Study describes singularities of height functions on specific singular surfaces.
problem Analyzing singularities of height functions on singular surfaces.
method Using geometric language and blowing-ups, investigate singularities of height functions and dual surfaces.
result Characterized singularities of height functions and dual surfaces on specific singular surfaces.
Study describes singularities of distance squared functions on singular surfaces.
problem Characterizing singularities of distance squared functions on singular surfaces.
method Using smooth map-germs Sk, Bk, Ck, and F4 singularities, the study describes singularities via blowing-ups. result Characterization of singularities of wave-fronts and caustics of singular surfaces.
The paper classifies singularities of plane congruences and affine distance functions.
problem Classifying singularities of plane congruences and affine distance functions.
method Classification through 2-parameter plane congruences in \(\mathbb{R^4}\) and affine normal plane congruences.
result Generic singularities of plane congruences and affine distance functions are classified.
Defines Perelman's functionals on manifolds with non-isolated conical singularities.
problem Defining functionals on manifolds with non-isolated conical singularities.
method Starting from a spectral point of view for the Perelman's λ-functional, defining the spectrum of Schrödinger operator and proving the existence of discrete eigenvalues.
result Proves the existence of the infimum of W-functional and obtains asymptotic behavior of eigenfunctions.
Study empty polar varieties' impact on singular function-germs.
problem Topology of singular function-germs with nonisolated singularities.
method Analysis of empty polar varieties.
result Topology implications of nonempty polar varieties.
New invariant distinguishes singular knots and links.
problem Classifying singular knots and links.
method Using oriented singquandles and weight functions at crossings.
result Distinguishes singular granny knot from singular square knot.
Study oscillatory integrals with degenerate singular points in multivariable phase functions.
problem Analyzing oscillatory integrals with degenerate singular points in phase functions.
method Using asymptotic expansions and results from one variable, the study examines multivariable phase functions.
result Asymptotic expansions of oscillatory integrals for multivariable phase functions with degenerate singular points.
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.
In this paper, we develop the theory of Perelman's W-functional on manifolds with isolated conical singularities. In particular, we show that the infimum of W-functional over a certain weighted Sobolev space on manifolds with isolated conical singularities is finite, and the minimizer exists, if the scalar curvatur…
Study on weighted cscK metrics on Kähler varieties with singularities.
problem Existence of singular weighted cscK metrics on Kähler varieties.
method Resolution of singularities, coercive weighted Mabuchi functional, construction of examples.
result Existence of singular weighted cscK metrics when the weighted Mabuchi functional is coercive.
Singularities of even smooth functions are studied. A classification of singular points which appear in typical parametric families of even functions with at most five parameters is given. Bifurcations of singular points near a caustic value of the parameter are also studied. A determinant for singularity types and con…
Study on contact Hamiltonian functions for singular contact structures.
problem Understanding infinitesimal contact transformations on singular contact structures.
method Showed injectivity and provided an explicit local formula for the inverse map.
result Explicit local formula for the inverse map when contact structure has singularities of the first type.
Develops methods to analyze manifold singularities using graph Laplacian.
problem Analyzing geometric properties of singularities in datasets.
method Theory and methods using the graph Laplacian to provide explicit bounds on manifold singularities.
result Explicit bounds on the graph Laplacian for functions near manifold singularities.
Study on singularities of specific polynomial functions.
problem Characterizing the topology of singularities of mixed functions.
method Introduced inner non-degenerate mixed functions and used Newton boundary to characterize links.
result Links of singularities can be completely characterized under certain conditions.
The paper proves the existence of singular cscK metrics on smoothable varieties.
problem Existence of singular cscK metrics on smoothable varieties.
method Developing a strong topology of pluripotential theory in families and uniform estimates for cscK metrics.
result Existence of singular cscK metrics on Q-Gorenstein smoothable klt varieties when the Mabuchi functional is coercive. Study Z2 harmonic functions with singularities on flat space.
problem Understanding Z2 harmonic functions with point singularities.
method Analyzes Z2 harmonic functions on R2 with point singularities. result Characterizes Z2 harmonic functions on R2 with point singularities. Study shows singular set of distance functions is delta-convex.
problem Understanding singular set of distance functions in Finsler manifolds.
method Proved singular set is delta-convex hypersurfaces or Jordan arcs up to exceptional sets.
result Optimal results in Finsler manifolds, even in Euclidean space.
The paper defines function spaces on manifolds with bounded or singular geometries.
problem Defining function spaces on manifolds with various geometries.
method Introduces and analyzes Sobolev, Besov, and Bessel potential spaces on uniformly regular and singular Riemannian manifolds.
result Demonstrates maximal regularity for a linear parabolic problem on singular manifolds.
This thesis studies moduli spaces of singular connections on 3-manifolds and manifolds with cylindrical ends. A Chern-Simons functional is defined for singular connections on 3-manifolds which are singular along a knot. The critical points of that Chern-Simons functional are flat singular connections. The Hodge-de Rham…
Two new methods improve forecasting of functional time series data.
problem Forecasting of functional time-dependent data.
method Functional Singular Spectrum Analysis (FSFA) based forecasting methods.
result Our methods outperform existing algorithms for periodic stochastic processes.
The aim of this article is to generalize the notion of the cut locus and to get the structure theorem for it. For this purpose, we first introduce a class of 1-Lipschitz functions, each member of which is called an {\it almost distance function}. Typical examples of an almost distance function are the distance function…
The paper defines wave-front singularities using explicit analytic functions.
problem Characterizing the images of wave-front singularities.
method Explicit resultant computations to construct main-analytic functions.
result Explicit formulas for main-analytic functions of wave-front singularities of types A, D, and E.
The minimizer of a volume function is unique for klt singularities.
problem Uniqueness of the minimizer of the normalized volume function for klt singularities.
method Defining stability thresholds for valuations and showing K-semistability.
result The minimizer of the normalized volume function for a klt singularity is unique up to rescaling.
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.
We define a generalization of convex functions, which we call δ-convex functions, and show they must satisfy interior Hölder and W1,p estimates. As an application, we consider solutions of a certain class of fully nonlinear equations in conformal geometry with isolated singularities, in the case of non-negative …
Analyzes singularities of convex hypersurfaces in hyperbolic space.
problem Understanding singularities of convex hypersurfaces with constant curvature.
method Analyzes the structure of singular sets using convex curvature functions.
result Describes the structure of singular sets in hyperbolic space.
Proves finitely generated graded rings for klt singularities.
problem Understanding the structure of klt singularities.
method Analyzes graded rings associated with minimizers of normalized volume functions.
result Graded rings are finitely generated for klt singularities.
In this paper, we prove that on a compact manifold with isolated conical singularity the spectrum of the Schrödinger operator −4Δ+R consists of discrete eigenvalues with finite multiplicities, if the scalar curvature R satisfies a certain condition near the singularity. Moreover, we obtain an asymptotic behavior fo…
Study curve shortening flow on Riemann surfaces with conic singularities.
problem Analyzing curve shortening flow on surfaces with conic singularities.
method Generalized Huisken's comparison function to Riemann surfaces and surfaces with conic singularities. Reproofed Gage-Hamilton-Grayson theorem. Proved CSF can't touch conic singularities with cone angles ≤ π.
result CSF can't touch conic singularities with cone angles ≤ π for embedded simple closed curves.
Lips and swallow-tails are generic local moves of singularities of a smooth map to a 2-manifold. We prove that these moves of singularities of the product map of two functions on a 3-manifold can be realized by isotopies of the functions.
The paper tackles singularities in diffusion models on submanifolds.
problem Analyzing singularities in diffusion models on lower-dimensional submanifolds.
method Small-time approximations of the Green's function and derivation of a new target function.
result The new target function remains bounded for singular data distributions.
The paper solves conditions for extending circle-valued Morse functions.
problem Conditions for extending circle-valued Morse functions on closed orientable surfaces.
method Provided necessary and sufficient conditions for the existence of a non-singular extension.
result Necessary and sufficient conditions for the existence of a non-singular extension of a circle-valued Morse function.
Study of singularities in two-dimensional Nijenhuis operators with non-zero trace differential.
problem Characterizing singularities of two-dimensional Nijenhuis operators.
method Analyzing the smoothness of functions related to the determinant of the operator.
result Complete description of singularities for certain function classes.
Study nondegenerate singularities in mean curvature flow.
problem Understanding the behavior of nondegenerate cylindrical singularities.
method New L2-distance monotonicity formula and discrete almost monotonicity. result Topology change agrees with level sets change near a critical point of a Morse function.
Growth functions of Coxeter groups and the Poincare series of Kleinian and Fuchsian singularities are 2-tangle q-fractions.
We give criteria for which a principal curvature becomes a bounded C∞-function at non-degenerate singular points of wave fronts by using geometric invariants. As applications, we study singularities of parallel surfaces and extended distance squared functions of wave fronts. Moreover, we relate these singularit…
Study shortest geodesics on flat cone spheres with conical singularities.
problem Understanding the distribution of shortest geodesics on flat cone spheres.
method Proved a recurrent relation on the distribution of the length of shortest geodesics with respect to Thurston's volume form.
result Proved a recurrent relation on the distribution of the length of shortest geodesics.
MFSSA improves reconstruction accuracy of multivariate functional time series.
problem Improving reconstruction accuracy of multivariate functional time series.
method Developed MFSSA, a functional extension of MSSA, for different dimensional domains.
result Better reconstruction accuracy of MFTS signals using MFSSA compared to other methods.
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 establish a correspondence on a Riemann surface between hyperbolic metrics with isolated singularities and bounded projective functions whose Schwarzian derivatives have at most double poles and whose monodromies lie in PSU(1,1). As an application, we construct explicitly a new class of hyperbolic metrics …
We study the blow-up behaviour of minimizing sequences for the singular Moser-Trudinger functional on compact surfaces. Assuming non-existence of minimum points, we give an estimate for the infimum value of the functional. This result can be applied to give sharp Onofri-type inequalities on the sphere in the presence o…
The paper constructs real algebraic functions with specific singularities and preimages.
problem Constructing real algebraic functions with prescribed singular points and preimages.
method Generalizing moment-like maps and Morse-Bott functions to real algebraic settings.
result Explicit construction of real algebraic functions with exactly one singular value and prescribed preimages.
Study weak super Ricci flow through neckpinch in metric measure spaces.
problem Understanding Ricci flow behavior at singularities.
method Introduce weak super Ricci flow and show conditions for continuation.
result Weak super Ricci flow properties at singularities.
Study on lengths and curvatures of harmonic functions on smooth and singular surfaces.
problem Investigate logarithmic convexity and isoperimetric inequalities of harmonic functions on surfaces.
method Analyzes geodesic curvature, uses Laplace-type equations, and studies growth estimates.
result Generalizes results on logarithmic convexity and isoperimetric inequalities for harmonic functions.
Study Euler obstruction of 1-forms on determinantal singularities.
problem Understanding the Euler obstruction of 1-forms on determinantal singularities.
method Investigation of connections between local Euler obstruction and PHN index.
result Explicit computations of Euler obstruction for specific singularities.
We prove Patterson's conjecture about the singularities of the Selberg zeta function associated to a convex-cocompact, torsion free group acting on a hyperbolic space.
For a Riemann surface with cusps we define a theta function using the eigenvalues of the Laplacian and the singularities of the scattering determinant. We provide its meromorphic continuation and discuss its singularities.
We study singularities of Gauss maps of fronts and give characterizations of types of singularities of Gauss maps by geometric properties of fronts which are related to behavior of bounded principal curvatures. Moreover, we investigate relation between a kind of boundedness of Gaussian curvatures near cuspidal edges an…