The minimal number of critical points is studied for smooth functions on closed manifolds.
problem Determining the minimal number of critical points for smooth functions on closed manifolds.
method Investigates cylindrical ball neighborhoods and exotic critical points, proving the conjecture for certain types of critical points.
result The minimal number of critical points is the same for smooth functions without exotic critical points on closed manifolds of dimension at least 6.
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.
This paper reverses a construction by merging boundary critical points into an interior one.
problem Pushing interior critical points to the boundary and splitting them into two boundary points.
method Specific assumptions allow merging two boundary critical points into one interior critical point.
result Merging two boundary critical points into a single interior critical point.
Upper bounds found for systole function critical points on surface moduli space.
problem Finding upper bounds for systole function critical points.
method Analyzing the systole function on the surface moduli space.
result Upper bounds for critical points of systole function and their systole values.
This paper focuses on the problem of topological equivalence of functions with isolated critical points on the boundary of a compact surface M which are also isolated critical points of their restrictions to the boundary. This class of functions we denote by Ω(M). Firstly, we've obtained the topological classificat…
Symmetric critical points lead to symmetry breaking in neural networks.
problem Understanding symmetry in critical points of invariant functions.
method Analyzing the symmetry of critical points and their neighbors in invariant nonconvex functions.
result Symmetric critical points in invariant nonconvex functions are generically followed by symmetry breaking adjacent points.
Ricci solitons as critical points of quadratic curvature functionals
problem Einstein metrics and Ricci solitons as critical points of quadratic Riemannian functionals
method Study of Ricci solitons as critical points of a special quadratic curvature functional
result Ricci solitons are non-Einstein critical points of these functionals
Study classifies Morse functions with 4 critical points on immersed 2-spheres.
problem Classifying Morse functions with 4 critical points on immersed 2-spheres.
method Used dual graph of immersion and Reeb graphs to classify functions.
result Found all possible structures of the functions.
Hard to approximate critical points for simple nonconvex functions.
problem Approximating critical points of nonconvex functions.
method Proving hardness results for polynomial-time approximation of critical points.
result Proving that approximating critical points is intractable for simple nonconvex functions.
To each isolated critical point of a smooth function on a 3-manifold we put in correspondence a tree (graph without cycles). We will prove that functions are topologically equivalent in the neighborhoods of critical points if and only if the corresponding trees are isomorphic. A complete topological invariant of functi…
Research characterizes critical points of scalar curvature functionals.
problem Characterizing critical points of scalar curvature functionals.
method Translation and analysis of a previous Russian paper.
result Provides insights into critical points of scalar curvature functionals.
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 functions with isolated critical points on a closed surface. We prove that in a neighborhood of a critical point the function conjugates with Rezk for the some nonnegative integer k. The full topological invariant of such functions is constructed.
Bound critical points for minimal Radó functions.
problem Counting interior critical points for minimal Radó functions.
method Bounding critical points in terms of boundary data and domain Euler characteristic.
result Bound the number of interior critical points.
Study on critical points in random neural networks, revealing three regimes based on activation function.
problem Investigating the expected number of critical points in random neural networks.
method Deriving asymptotic formulas for critical points under infinite-width limit and suitable regularity conditions.
result Three distinct regimes of critical points behavior depending on activation function.
The paper studies critical points in overparameterized neural networks, identifying a star locus and degenerate critical points.
problem Understanding the geometry of loss functions in overparameterized neural networks.
method Identifying and analyzing components of the critical locus of the loss function L for overparameterized feedforward neural networks of depth ℓ≥4. result For very wide networks, all critical points are degenerate, and lower bounds on the number of zero eigenvalues of the Hessian are given.
Study finds critical points in perimeter functional for fixed volume sets.
problem Finding critical points in perimeter functional for sets of fixed volume.
method Utilizes Mazurwoski--Zhou techniques and new Cacciopoli set connectedness results.
result Constructs smooth almost embedded hypersurfaces with non-zero constant mean curvature.
Distance function to a finite set is a topological Morse function.
problem Characterizing the topological Morse function of a finite set.
method Analyzing the distance function to a finite set in \(\mathbb{R}^n\).
result Distance function is a topological Morse function, with precise critical points and indices.
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.
Paper proves Łojasiewicz inequalities near simple bubble trees on surfaces.
problem Proving Łojasiewicz inequalities for critical points on surfaces.
method Deriving sufficient conditions for Łojasiewicz inequalities near almost-critical points in a Hilbert space.
result Sequences of almost critical points satisfy Łojasiewicz inequalities as they approach the first non-trivial bubble tree.
The article studies critical points of a new energy functional in higher dimensions.
problem Investigating critical points of a new energy functional in higher dimensions.
method Holomorphic deformations, closed and open properties, differential of the functional.
result Properties of critical points under holomorphic deformations are closed and open.
Characterizes infinite harmonic maps using 1-currents.
problem Defines critical points of a non-differentiable functional.
method Uses subdifferential and geometric condition in terms of 1-currents.
result Geometric condition equivalent to criticality in terms of 1-currents.
Proves planar Lipschitz critical points of area functional are smooth.
problem Lawson-Osserman conjecture about smoothness of critical points.
method Outer variations to prove smoothness of critical points.
result Proves conjecture for planar case.
Let (W,M,M'), dim W > 5, be a non-trivial h-cobordism (i.e., the Whitehead torsion of (W,V) is non-zero). We prove that every smooth function f: W --> [0,1], f(M)=0, f(M')=1 has at least 2 critical points. This estimate is sharp: W possesses a function as above with precisely two critical points.
Classical Ljusternik-Schnirelmann category is upper bounded by the number of critical points of any bounded from below differentiable functions of Palais-Smale type. Here we achieve an adaptation of this result for the tangential category of foliations. We introduce a weaker type of Palais-Smale function, obtaining a s…
We define a new notion---the sub-index of a critical point of a distance function. We show how sub-index affects the homotopy type of sublevel sets of distance functions.
We study critical points of the Ginzburg-Landau (GL) functional and the abelian Yang-Mills-Higgs (YMH) functional on the sphere and the complex projective space, both equipped with the standard metrics. For the GL functional we prove that on Sn with n≥2 and CPn with n≥1, stable critical…
We prove that the number of critical points of a Li-Tam Green's function on a complete open Riemannian surface of finite type admits a topological upper bound, given by the first Betti number of the surface. In higher dimensions, we show that there are no topological upper bounds on the number of critical points by con…
The distance function to a generic submanifold behaves well under small perturbations.
problem The critical points of the distance function to a generic submanifold can be poorly behaved.
method Listed and proved regularity conditions on critical and μ-critical points of a submanifold, and showed they are generically satisfied and stable under small C2 perturbations. result The distance function to a submanifold satisfies Morse-like conditions when the regularity conditions are fulfilled.
The paper calculates critical points of systole function on Teichmüller space.
problem Critical points with pathological feature in surfaces of large genus.
method Integer linear programming with symmetry breaking technique.
result Found minimal filling sets of systoles in genus 5 with 8 geodesics.
Due to the success of deep learning to solving a variety of challenging machine learning tasks, there is a rising interest in understanding loss functions for training neural networks from a theoretical aspect. Particularly, the properties of critical points and the landscape around them are of importance to determine …
The study examines higher-order modern portfolio theory with complex critical points and feasible portfolio variety.
problem Understanding the complex critical points and feasible portfolio variety in higher-order modern portfolio theory.
method Established genericity conditions for utility functions with higher-order cumulants, analyzed discriminant loci, and determined the dimension and degree of the feasible portfolio variety.
result The utility function has a constant number of complex critical points under genericity conditions, and the feasible portfolio variety has a determined dimension and degree.
The paper studies critical points of horizontal energy functional in Riemannian foliations.
problem Analyzing critical points of horizontal energy functional in Riemannian foliations.
method Utilizing stress-energy tensor, establishing monotonicity formulas, and Jin-type theorems.
result Established monotonicity formulas for horizontally harmonic maps and transversally harmonic maps.
New spinorial functional connects Perelman's W- and F-functionals.
problem Unifying Perelman's functionals for spin manifolds.
method Introduced a new energy functional on spin manifolds, computed its first variation, and established a gradient flow.
result Critical points of the functional are twisted Ricci solitons and eigen-spinsors.
The paper explores the shape of filling-systole subspace in surface moduli space and critical points of systole function.
problem Understanding the structure and critical points of the filling-systole subspace in surface moduli space.
method Analyzing Teichmüller and Weil-Petersson distances to determine the proximity of points to the subspace.
result Most points in Mg are within a specific Teichmüller distance from Xg and have a certain distance from the thick part of Mg. The paper proves new rigidity results for critical metrics of quadratic curvature functionals.
problem Proving rigidity of critical metrics for specific quadratic curvature functionals.
method Rigidity results for conformal vector fields, ODE argument, and new pointwise and integral estimates.
result Critical metrics are rigid under specific conditions.
We study a functional, whose critical points couple Dirac-harmonic maps from surfaces with a two form. The critical points can be interpreted as coupling the prescribed mean curvature equation to spinor fields. On the other hand, this functional also arises as part of the supersymmetric sigma model in theoretical physi…
Study uncovers complex critical points in tensor decomposition.
problem Nonconvex optimization of symmetric tensor decomposition.
method Utilized symmetry to construct critical points and analyze Hessian.
result Obtained precise analytic estimates on objective function and Hessian.
Paper proves conjecture about Einstein metrics on manifolds with positive isotropic curvature.
problem Proving the Besse conjecture for metrics with positive isotropic curvature.
method Analyzing the critical point equation and using properties of metrics with positive isotropic curvature.
result The Besse conjecture is true for metrics with positive isotropic curvature.
The Morse function f near a non-degenerate critical point p is understood topologically, in the light of Morse's lemma. However, Morse's lemma standardizes the function f itself, providing little information of how the gradient ∇f behaves. In this paper, we prove an analytical analogue of Morse's lemma, s…
We study the level sets of the distance function from a boundary point of a convex set in Euclidean space. We provide a lower bound for the range of connectivity of the level sets, in terms of the critical points of the distance function in the sense of Grove-Shiohama-Gromov-Cheeger.
Paper proves convex domains have one maximum for semi-stable solutions.
problem Analyzing critical points of semi-stable solutions on convex domains.
method Relating critical points to an auxiliary function and using topological degree.
result Positive, semi-stable solutions have exactly one non-degenerate critical point.
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.
Constrained Willmore surfaces are conformal immersions of Riemann surfaces that are critical points of the Willmore energy W=∫H2 under compactly supported infinitesimal conformal variations. Examples include all constant mean curvature surfaces in space forms. In this paper we investigate more generally the crit…
Stochastic subgradient descent avoids critical points in definable functions.
problem Finding local minima in definable functions.
method Stochastic subgradient descent with density-like perturbation.
result SGD converges to a local minimum in definable functions.
According to Pixton, there are Morse-Smale diffeomorphisms of the 3-sphere which have no energy function, that is a Lyapunov function whose critical points are all periodic points of the diffeomorphism. We introduce the concept of quasi-energy function for a Morse-Smale diffeomorphism as a Lyapunov function with the le…
In this paper, we investigate critical points of the Laplacian's eigenvalues considered as functionals on the space of Riemmannian metrics or a conformal class of metrics on a compact manifold. We obtain necessary and sufficient conditions for a metric to be a critical point of such a functional. We derive specific con…
Characterizes CR manifolds as critical points of an energy functional.
problem Understanding homogeneous three-dimensional CR manifolds.
method Uses an energy functional dependent on Webster curvature and torsion.
result Identifies Rossi spheres as a specific type of critical point.