understanding the behavior of neural networks in the asymptotic and non-asymptotic limits.
method Mapping neural networks to Wilsonian effective field theory, using Gaussian processes and Feynman diagrams.
result Established a direct connection between overparameterization and simplicity of neural network likelihoods.
This paper provides performance guarantees for neural estimation of statistical distances.
problem Developing performance guarantees for neural estimation of statistical distances.
method Non-asymptotic error bounds using function approximation theorems and empirical process theory.
result Established a fundamental tradeoff between approximation and estimation errors in neural estimation of statistical distances.
The paper analyzes short maturity Asian options using large deviations theory.
problem Efficiency of existing methods for small maturities and volatilities.
method Large deviations theory and a local volatility model with a jump term.
result Asymptotics for Asian options are derived, showing rare event behavior for out-of-the-money options and more complex behavior for at-the-money options.
We prove positive mass theorem with angular momentum and charges for axially symmetric, simply connected, maximal, complete initial data sets with two ends, one designated asymptotically flat and the other either (Kaluza-Klein) asymptotically flat or asymptotically cylindrical, for 4-dimensional Einstein-Maxwell theory…
Study geometric operators on Tian-Yau spaces, finding L2 harmonic forms and asymptotic regularity.
problem Analyzing geometric elliptic operators on Tian-Yau spaces.
method Use a-pseudodifferential calculus to determine L2 harmonic forms and asymptotic regularity. result Determine the space of L2 harmonic forms and refined asymptotic regularity of ALH* structures. We analyze SGAs for statistical inference via asymptotics, improving tuning methods.
problem Improper tuning of SGAs for optimization and sampling.
method Characterize large-sample asymptotics of SGAs via step-size and sample-size scaling limits.
result Iterate averaging with large step size is robust and asymptotically has covariance proportional to MLE's.