Develops resolvent degree theory for algebraic geometry problems.
problem Hilbert's 13th Problem and related conjectures.
method Extends Brauer's resolvent degree theory to algebraic geometry.
result Hilbert's 13th Problem and related conjectures are equivalent to enumerative geometry problems.
Paper proves computational hardness for graph matching and detection problems.
problem Computational hardness for graph matching and detection problems in correlated random graphs.
method Algorithmic contiguity and low-degree advantage bounds.
result No efficient algorithms exist for certain graph matching and detection problems.
New findings on computational limits for estimating hidden structures.
problem Estimating hidden structures in noisy data.
method Use of low-degree polynomials as a restricted model of computation.
result Established low-degree hardness of recovery problems for easy detection problems.
The paper solves a problem related to curvature in complex geometry.
problem Resolving the prescribed Chern scalar curvature problem.
method Divided into three cases based on the sign of the Gauduchon degree, analyzed separately.
result Proves that certain functions are Chern scalar curvatures of conformal metrics.
Study on Kähler manifolds connects curvature decay with growth of holomorphic functions.
problem Analyzing properties of Kähler manifolds with nonnegative bisectional curvature.
method Established precise relations among minimal degree, volume growth, and scalar curvature decay.
result Unified understanding of Kähler-Ricci flow through polynomial growth holomorphic functions.
Proposes a test for stochastic block models with bounded degrees.
problem Testing Erdös-Rényi model versus bisection stochastic block model with bounded degrees.
method Likelihood-ratio (LR) type procedure based on regularization.
result Limit distributions as power Poisson laws under null and alternative hypotheses.
Graphs with nonnegative Bakry-Émery curvature have volume doubling and Poincaré inequalities.
problem Proving properties of graphs with specific curvature conditions.
method Graph-theoretic modified nonlinear heat-flow method, including point-mass consequences and diffusive exit-time control.
result Volume doubling and Poincaré inequalities for graphs with nonnegative Bakry-Émery curvature.
Christoffel function characterizes the corruption a bounded-degree certificate cannot remove in robust halfspace learning.
problem Robust halfspace learning under malicious noise
method Sum-of-Squares degree of outlier-removal certificate
result Christoffel function bounds the corruption a bounded-degree certificate cannot remove
The proliferation of models for networks raises challenging problems of model selection: the data are sparse and globally dependent, and models are typically high-dimensional and have large numbers of latent variables. Together, these issues mean that the usual model-selection criteria do not work properly for networks…
We find explicit models for the PSL(2,C)- and SL(2,C)-character varieties of the fundamental groups of complements in S^3 of an infinite family of two-bridge knots that contains the twist knots. We compute the genus of the components of these character varieties, and deduce upper bounds on the degree of the associated …
Projective resolves symplectic Steinberg module for number rings.
problem Constructing a projective resolution for symplectic Steinberg module.
method Similar to special linear group, but more complex construction.
result Computed top degree cohomology of congruence subgroups.
GRAMPA algorithm recovers latent vertex correspondence in correlated graphs with high probability.
problem Recovering latent vertex correspondence between unlabeled, edge-correlated weighted graphs.
method Spectral graph matching algorithm, GRAMPA, with exact recovery guarantees for Erdős-Rényi graphs.
result GRAMPA exactly recovers latent vertex correspondence with high probability for Erdős-Rényi graphs with edge correlation coefficient 1−σ2 and average degree at least polylog(n) when σ≲1/polylog(n). A new complex space resolves projective structures on surfaces.
problem Understanding projective structures on compact surfaces.
method Proposed a complex analytic space Pg and analyzed it for g=1. result The space Pg naturally resolves the orbifold locus of Ag=1. Spectral pruning compresses deep networks by reducing degrees of freedom.
problem Efficiently compress deep neural networks for edge devices.
method Develops a new theoretical framework and spectral pruning method.
result Shows a sharp generalization error bound for compressed models.
New algebraic theory classifies symplectic curves in complex projective space.
problem Classifying symplectic curves with specific singularities.
method Developed a novel algebraic theory of positive braids and conjugacy classes in the braid group.
result Established a complete classification of isotopy classes of degree three symplectic curves with An-singularities. By studying the development of shock waves out of discontinuity waves, in 1954 P. Lax discovered a class of PDEs, which he called 'completely exceptional', where such a transition does not occur after a finite time. A straightforward integration of the completely exceptionality conditions allowed Boillat to show that s…
Researchers resolve string theory ambiguities and define a new metric for massless spectrum.
problem Ambiguities in string theory regarding spin connection and Hodge decomposition.
method Constructing a vector bundle Q and operators D and D† to define a metric and gauge fixing.
result Massless spectrum are harmonic representatives of the operator D, resolving previous complications.
Researchers create functors to match colored homologies of knots and links.
problem Equivalence of colored HOMFLYPT homologies for links and knots.
method Constructing functors on singular Soergel bimodules to identify homologies.
result Established parity results for intrinsic column-colored homology of positive torus knots.
Ray-Singer torsion measures light degrees of freedom in black hole entropy.
problem Entropy of small black holes with many light particles.
method Computes partition function at genus one using mirror symmetry and Ray-Singer torsion.
result Ray-Singer torsion provides an effective quantum gravity cutoff known as the species scale.
Researchers found counterexamples to a 2-jet determination theorem in higher codimension.
problem Counterexample construction to the 2-jet determination Chern-Moser Theorem in higher codimension.
method Constructed counterexamples of quadratic submanifolds with specific properties.
result Generated counterexamples to the 2-jet determination Chern-Moser Theorem in higher codimension.
A new approach of solving the ill-conditioned inverse problem for analytical continuation is proposed. The root of the problem lies in the fact that even tiny noise of imaginary-time input data has a serious impact on the inferred real-frequency spectra. By means of a modern regularization technique, we eliminate redun…
Study extends resolvent estimates for non-even metrics on hyperbolic spaces.
problem Estimating resolvent for non-even metrics on asymptotically hyperbolic spaces.
method Extends Vasy's method for non-trapping geodesic flow, proving same strip size as Guillarmou.
result Same strip size for meromorphic continuation of resolvent as Guillarmou's result.
Paper compares GCNs and MPNNs, finding GCNs are one step ahead of WL algorithm.
problem Comparing graph convolutional networks (GCNs) and message-passing neural networks (MPNNs).
method Casts GCNs and MPNNs as MPNNs, studies distinguishing power of different architectures.
result GCNs are one step ahead of the Weisfeiler-Lehman (WL) algorithm in distinguishing power.
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…
The study examines compact spaces resolvable by p-adic actions.
problem Resolving compact spaces by p-adic actions.
method Free p-adic actions on compact spaces of lower dimension.
result Compact spaces with cohomological dimension 1 under Z[1/p].
Study low energy resolvent behavior on fibred boundary metrics.
problem Analyze the resolvent of Hodge Laplacian on manifolds with fibred boundary metrics.
method Develop a 'split' pseudodifferential calculus to handle different asymptotic behaviors.
result Precise asymptotic behavior of resolvent as a fibred boundary pseudodifferential operator.
New findings on tensor decomposition complexity, showing polynomial functions can estimate the largest component under certain conditions.
problem The complexity of tensor decomposition, especially for low-degree polynomials.
method Modeling a slightly larger component in a random tensor decomposition and using polynomial functions to estimate it.
result Polynomial functions can accurately estimate the largest component when r≪n3/2 but fail when r≫n3/2. We show that for any n > 3 there exists an equivalence functor from the category of n-fold connected simple coverings of B^3 x [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, to the category Chb^{3+1} of orientable relative 4-dimensional 2-handlebody cobordisms up to 2-deformations. As a c…
The paper generalizes relations between dynamical series and resolvents of vector fields.
problem Analyzing dynamical series using resolvents of vector fields.
method Derives the general form of relations involving intersection of kernel with integration currents for any smooth flow.
result Computes values of dynamical series and their relation with topological invariants.
In this paper, which is a natural continuation of our previous paper math.DG/0504557, we describe some special Lagrangians of cohomogeneity one in the resolved conifold. Our main result gives a foliation of the resolved conifold by T^2-invariant special Lagrangians, where the generic leaf is topologically T^2 X R. We a…
Researchers create a parametrix for resolvents on manifolds with ends.
problem Essential self-adjointness of elliptic symmetric differential operators on manifolds with ends.
method Introduced semiclasical pseudodifferential operators compatible with the end structure.
result Essential self-adjointness of elliptic symmetric differential operators proved.
AdamCB optimizes neural network training by adaptively selecting samples.
problem Inefficient convergence due to unequal influence of different data samples.
method Integrates combinatorial bandit techniques into Adam to adaptively select samples.
result AdamCB achieves faster convergence and better performance than existing methods.
Paper proposes methods to predict hard drive health using machine learning, improving accuracy and predictive time.
problem Predicting hard drive health with high accuracy from imbalanced SMART datasets.
method Layerwise perturbation-based adversarial training and semi-supervised learning.
result The model can predict hard drive health status 5-15 days in advance.
The resolved conifold geometry is linked to a special Kähler manifold and an instanton-corrected hyperkähler manifold.
problem Understanding the geometry of the resolved conifold and its associated structures.
method Explicit description of ASK and instanton-corrected HK manifolds, relating them to twistor coordinates and solving Riemann-Hilbert problems.
result The instanton-corrected hyperkähler manifold realizes a smoothing of the semi-flat HK metric associated with the ASK geometry.
Price and return predictions are limited by economic complexity, not just volatility.
problem Limited accuracy of price and return probability forecasts by Gaussian distributions.
method Analyzes economic reasons behind limitations in predicting price and return statistical moments.
result Predictions of price and return probabilities by Gaussian distributions are inaccurate due to economic complexity.
Deep learning can learn compositional functions more efficiently by breaking them into stages.
problem Understanding why deep learning performs better than shallow models in learning compositional functions.
method Analyzed learnability of compositional target functions using a three-layer fitting model trained with layer-wise spectral estimators.
result Learning compositional functions can be simplified by breaking them into stages, reducing the complexity of the learning problem.
Mathematical structures link Gromov-Witten to Donaldson-Thomas invariants.
problem Understanding non-perturbative topological string theory.
method Borel summation of Gromov-Witten potential and analysis of Stokes phenomena.
result Stokes phenomena encode Donaldson-Thomas invariants of the resolved conifold.
This paper develops a theory of graded manifolds in differential geometry.
problem Defining consistent global descriptions of graded manifolds with mixed graded coordinates.
method Using sheaves of graded commutative associative algebras on topological spaces.
result Resolved known issues in the definition of graded manifolds, especially those involving mixed graded coordinates.
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.
Model resolves asset pricing puzzles with price-impact.
problem Asset pricing puzzles like interest rate, stock-price volatility, and equity premium.
method Closed-form equilibrium model with exponential investors trading continuously and experiencing price-impact.
result Price-impact amplifies risk-sharing distortions, resolving puzzles.
New method combines FMEA and Bayesian Network for root cause analysis in lithium-ion battery production.
problem Complex cause-effect relationships in lithium-ion battery production.
method Combining FMEA with Bayesian Network to detect and resolve inconsistencies.
result Holistic method builds large-scale cross-process Bayesian Failure Network for root cause analysis.
Study resolvents of Bochner Laplacians on compact manifolds.
problem Analyzing the resolvents of Bochner Laplacians in the semiclassical limit.
method Introducing Heisenberg semiclassical pseudodifferential operators to study sections of line bundles.
result Resolvents and spectral projections of Bochner Laplacians are studied in the large power limit.
In this paper we continue our program of extending the methods of geometric scattering theory to encompass the analysis of the Laplacian on symmetric spaces of rank greater than one and their geometric perturbations. In our previous work we described the resolvent, and specifically the asymptotic behavior of the Green'…
In this paper we examine the Laplacian on the product of two asymptotically hyperbolic (or conformally compact, as they are often called) spaces from the point of view of geometric scattering theory. In particular, we describe the asymptotic behavior of the resolvent applied to Schwartz functions and that of the resolv…
This chapter tackles class imbalance in datasets to promote data democracy.
problem Class imbalance in datasets leading to biased decisions and policies.
method Statistical measures and data-level methods (oversampling, undersampling, etc.) applied to a real dataset.
result Popular data-level methods improve performance in handling class imbalance.
Study resolves conjectures on hypercritical deformed Hermitian-Yang-Mills equation.
problem Resolving conjectures on hypercritical deformed Hermitian-Yang-Mills equation.
method Study compact Kähler manifolds and resolves conjectures of Collins-Yau.
result Resolves two conjectures of Collins-Yau.
New method tests causal relationships from data without needing to learn the entire graph.
problem Testing if a causal graph belongs to a specific Markov equivalence class from observational data.
method Established bounds on the number of independence tests required and provided an algorithm that matches these bounds.
result Testing requires exponentially less independence tests compared to learning, especially in graphs with high in-degrees and small clique sizes.
Efficient method for resampling problems using vector approximate message passing.
problem Computational demand in resampling techniques for statistical inference and ensemble learning.
method Combination of replica method from statistical physics and vector approximate message passing from information theory.
result Fast convergence and high approximation accuracy for variable selection problems.