Algorithm classifies saddle-focus singularities in Hamiltonian systems.
problem Classifying nondegenerate saddle-focus singularities in integrable Hamiltonian systems.
method Developed an algorithm based on semi-local equivalence to represent singularities as almost direct products.
result Obtained complete lists of saddle-focus singularities of complexities 1, 2, and 3.
Study focuses on classifying special geometric structures.
problem Classify singular affine structures of integrable systems.
method Classification through simple semitoric systems equivalence.
result Counterexamples exist for multiple pinched fibers.
We show a natural relation between the monodromy formula for focus-focus singularities of integrable Hamiltonian systems and a formula of Duistermaat-Heckman, and extend the main results of our previous note on focus-focus singularities ($\bbS^1$-action, monodromy, and topological classification) to the degenerate case…
We give a topological and geometrical description of focus-focus singularities of integrable Hamiltonian systems. In particular, we explain why the monodromy around these singularities is non-trivial, a result obtained before by J.J. Duistermaat and others for some concrete systems.
Study convexity in Hamiltonian systems with focus singularities.
problem Convexity of singular affine structures in Hamiltonian systems with focus singularities.
method Systematic study of symplectic convexity for integrable Hamiltonian systems with focus-focus singularities.
result Local and global convexity properties of singular integral affine base spaces, with conditions for convexity breakdown.
Eliashberg simplifies singularities in geometry.
problem Complex singularities in geometric structures.
method Philosophy of the h-principle and simplification techniques.
result Simplified understanding of singularities in geometry.
Surveying stability of klt singularities with new solutions.
problem Stability of klt singularities.
method Survey and solution of the stable degeneration conjecture.
result Solution to the stable degeneration conjecture.
This paper introduces a new invariant for symplectic semitoric manifolds.
problem Classifying and understanding symplectic semitoric manifolds.
method Introducing the helix as a symplectic analogue of the fan in algebraic geometry.
result Classification of minimal models of symplectic semitoric manifolds.
Survey on singularity theory and its relation to minimal model program.
problem Understanding singularities and their role in minimal model program.
method Construction and analysis of dual complexes, proof of ACC conjecture, local stability theory.
result Recent progress on local stability theory of Kawamata log terminal singularities.
Lectures on surface evolution through singularities.
problem Analyzing the mean curvature flow of surfaces and their singularities.
method Analysis of neck and conical singularities, using monotonicity formulas, epsilon-regularity, weak solutions, and blowup techniques.
result Unique evolution through neck singularities, nonuniqueness through conical singularities.
Study isotopy of rational cuspidal curves in 4-manifolds.
problem Isotopy of rational cuspidal curves in 4-manifolds.
method Tame symplectic curves, pseudo-holomorphic curves, log pairs, 4-dimensional topology.
result Every rational cuspidal curve is isotopic to a complex curve in degrees up to 5.
Study on homology groups of cDV singularity links, identifying their topology.
problem Identify the topology of links of cDV singularities of types cAn and cDn. method Analyzing the second integral homology group of the links, using results from Smale and Thom-Sebastiani sums.
result The homology groups of the links are determined for cDV singularities of types cAn and cDn. Study on the geometry of limit spaces of manifolds with boundary.
problem Understanding the geometry of limit spaces of manifolds with boundary.
method Developed infinitesimal geometry for limit spaces under curvature and diameter bounds.
result Determined the infinitesimal structure and Hausdorff dimensions of boundary singular sets.
A generalized semitoric system F:=(J,H): M --> R^2 on a symplectic 4-manifold is an integrable system whose essential properties are that F is a proper map, its set of regular values is connected, J generates an S^1-action and is not necessarily proper. These systems can exhibit focus-focus singularities, which corresp…
Study the monodromy and center-focus problems for rational maps defined by products of generic lines.
problem Monodromy and center-focus problems for rational maps defined by products of generic lines.
method Analyze the 1-homology group and meromorphic 1-forms to characterize vanishing Abelian integrals.
result Characterize meromorphic 1-forms whose Abelian integrals vanish on cycles around a center singularity.
We consider the general Kähler-Ricci flows which exist for all time. The zeroth order control on the flow metric potential for various infinite time singularities is the focus. The possible semi-amplness for numerically effective classes serves as the main motivation.
Study of singularities in mean curvature flow with focus on S3imesR.
problem Understanding singularities in mean curvature flow.
method Detailed analysis of singularities modeled on S3imesR, using normal form transformations and rescaled MCF. result Proves mean convexity and singularity isolation in a small neighborhood, conjectures singularity formation in entire neighborhood.
The paper studies constraint maps with singularities and free boundaries, proving continuity near singularities and optimality.
problem Analyzing the structure of constraint maps with singularities and free boundaries.
method Establish continuity near singularities using a new quantitative unique continuation principle, and investigate the presence of branch points leading to new singularities.
result Topological singularities can only lie in the interior of the contact set in the uniformly convex setting, and the optimality of this result is proven.
We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group SL2(Z) to its preimage in the universal cover of SL2(R). With this method we recover the classification of two-dimensional toric fans, and obtain a des…
Surveying Ricci flow and its applications to Ricci limit spaces.
problem Understanding Ricci flows with unbounded curvature.
method Analyzing Ricci flows that can have infinite curvature at infinity and singular times.
result Developed insights into Ricci limit spaces.
Study Ricci flow on Rn+1, focusing on asymptotic behavior and singularities.
problem Analyzing the behavior of Ricci flow on Rn+1, especially near singularities. method Examined the flow starting from rotationally symmetric metrics, considering asymptotic curvature and pinched necks.
result Proved the existence of Type-II and Type-I singularities under specific conditions.
Quantum statistical models with singularities are studied for state estimation and model selection.
problem Understanding statistical properties of quantum singular models.
method Classical singular learning theory extended to quantum state estimation and model selection using algebraic geometrical methods.
result Asymptotically unbiased estimator (QWAIC) for quantum generalization loss constructed.
We study the scalar curvature of Kähler metrics that have cone singularities along a divisor, with a particular focus on certain specific classes of such metrics that enjoy some curvature estimates. Our main result is that, on the projective completion of a pluricanonical bundle over a product of Kähler--Einstein Fano …
Optimal rank-adaptive matrix estimation from linear measurements.
problem Estimating high-dimensional matrices from linear measurements with adaptive rank selection.
method Combines Least-Squares estimator with universal singular value thresholding.
result Algorithm performance nearly matches fundamental limits.
Penrose's conjecture links black holes and gravitational singularities.
problem Deterministic nature of General Relativity and black holes.
method Modern techniques in partial differential equations applied to gravitational collapse.
result Recent advances in understanding Strong Cosmic Censorship.
We consider spacetimes consisting of a manifold with Lorentzian metric and a weight function or scalar field. These spacetimes admit a Bakry-Émery-Ricci tensor which is a natural generalization of the Ricci tensor. We impose an energy condition on the Bakry-Émery-Ricci tensor and obtain singularity theorems of a cosmol…
This is a survey concerning the relationship between Lie Groupoids (and their morphisms) and singular foliations in the sense of Sussmann-Stefan (considered from a purely geometrical point of view). We focus on the interaction between the algebraic and differentiable structures underlying Lie groupoids, and between gro…
Investigate scalar curvature under geometric flows
problem Behavior of scalar curvature under geometric flows
method Three specific cases: Ricci flow, Kähler-Ricci flow, Laplacian flow
result Long-time existence of flows
The truncated singular value decomposition (SVD) of the measurement matrix is the optimal solution to the_representation_ problem of how to best approximate a noisy measurement matrix using a low-rank matrix. Here, we consider the (unobservable)_denoising_ problem of how to best approximate a low-rank signal matrix bur…
Faster matrix completion through randomized SVD algorithms.
problem Efficiently completing large sparse matrices for applications like image inpainting and recommender systems.
method Proposed two fast randomized algorithms (rSVD-PI and rSVD-BKI) and a new subspace recycling technique to accelerate singular value thresholding (SVT) method.
result The proposed algorithms achieve up to 15X faster computation time for image inpainting and movie rating estimation problems.
Study of hyperkahler spaces from Lie algebras.
problem Understanding hyperkahler quotients of cotangent bundles.
method Analyzing stratified spaces of cotangent bundles of complex semisimple Lie groups.
result Explicit description of partial order on strata using Lie theoretic data.
This paper is devoted to studying the structure of codimension one singular holomorphic foliations on (C3,0) without invariant germs of analytic surface. We focus on the so-called CH-foliations, that is, foliations without saddle nodes in two dimensional sections. Considering a reduction of singularities, …
This paper is concerned with the construction of special metrics on non-compact 4-manifolds which arise as resolutions of complex orbifold singularities. Our study is close in spirit to the construction of the hyperkaehler gravitational instantons, but we focus on a different class of singularities. We show that any re…
Proposes a regularization method for unsupervised domain adaptation that aligns predictions with target data's top singular vectors.
problem Domain adaptation challenges in high joint error scenarios.
method Regularizes classifier to align with unsupervised target data guided by label alignment property (LAP).
result The method improves performance in MNIST-USPS domain adaptation and cross-lingual sentiment analysis.
For almost half of the one hundred year history of Einstein's theory of general relativity, Strong Cosmic Censorship has been one of its most intriguing conjectures. The SCC conjecture addresses the issue of the nature of the singularities found in most solutions of Einstein's gravitational field equations: Are such si…
Lectures on symplectic aspects of surface degenerations at KIAS.
problem Exploring symplectic structures in surface degenerations.
method Expository account of symplectic aspects of cyclic quotient surface singularities.
result Discussion of symplectic structures in surface degenerations.
New method for causal effect estimation with hidden confounders.
problem Estimating causal effects in the presence of hidden confounders.
method Singular value decomposition of a conditional expectation operator followed by saddle-point optimization.
result Our method outperforms existing methods on common benchmarks.
Second paper in series solves Einstein vacuum equations for three impulsive waves.
problem Solving local Cauchy problem for impulsive gravitational waves.
method Geometric commutators for energy estimates, fractional-derivative regularity, anisotropic Sobolev embedding.
result Scalar field becomes everywhere Lipschitz and C1,θ away from singular region. Study high codimension mean curvature flow in Riemannian manifolds, proving limiting flow in Euclidean space.
problem Analyzing mean curvature flow in high codimension Riemannian manifolds.
method Establishing codimension estimate, using quadratic pinching condition, gradient estimates.
result Existence of limiting flow in Euclidean space under cylindrical pinching condition.
The study analyzes XRP transaction networks to understand market dynamics.
problem Understanding market dynamics of XRP through transaction data.
method Weekly weighted directed networks are embedded into a vector space using network embedding techniques. A correlation tensor is calculated and analyzed using singular value decomposition.
result The correlation tensor provides insights into the system's behavior and dependence on model parameters.
We initiate the systematic study of G2-instantons with SU(2)2-symmetry. As well as developing foundational theory, we give existence, non-existence and classification results for these instantons. We particularly focus on R4×S3 with its two explicitly known distinct holonomy G2 metrics, whic…
Study submersions with definite folds on manifolds with boundary into Euclidean spaces.
problem Understanding differential-topological properties of manifolds with boundary under submersions with definite folds.
method Analyzing submersions with definite folds on manifolds with boundary into Euclidean spaces, focusing on restrictions to the boundary and using results for m-functions.
result Obtained restrictions on the diffeomorphism types of the source manifolds and studied the diffeomorphism types and Euler characteristics of manifolds admitting such maps.
The paper provides uniform length estimates for trajectories on flat cone surfaces.
problem Estimating the length of trajectories on flat cone surfaces.
method Using self-intersection numbers and constants depending only on the flat metric, the paper focuses on convex flat cone spheres with a positive curvature gap and a fixed number of singularities.
result Uniform two-sided estimates for trajectory lengths on convex flat cone spheres are obtained.
The paper presents fundamental groups of complements of shadows in 4-balls.
problem Understanding fundamental groups of 4-manifold complements.
method Similar to Wirtinger presentation, focusing on contractible shadows.
result A presentation of fundamental groups of subpolyhedra in 4-balls.
Paper compares higher torsions and removes fiberwise Morse function assumption.
problem Comparing higher torsions from analytic and topological perspectives.
method Introduced fiberwise generalized Morse functions (GMFs) and excised neighborhoods around birth-death points.
result Established a generalized version of the higher Cheeger-Müller/Bismut-Zhang theorem.
We continue our previous work studying critical exponent semilinear elliptic (and subelliptic) problems which generalize the classical Yamabe problem. In [3] the focus was on metric-measure spaces with an `almost smooth' structure, with stratified spaces furnishing the key examples. The criterion for solvability there …
The study finds PK cone metrics on complex manifolds near hyperplane arrangements.
problem Finding metrics on complex manifolds near singularities.
method Analyzing flat torsion-free meromorphic connections on \(\mathbb{C}^n\) with simple poles at hyperplanes.
result Metric completion of certain connections yields PK cone metrics on \(\mathbb{C}^n\).
In this paper we study solutions to elliptic linear equations L(u)=∂i(aij(x)∂ju)+bi(x)∂iu+c(x)u=0, either on Rn or a Riemannian manifold, under the assumption of Lipschitz control on the coefficients aij. We focus our attention on the critical set $Cr(u)\equiv\{x:|\nabla u|…