In this paper we investigate the singularities of Lagrangian mean curvature flows in by means of smooth singularity models. Type I singularities can only occur at certain times determined by invariants in the cohomology of the initial data. In the type II case, these smooth singularity models are asympto…
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The Kähler-Ricci flow near conical singularities is described with a curvature bound.
problem Describing the Kähler-Ricci flow near conical singularities.
method Showed a curvature bound and used the unique Kähler-Ricci expander.
result The flow near each singular point is modelled on the unique Kähler-Ricci expander.
We introduce thermodynamic response functions for singular Bayesian models.
problem Singular Bayesian models violate regular asymptotics due to non-identifiability and degenerate Fisher geometry.
method Posterior tempering induces thermodynamic response functions, linking WAIC, WBIC, and singular fluctuation.
result WAIC, WBIC, and singular fluctuation are unified within a thermodynamic response framework.
Bayesian models' singular fluctuation is shown to be akin to specific heat, influencing model complexity and generalization.
problem Understanding the thermodynamic interpretation of singular fluctuation in Bayesian models.
method Showed singular fluctuation as the curvature of Bayesian free energy and variance of log-likelihood observable under a Gibbs posterior.
result Singular fluctuation is the statistical analogue of specific heat, controlling model complexity and generalization.
Ricci flow modelled on specific singularities on closed manifolds.
problem Analyzing singularities in Ricci flows.
method Closed manifold Ricci flow with singularity modeled on asymptotically conical shrinkers.
result Ricci flow solution forms a singularity that matches the given asymptotically conical shrinker.
Study analyzes perturbations in singular subspaces under random noise.
problem Understanding singular vector and subspace changes in signal-plus-noise models.
method Generalized Davis-Kahan-Wedin theorem for any unitarily invariant norm, considering and bounds.
result Fine-grained insights into singular vector and subspace perturbations, including and bounds.
Survey classifies singularity models in 3D Ricci flow.
problem Understanding singularity formation in 3D Ricci flow.
method Analysis of ancient κ-solutions and steady gradient Ricci solitons.
result Complete classification of singularity models in 3D.
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.
Singularities of the mean curvature flow of an embedded surface in R^3 are expected to be modelled on self-shrinkers that are compact, cylindrical, or asymptotically conical. In order to understand the flow before and after the singular time, it is crucial to know the uniqueness of tangent flows at the singularity. In …
We investigate Riemannian (non-Kahler) Ricci flow solutions that develop finite-time Type-I singularities and present evidence in favor of a conjecture that parabolic rescalings at the singularities converge to singularity models that are shrinking Kahler-Ricci solitons. Specifically, the singularity model for these so…
The Weyl curvature hypothesis of Penrose attempts to explain the high homogeneity and isotropy, and the very low entropy of the early universe, by conjecturing the vanishing of the Weyl tensor at the Big-Bang singularity. In previous papers it has been proposed an equivalent form of Einstein's equation, which extends i…
We extend the state models for Jones and Alexander polynomials of classical links to state models of 2-variable polynomials in the case of singular links. Moreover, we extend both of them to polynomials with d+1 variables for long singular knots with exactly d double points. These extensions can detect non-invertibilit…
Study shows instability of naked singularities in perfect fluid models.
problem Instability of naked singularities in Einstein equations coupled with isothermal perfect fluid.
method Investigated spherically symmetric self-similar naked singularities under perturbations of an external massless scalar field.
result Spherically symmetric self-similar naked singularities are unstable to trapped surface formation.
Small time heat expansion of the laplacian on an analytic hypersurface with an isolated singularitymath.AP
Proves heat expansion for Laplacian on a singularity.
problem Analytic hypersurface with isolated singularity and Laplacian heat expansion.
method Local parametrization, Newton scheme, quasihomogeneous tangent cone, local models with irregular singularities.
result Existence of small time heat expansion for Laplace operator.
The paper examines Ricci flows with closed and smooth tangent flows, proving uniqueness and characterizing ancient flows.
problem Characterizing and understanding Ricci flows with closed and smooth tangent flows.
method Analyzing ancient and finite-time singularity Ricci flows to prove uniqueness and characterizations.
result The tangent flow is unique and characterizes ancient and finite-time singularity flows.
Novel singularity models for 4D harmonic forms and spinors from polytopes.
problem Understanding harmonic forms and spinors in 4D.
method Homogeneous singularity models based on regular 4-polytopes.
result Models describe cones on the 1-skeletal of polytopes.
Hypothesis testing in singular models is fundamentally about identifiable vs. non-identifiable parameters.
problem Testing in singular models is inherently problematic due to non-identifiability and degeneracy of Fisher information.
method Formalized the overlap obstruction and showed that hypotheses over non-identifiable parameters are untestable, while those over identifiable parameters reduce to classical testing.
result Hypotheses over non-identifiable parameters are untestable, while those over identifiable parameters reduce to classical testing.
This paper sets lower bounds for scalar curvatures in Ricci flow singularity models.
problem Understanding scalar curvatures in Ricci flow singularity models.
method Developed high-dimensional theory of Hamilton's Ricci flow, including new monotonicity formulas, compactness theorem, and partial regularity theory.
result Obtained a quadratic decay lower bound for the scalar curvature in 4-dimensional non-Ricci-flat steady soliton singularity models.
The paper extends a geometric model using singular curves.
problem Understanding abnormal extremals in sub-Riemannian geometry.
method Analysis of singular curves and construction of a graded Lie algebra.
result A nilpotent graded Lie algebra is constructed isomorphic to .
The paper derives an equation linking WAIC and WBIC for singular models.
problem In singular models, conventional criteria fail due to likelihood and posterior breakdown.
method Theoretical derivation linking WAIC and WBIC.
result An asymptotic equation linking WAIC and WBIC for singular models.
The paper studies complex affine structures near irregular singularities.
problem Understanding complex affine structures near irregular singularities.
method Introducing local invariants and a Delaunay decomposition.
result Upper bounds on the complexity of the Delaunay decomposition.
The Type IIA flow converges on symplectic manifolds, with singularity models identified.
problem Little was known about the singularities of the Type IIA flow.
method Formulated and proved convergence theorems for the Type IIA flow.
result Identified singularity models for the Type IIA flow.
Constructs Kahler-Einstein metrics near isolated log canonical singularities.
problem Constructing metrics near singularities in complex geometry.
method Constructs Kahler-Einstein metrics with negative scalar curvature near isolated log canonical singularities.
result Metrics are complete near the singularity if the underlying space has complex dimension 2 or if the singularity is smoothable.
LS improves model selection for singular statistical models.
problem Challenges in model selection for singular statistical models.
method Integrates empirical loss from WAIC and sBIC penalty term.
result Enhanced utility for model selection without regularity constraints.
We show that the Big Bang singularity of the Friedmann-Lemaitre-Robertson-Walker model does not raise major problems to General Relativity. We prove a theorem showing that the Einstein equation can be written in a non-singular form, which allows the extension of the spacetime before the Big Bang. The physical interpret…
Study shows instability of naked singularities in scalar field models.
problem Stability of naked singularities in spherically symmetric Einstein-Scalar field systems.
method Analysis of a family of incoming null cones becoming increasingly singular.
result Naked singularities are unstable to black hole formation under certain perturbations.
A new method reduces task interference in model merging.
problem Model merging overlooks structural information and is susceptible to task interference.
method Task Singular Vectors (TSV) and TSV-Compress for compression and interference reduction.
result TSV-Merge significantly outperforms existing methods.
Proves smoothness of conical singularities in mean curvature flow.
problem Resolving singularities in mean curvature flow.
method Analyzes smooth hypersurfaces with isolated conical singularities.
result Smoothness of level set flow through asymptotically conical singularities.
Study of light function singularities on surfaces.
problem Characterizing singularities of the slant function on surfaces.
method Analyzing the differential geometry of the parabolic set and its spherical image under the Gauss map.
result The type of singularities of the slant function is determined by the geometry of the parabolic set and its spherical image.
The paper solves a 5-manifold foliation problem using a Sasaki-Ricci flow.
problem Solving foliation singularities on Sasakian 5-manifolds.
method Applying the Sasaki-Ricci flow to resolve cyclic quotient foliation singularities.
result Proves a Sasaki analogue of the analytic minimal model program.
New method for analyzing learning dynamics in singular models.
problem Challenges in analyzing learning of singular models with no one-to-one parameter space.
method Relative reparameterization technique to extract regular sub-models.
result Demonstrated differences in convergence behavior due to algorithmic and intrinsic aspects.
Robert Bryant (Theorie des varietes minimales et applications, 1988, 154: 321-347) proved that an isolated singularity of a conformal metric of positive constant curvature on a Riemann surface is a conical one. Using Complex Analysis, we find all of the local models for an isolated singularity of a flat metric whose ar…
New singularities and fibrations in non-orientable 4-manifolds.
problem Understanding singularities and fibrations in non-orientable 4-manifolds.
method Introducing -singularities and -fibrations, studying their handle decompositions and orientation double coverings.
result Relations among crosscap transpositions give rise to -fibrations on non-orientable 4-manifolds.
We give a simple direct proof of uniqueness of tangent cones for singular projectively Hermitian Yang-Mills connections on reflexive sheaves at isolated singularities modelled on -polystable holomorphic bundles over .
Unified framework for singular statistical models using observable charts.
problem Non-identifiability and breakdown of classical asymptotic theory in singular models.
method Invariant framework based on observable charts to define local coordinate systems in model space.
result Observable order provides a lower bound on KL divergence vanishing rate in singular models.
We construct local models of isolated singularities for special Kähler structures in real dimension two assuming that the associated holomorphic cubic form does not have essential singularities. As an application we compute the holonomy of the flat symplectic connection, which is a part of the special Kähler structure.
An Elementary Remark on a Lower bound for the -Invariant of Singularity Models for Ricci Flowmath.DG
Expository observation on the -invariant of singularity models for Ricci Flow.
We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a di…
Singular cotangent modelmath.SG
Any singular level of a completely integrable system (c.i.s.) with non-degenerate singularities has a singular affine structure. We shall show how to construct a simple c.i.s. around the level, having the above affine structure. The cotangent budle of the desingularised level is used to perform the construction, and th…
We survey some recent topics on singularities, with a focus on their connection to the minimal model program. This includes the construction and properties of dual complexes, the proof of the ACC conjecture for log canonical thresholds and the recent progress on the `local stability theory' of an arbitrary Kawamata log…
Seminar held at JINR, Dubna, May 15, 2012. In General Relativity, spacetime singularities raise a number of problems, both mathematical and physical. One can identify a class of singularities - with smooth but degenerate metric - which, under a set of conditions, allow us to define proper geometric invariants, and to w…
In this paper, we study the singularities of two extended Ricci flow systems --- connection Ricci flow and Ricci harmonic flow using newly-defined curvature quantities. Specifically, we give the definition of three types of singularities and their corresponding singularity models, and then prove the convergence. In add…
This paper studies Gorenstein singularities and their applications in moduli spaces of holomorphic differentials.
problem Understanding Gorenstein singularities and their moduli spaces.
method Construction of Gorenstein curve singularities via test configurations and miniversal deformation spaces.
result Classification of Gorenstein singularities and compactification of nonvarying strata.
Geometric models for Lie algebras from simple singularities.
problem Classifying simply-laced simple Lie algebras.
method Using polygonal wheels derived from Milnor fibers of simple singularities.
result Geometric root systems are isomorphic to Lie algebras.
Extends Serre-Swan theorem to all finitely generated modules over smooth functions.
problem Classical Serre-Swan theorem limitations.
method Introduces tepui fibrations and singular vector bundles.
result Realizes all finitely generated modules over smooth functions.
We survey what is known about singularities of special Lagrangian submanifolds (SL m-folds) in (almost) Calabi-Yau manifolds. The bulk of the paper summarizes the author's five papers math.DG/0211294, math.DG/0211295, math.DG/0302355, math.DG/0302356, math.DG/0303272 on SL m-folds X with isolated conical singularities.…
The paper proves a precise SYZ conjecture for toric Calabi-Yau manifolds with singular fibers.
problem Proving the SYZ conjecture for Calabi-Yau manifolds with singular fibers.
method Using family Floer mirror construction and Gross Lagrangian fibration.
result The dual singular fibration is compatible with the family Floer mirror construction.
Detects singularities in complex data to improve machine learning models.
problem Real-world data often contains non-manifold structures (singularities) that can mislead machine learning models.
method Develops a topological framework to quantify local intrinsic dimension and Euclidicity score for multiple scales.
result Identifies singularities and captures local geometric complexity in image data.