The Brasselet number helps calculate function germs with one-dimensional critical sets.
problem Calculating topological information of function germs with nonisolated singularities.
method Using the Brasselet number, the paper presents formulas for function germs with a one-dimensional critical locus.
result Formulas for function germs with a one-dimensional critical locus.
The paper constructs solutions to a critical Dirac equation on spheres.
problem Solving the critical Dirac equation on spheres with singularities.
method Constructing Delaunay-type solutions and another kind of singular solutions.
result The constructed solutions are building blocks for singular solutions on Spin manifolds.
Extends Eliashberg's principle to maps with specific singularities.
problem Homotoping maps of surfaces with prescribed singularities.
method Extends Eliashberg's h-principle to surfaces with cusp and fold singularities. result Necessary and sufficient conditions for maps with given singularities.
This paper bounds the volume of singular and critical sets for elliptic equations with Hölder coefficients.
problem Bounding the volume of singular and critical sets for elliptic equations with Hölder coefficients.
method Proves explicit bounds for (n−2)-dimensional Minkowski estimates of singular and critical sets using Hölder continuity and new almost monotonicity formula. result Optimal improvement on Cheeger-Naber-Valtorta's volume estimates on each quantitative stratum.
Solves a 30-year-old problem on singular solutions for Yamabe equations.
problem Constructing singular solutions to semilinear equations without phase-plane analysis.
method Careful gluing in weighted L∞ spaces, exploiting semilinearity and stability of linearized operator. result Provides an alternative construction for the Yamabe problem with maximal singular set.
Study local topology of a function-germ deformation with a one-dimensional critical set.
problem Analyze the local topology of a deformation of a function-germ with a one-dimensional critical set.
method Use the Brasselet number to study the local topology of a deformation of a function-germ.
result Present a new proof of the Lê-Iomdin formula for the Brasselet number.
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|…
We propose a strategy for approximating Pareto optimal sets based on the global analysis framework proposed by Smale (Dynamical systems, New York, 1973, pp. 531-544). The method highlights and exploits the underlying manifold structure of the Pareto sets, approximating Pareto optima by means of simplicial complexes. Th…
Level set flow's singularities are type I under 2-convexity, leading to specific curvature blow-up rates.
problem Understanding the nature and behavior of singularities in level set flow.
method Analytical approach using Lojasiewicz inequality and curvature blow-up rates.
result The arrival time is C2 near a critical point if and only if it satisfies a Lojasiewicz inequality. Study on critical Lagrangian phase singularities in mean curvature flow.
problem Analyzing singularities in the Lagrangian mean curvature flow at the critical phase.
method Developed new method to prove C2,α estimates by using concave operators. result Established interior estimates for critical Lagrangian phase singularities.
The paper improves estimates on singular sets in manifolds with integral curvature bounds.
problem Estimating the singular set of manifolds with integral curvature bounds.
method Using Gromov-Hausdorff limits and Cheeger-Naber methods.
result Improved Minkowski dimension estimate for singular sets.
Improved remainder estimate for eigenvalue asymptotics on manifolds with group actions.
problem Asymptotic distribution of eigenvalues of invariant elliptic operators.
method Refined stationary phase approximation and singular critical sets analysis.
result Asymptotic multiplicity formula for families of irreducible representations.
New action-angle coordinates found for singular symplectic manifolds.
problem Existence of action-angle coordinates for singular symplectic manifolds.
method Action-angle theorem for folded symplectic integrable systems.
result New topological obstructions found for global existence of action-angle coordinates.
Formula calculates homology groups of Milnor fibres for real hypersurface singularities.
problem Calculating homology groups of Milnor fibres for real hypersurface singularities.
method Established a formula for homology groups of Milnor fibres relative to their boundaries.
result Formula provides a method to compute homology groups of Milnor fibres.
Solves Dirichlet problem for Lagrangian phase equation with critical and supercritical phase.
problem Solving Dirichlet problem for Lagrangian phase equation with critical and supercritical phase.
method Uses interior C2 estimate. result Result is sharp, showing existence of singular solutions in subcritical phase.
Proves conditions for gradient flow near critical points in analytic functions.
problem Gradient flow behavior near unstable sets of critical points.
method General result showing first three conditions imply fourth for locally compact spaces.
result Proves conditions for analytic functions on locally compact spaces.
Study finds non-uniqueness in sphere metrics with constant fractional curvature.
problem Non-uniqueness of metrics with constant positive fractional curvature on spheres.
method Bifurcation techniques applied to non-local equations with critical non-linearity.
result Non-uniqueness results for complete metrics on Sn∖Sk. We use numerical techniques to study the formation of singularities in Ricci flow. Comparing the Ricci flows corresponding to a one parameter family of initial geometries on S^3 with varying amounts of S^2 neck pinching, we find critical behavior at the threshold of singularity formation.
Study the singular limit of a boundary reaction equation, showing energy concentration and varifold support.
problem Analyzing the singular limit of a boundary reaction equation.
method Investigates the critical points of the boundary reaction equation \((-Δ)^{\frac{1}{2}}u = \frac{1}{\varepsilon}(u-u^3)\) in \(U \subset \mathbb{R}^n\).
result Shows existence of an (n−1)-rectifiable energy concentration set and associates limit energy measures to a stationary varifold. The paper simplifies complex 2D functions near their critical points.
problem Simplifying smooth functions on 2-manifolds near critical points.
method Explicit construction of coordinate changes to canonical form.
result Estimates the radius of required neighbourhoods for specific singularity types.
We consider the mean curvature evolution of rotationally symmetric surfaces. Using numerical methods, we detect critical behavior at the threshold of singularity formation resembling the one of gravitational collapse. In particular, the mean curvature simulation of a one-parameter family of initial data reveals the exi…
Classifies solutions to critical sixth order equations with a singularity.
problem Classifying entire positive singular solutions to critical sixth order equations.
method Integral sliding methods, qualitative analysis of ODEs, topological two-parameter shooting technique.
result Solutions are given by a singular radial factor times a periodic solution to a sixth order IVP with constant coefficients.
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.
The study of equidistants for families of surfaces, focusing on specific ratios of tangent planes.
problem Understanding the geometric properties of surfaces and their tangent planes.
method Local study of affine equidistants, critical value analysis of 2-parameter unfoldings, geometric classification of singularities.
result Classification of singularities in equidistants near the supercaustic chord.
We describe how to compute topological objects associated to a polynomial map of several complex variables with isolated singularities. These objects are: the affine critical values, the affine Milnor numbers for all irregular fibers, the critical values at infinity, and the Milnor numbers at infinity for all irregular…
Harmonic map flow's singularity properties proven with Lojasiewicz inequalities.
problem Finite-time singularities of harmonic map flow in critical dimensions.
method Proving a weighted Lojasiewicz inequality.
result Continuity of body map and no-neck property for bubble-tree decompositions.
We prove that the level sets of a real C^s function of two variables near a non-degenerate critical point are of class C^[s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at hyperbolic points or elliptic points, and in particular at umbilic points. We al…
Study on solutions to conformally invariant fourth order equations, classifying their properties.
problem Classify qualitative properties of solutions to conformally invariant fourth order equations.
method Analyze two cases: removable and non-removable singularities, using Pohozaev-type invariant.
result Non-existence of semi-singular solutions, classifying them as multiples of the Emden--Fowler solution.
The paper describes the topology of spaces of functions with specific singularities on surfaces.
problem Understanding the homotopy type and structure of spaces of functions with prescribed singularities.
method Analyzes the homotopy type of spaces of functions with specific singularities on surfaces, considering the action of coordinate transformations.
result Describes the homotopy type and decomposition of spaces of functions with prescribed singularities on surfaces.
Study on Gaussian random fields' singularities on manifolds.
problem Understanding singularities of Gaussian random fields on manifolds.
method Computed expected values of singularities under various conditions.
result Explicit formulae for singularities under different constraints.
The paper explores the correspondence between gradient flow lines of a function and its Lagrange multiplier functional.
problem Detecting critical points of a function subject to constraints.
method Adiabatic limit technique and singular version of the implicit function theorem.
result A one-to-one correspondence between gradient flow lines connecting critical points of Morse index difference one.
The paper studies the smoothness of critical points of variational integrals on Hessian spaces.
problem The study focuses on the regularity of critical points of variational integrals defined on Hessian spaces.
method The approach involves solving a fourth order nonlinear equation and analyzing the Hessian of the critical points.
result Smooth critical points with bounded Hessian are shown to be smooth provided their Hessian has small BMO.
Research extends a function to a non-orientable 3-manifold without critical points.
problem Extending a function to a non-orientable 3-manifold without critical points.
method Analyzing the Reeb graph condition for extension.
result Necessary and sufficient condition for extension exists.
Paper develops methods for analyzing forms with synchronized singularities.
problem Analyzing forms with synchronized singularities.
method Exact reduction, analytic transfer, and geometric recomposition.
result Transfer of sparse domination principle to synchronized singular forms.
We are going to use the Euler's vector fields in order to show that for real quasi-homogeneous singularities with isolated critical value, the Milnor's fibration in a "thin" hollowed tube involving the zero level and the fibration in the complement of "link" in sphere are equivalents, since they exist. Moreover, in ord…
Smooth functions on manifolds with degenerate singular submanifolds
problem Existence of functions with minimal number of submanifolds
method Sufficient conditions for existence
result Minimal number of submanifolds
Study classifies positive solutions to a biharmonic equation near singularities.
problem Classifying positive solutions to a nonlinear biharmonic equation with critical exponent.
method Analyzes periodic functions of the logarithm of the distance to the origin.
result Identifies periodic functions of the logarithm of the distance to the origin as solutions.
Using a method developped in [1] and [2], we prove the existence of weak non trivial solutions to fourth order elliptic equations with singularities and with critical Sobolev growth.
The paper uses singularity theory to find normal forms for sub-Riemannian exponential maps.
problem Finding normal forms near critical points of sub-Riemannian exponential maps.
method Singularity theory applied to sub-Riemannian structures.
result Normal forms for sub-Riemannian exponential maps in specific cases.
Develops virtual Morse-Bott indices for four-manifolds, proving inequalities.
problem Proving inequalities for four-manifolds of Seiberg-Witten simple type.
method Uses virtual Morse-Bott indices and Hirzebruch-Riemann-Roch Theorem.
result Proves positivity of virtual Morse-Bott indices, leading to inequalities.
New insights into matrix factorization show strict saddles have bounded eigenvalues.
problem Understanding the nature of critical points in matrix factorization.
method Analyzing orbits of critical points under the general linear group and identifying canonical points.
result Minimum eigenvalue of strict saddles is not uniformly bounded below zero.
Study proves stability of big bang singularity in complex system.
problem Stability of Kasner solutions in Einstein-Maxwell-scalar field-Vlasov system.
method Detailed mathematical structures and new delicate arguments.
result Nonlinear stability with Kasner exponents in full strong sub-critical regime.
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…
The monodromy action in the homology of level sets of Morse functions on stratified singular analytic varieties is studied. The local variation operators in both the standard and the intersection homology groups defined by the loops around the critical values of such functions are reduced to similar operators in the ho…
We characterize those closed 2k-manifolds admitting smooth maps into (k+1)-manifolds with only finitely many critical points, for k∈{2,4}. We compute then the minimal number of critical points of such smooth maps for k=2 and, under some fundamental group restrictions, also for k=4. The main ingredients ar…
Our objective is to develop a stratified Morse theory with tangential conditions. We define a continuous strata-wise smooth Morse function on an abstract stratified space by using control conditions and radiality assumptions on the gradient vector field. For critical points of a Morse function one can show that the loc…
Study singular curves on subriemannian spaces that don't affect homotopy types.
problem Understanding how singular curves affect the topology of horizontal paths.
method Analyzing the subriemannian energy and endpoint map to identify homotopically invisible curves.
result For d≥3, generic subriemannian structures have only homotopically invisible singular curves. The study confirms a conjecture about critical points of smooth functions.
problem Understanding isolated critical points of smooth functions.
method Investigated cone-like, reasonable, and Rothe H hypothesis critical points.
result The conjecture holds true for certain critical points.