Improves Bernstein theorem for zero mean curvature hypersurfaces in Lorentz-Minkowski space.
problem Proving entire zero mean curvature graphs are hyperplanes in Lorentz-Minkowski space.
method Using line theorems at degenerate light-like points to generalize Bernstein theorem.
result Entire zero mean curvature graphs in Lorentz-Minkowski space are hyperplanes if they only contain space-like or light-like points.
New method avoids saddle points without gradients.
problem Optimizing non-convex functions efficiently.
method Zero-order derivative-free algorithm using only function evaluations.
result Converges to second-order stationary points efficiently.
Transforms study meromorphically isothermic surfaces near singular points.
problem Analyzing meromorphically isothermic surfaces near singular points.
method Apply Darboux and Calapso transformations to patches of surfaces.
result Transformed patches exhibit continuity around singular points.
The article discusses localization formulas for Killing vector fields on zero points.
problem Localization formulas for equivariant cohomology of Killing vector fields.
method Equivariant cohomology and zero points analysis.
result Formulas for characteristic numbers and Duistermaat-Heckman type formula.
For a conformal vector field ξ on a Riemannian manifold, we say that a point is essential if there is no local metric in the conformal class for which ξ is Killing. We show that the only essential points are isolated zeros of ξ. As an application, we show that every connected component of the zero set of ξ is t…
Article studies continuous closed 1-forms and their zero points on CW-complexes.
problem Understanding zero points of continuous closed 1-forms on topological spaces.
method Defines continuous closed 1-forms, constructs a category, and applies dynamical systems.
result Relates the number of zero points to the category with respect to a cohomology class.
Constructs all real analytic germs of zero mean curvature surfaces in Lorentz-Minkowski 3-space.
problem Analyzing surfaces with light-like points in Lorentz-Minkowski 3-space.
method Applying the Cauchy-Kovalevski theorem for partial differential equations.
result Surfaces with light-like points in Lorentz-Minkowski 3-space contain a light-like line when they do not change causal types.
The study proves the finiteness of moments for Gaussian field zeros and critical points.
problem Finiteness of moments for Gaussian field zeros and critical points.
method Definition and study of multijets, construction of p-multijet bundles.
result Linear statistics of Gaussian field zeros have finite p-th moments for p ≥ 1.
Improves Bernstein theorem for space-like graphs in Lorentz-Minkowski space.
problem Proves a new Bernstein-type theorem for space-like zero mean curvature graphs.
method Uses fluid mechanical duality between minimal surfaces and maximal surfaces.
result Shows that a zero mean curvature graph with only space-like and light-like points is a plane.
The study finds rational points on specific types of hypersurfaces.
problem Identifying rational points on generic marked hypersurfaces.
method Analyzing hypersurfaces in projective spaces over various fields.
result Conditions for the existence of rational points on hypersurfaces.
The paper studies critical metrics on compact manifolds with zero radial Weyl curvature.
problem Finding critical metrics on compact manifolds with specific curvature properties.
method Analyzing the critical point of the total scalar curvature functional under zero radial Weyl curvature condition.
result CPE metrics with nonnegative sectional curvature in 3-dimension are isometric to a standard 3-sphere.
The paper proves conditions for zero Gaussian curvature convex hypersurfaces to be hyperplanes.
problem Conditions for zero Gaussian curvature convex hypersurfaces to be hyperplanes.
method Proving Bernstein type theorems for entire convex graphical hypersurfaces with zero Gaussian curvature in Euclidean and Minkowski contexts.
result Zero Gaussian curvature convex hypersurfaces must be hyperplanes if the mean curvature goes to zero at infinity.
We discuss Ghys' theorem on 4 zeroes of the Schwarzian derivative and its relation with flattening points of Legendrian curves and Sturm theory.
The paper explores positivity and irreducibility in Hurwitz spaces related to differentials of the second kind.
problem Positivity and irreducibility in Hurwitz spaces of certain covers of the projective line.
method Analyzes strata of differentials of the second kind with fixed multiplicities of zeros and poles, and applies this to show positivity and irreducibility in Hurwitz spaces.
result The Hurwitz spaces of degree d, genus g covers of P1 with pure branching at all but possibly one branch point are irreducible under certain conditions. Proposes a new algorithm to find local Nash equilibria in zero-sum games.
problem Cannot guarantee convergence to local Nash equilibria with previous methods.
method Local symplectic surgery, a two-timescale procedure.
result Local Nash equilibria are the only attracting fixed points.
A little complement concerning the dynamics of non-metric manifolds is provided, by showing that any flow on an ω-bounded surface with non-zero Euler character has a fixed point.
Study shows equivalence of two methods for solving scalar curvature problem.
problem Prescribing scalar curvature of closed Riemannian manifolds.
method Subcritical approximations or negative pseudo gradient flows.
result Equivalence of both approaches with respect to zero weak limits.
This paper tackles the computational complexity of finding approximate stationary points in non-convex optimization.
problem Finding approximate stationary points in non-convex optimization problems.
method PLS-completeness, zero-order algorithms, and gradient queries.
result The query complexity of finding approximate stationary points is Θ(1/ε) for d=2.
We examine the fixed points to first-order RG flow of a non-linear sigma model with background metric, dilaton and tachyon fields. We show that on compact target spaces, the existence of fixed points with non-zero tachyon is linked to the sign of the second derivative of the tachyon potential V′′(T) (this is the anal…
Uncertainty sampling is explained as a gradient step on a smoothed loss, leading to better parameters.
problem Reducing the amount of data required to learn a classifier.
method Interprets uncertainty sampling as a preconditioned stochastic gradient step on a smoothed zero-one loss.
result Uncertainty sampling converges to stationary points of the smoothed population zero-one loss.
We construct, for any ``good'' Cantor set F of Sn−1, an immersion of the sphere Sn with set of points of zero Gauss-Kronecker curvature equal to F×D1, where D1 is the 1-dimensional disk. In particular these examples show that the theorem of Matheus-Oliveira strictly extends two results by do C…
A self-dual harmonic 2-form on a 4-dimensional Riemannian manifold is symplectic where it does not vanish. Furthermore, away from the form's zero set, the metric with the 2-form give a compatible almost complex structure and thus pseudo-holomorphic subvarieties. Such a subvariety is said to have finite energy when the …
Study Nash equilibrium in non-zero-sum game with Bermudan strategies.
problem Optimizing pay-offs in non-linear non-zero-sum games.
method Recursive construction to find Nash equilibrium.
result Existence of Nash equilibrium in non-zero-sum game.
Study finds Scherk type surfaces as extremals for zero-curvature minimal graphs.
problem Finding extremal minimal graphs with zero Gaussian curvature at the center.
method Analyzing Scherk surfaces and their properties.
result Scherk type minimal surfaces are extremals for zero-curvature minimal graphs.
New examples of mixed-type zero-curvature graphs found.
problem Finding new examples of zero-curvature graphs in Lorentz-Minkowski space.
method Using Konderak's representation formula to construct entire zero-curvature graphs over specific planes.
result Existence of new types of entire zero-curvature graphs in mixed-type in Lorentz-Minkowski space.
A Teichmuller lattice is the orbit of a point in Teichmuller space under the action of the mapping class group. We show that the proportion of lattice points in a ball of radius r which are not pseudo-Anosov tends to zero as r tends to infinity. In fact, we show that if R is a subset of the mapping class group, whose e…
Study localizes integrals at isolated degenerate zeros.
problem Localization of Futaki-Morita integrals at isolated degenerate zeros.
method Streamlined exposition in the spirit of Bott, localization procedure for a holomorphic vector field on CPn. result Essentially unique formula for Futaki-Morita integral invariants.
The connected components of the zero set of any conformal vector field v, in a pseudo-Riemannian manifold (M,g) of arbitrary signature, are of two types, which may be called `essential' and `nonessential'. The former consist of points at which v is essential, that is, cannot be turned into a Killing field by a lo…
Theory of point vortices extended to closed surfaces.
problem Extending point vortex dynamics to closed surfaces.
method Unified theory of point vortex dynamics on the plane, sphere, and closed surfaces.
result Comprehensive guide to point vortex dynamics on closed surfaces with genus zero and vanishing total vorticity.
Let T be the standard torus of revolution in R^3 with radii b and 1, 0<b<1. Let αbe a (p,q) torus curve on T. We show that there are points of zero curvature on αfor only one value of the variable radius of T, b=p^2/(p^2+q^2). The curve αhas non-vanishing curvature for all other values of b. Moreover, for this value of…
We consider the dimensionality-reduction problem (finding a subspace approximation of observed data) for contaminated data in the high dimensional regime, where the number of observations is of the same magnitude as the number of variables of each observation, and the data set contains some (arbitrarily) corrupted obse…
Study locates divisors in Hodge bundle with specific properties.
problem Locating effective divisors in the projectivized Hodge bundle.
method Computing the class of closures of loci of canonical divisors with specific conditions.
result Strata of canonical and bicanonical divisors with double zeros span extremal rays of pseudoeffective cones.
Models for 3D harmonic 1-forms and spinors near singular points.
problem Constructing models for Z/2 harmonic 1-forms and spinors in 3D near singular points. method Using symmetries of tetrahedron, octahedron, and icosahedron to construct local models on R3. result Local models are Z/2 harmonic 1-forms or spinors on R3 with zero locus consisting of rays from the origin. We study robust properties of zero sets of continuous maps f:X→Rn. Formally, we analyze the family Zr(f)={g−1(0):∥g−f∥<r} of all zero sets of all continuous maps g closer to f than r in the max-norm. The fundamental geometric property of Zr(f) is that all its zero sets lie outside o…
We formulate a theory of pointed manifolds, accommodating both embeddings and Pontryagin-Thom collapse maps, so as to present a common generalization of Poincaré duality in topology and Koszul duality in En-algebra.
Zero loss is achievable in overparametrized DL networks under specific conditions.
problem Achieving zero loss in overparametrized deep learning networks.
method Determine sufficient conditions for zero loss attainability and present an explicit construction of zero loss minimizers.
result Explicit minimizers for zero loss in overparametrized DL networks are constructed without gradient descent.
Study describes how compact Ricci solitons degenerate as cone angles approach zero.
problem Understanding degenerations of compact Ricci solitons as cone angles approach zero.
method Completely describes the degenerations of compact Ricci solitons, including the Gromov--Hausdorff limit of cigar solitons from conical teardrop solitons.
result Gromov--Hausdorff limit of cigar solitons from conical teardrop solitons.
Mutation improves FTRL convergence in zero-sum games.
problem Lack of last-iterate convergence in FTRL variants.
method Introduced mutation to perturb action probabilities in FTRL.
result M-FTRL converges to Nash equilibria under full-information feedback.
Estimates point counts on Riemannian varieties over finite fields.
problem Counting points on geometrically connected varieties over finite fields.
method Estimates point counts using Riemannian curvature and diameter.
result If sectional curvature and diameter grow, point counts diverge.
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.
We consider the closely related problems of bandit convex optimization with two-point feedback, and zero-order stochastic convex optimization with two function evaluations per round. We provide a simple algorithm and analysis which is optimal for convex Lipschitz functions. This improves on \cite{dujww13}, which only p…
This work finds mixed equilibria in zero-sum games using interacting particle dynamics.
problem Finding mixed equilibrium points in continuous minmax games.
method A method based on entropic regularisation of two-layer zero-sum games with interacting particle dynamics.
result The sequence of empirical measures of the particle system satisfies a large deviation principle as the number of particles grows to infinity, implying convergence of the empirical measure and the Nikaidô-Isoda error.
Policy optimization converges to Nash equilibria in zero-sum LQ games.
problem Finding Nash equilibria in zero-sum linear quadratic games.
method Developed three projected nested-gradient methods to converge to NE.
result Policy optimization methods converge to Nash equilibria in zero-sum LQ games.
We study the problem of isometrically embedding a two-dimensional Riemannian manifold into Euclidean three-space. It is shown that if Gaussian curvature vanishes to finite order and its zero set consists of two smooth curves tangent at a point, then local sufficiently smooth isometric embedding exists.
Consider degenerations of Abelian differentials with prescribed number and multiplicity of zeros and poles. Motivated by the theory of limit linear series, we define twisted canonical divisors on pointed nodal curves to study degenerate differentials, give dimension bounds for their moduli spaces, and establish smootha…
The energy function associated to harmonic maps between surfaces is convex at critical points.
problem Proving convexity of the energy function for harmonic maps between surfaces.
method Analyzing the energy function on Teichmüller space and proving convexity at critical points.
result The energy function is convex at critical points and strictly convex under certain conditions.
New probabilistic method constructs Kähler-Einstein metrics and suggests zero-free properties of zeta functions.
problem Existence and explicit formulas for Kähler-Einstein metrics on Fano varieties.
method Probabilistic construction involving canonical random point processes.
result Zero-free properties of Archimedean zeta functions and their relation to Langlands program.
New ε-harmonic maps of low degree are rigid under certain energy bounds.
problem Understanding the rigidity of ε-harmonic maps of low degree. method Analysis of ε-harmonic maps and their critical points. result Non-trivial ε-harmonic maps of degree zero exist with energy above 8π.