Minimal generating sets of Reidemeister moves identified and classified.
problem Classifying minimal generating sets of Reidemeister moves.
method Determined minimal generating sets, provided classifications, and identified candidates.
result 12 out of 16 candidates for minimal generating sets were proven minimal.
Minimal sets of moves for isotopic knots and trivalent graphs identified.
problem Identifying minimal sets of moves for isotopic knots and trivalent graphs.
method Provided and proved the existence of minimal generating sets of oriented Reidemeister moves for isotopic knots and spatial trivalent graphs.
result Twelve minimal generating sets of oriented Reidemeister moves for isotopic knots and ten for spatial trivalent graphs identified.
The paper finds minimal generating sets and abelianizes the quasitoric braid group.
problem Understanding the structure of quasitoric braids and their subgroup properties.
method Provided two minimal generating sets and determined the abelianization.
result Minimal generating sets and abelianization of the quasitoric braid group were determined.
Paper solves the minimal generating set problem for singular Reidemeister moves.
problem Determine minimal generating sets of oriented singular Reidemeister moves.
method Introduced new invariant for singular links to detect type IV moves and provide obstructions.
result Proved exactly 96 distinct inclusion-minimal generating sets for singular moves.
Proves a minimal generating set for a specific group of mapping classes.
problem Finding a minimal generating set for a specific group of mapping classes.
method Proved the group is generated by four elements, with minimal exceptions.
result Minimal generating set for the balanced superelliptic mapping class group.
Minimal generating sets found for Kim-Manturov groups.
problem Understanding the structure of groups related to surface triangulations.
method Provided minimal generating sets and determined abelianizations.
result Minimal generating sets and abelianization results for the groups.
Minimal moves for surfaces in 4D identified.
problem Classifying surfaces embedded in 4D space.
method Derived minimal generating set of planar moves.
result Identified minimal moves for surfaces in 4D.
Regret minimization is a powerful tool for solving large-scale problems; it was recently used in breakthrough results for large-scale extensive-form game solving. This was achieved by composing simplex regret minimizers into an overall regret-minimization framework for extensive-form game strategy spaces. In this paper…
Polyak proved that the set {Ω1a,Ω1b,Ω2a,Ω3a} is a minimal generating set of oriented Reidemeister moves. One may distinguish between forward and backward moves, obtaining 32 different types of moves, which we call directed oriented Reidemeister moves. In this article we prove that the set of $…
Study shows area-minimizing submanifolds are mostly rough, not smooth.
problem Understanding the smoothness of area-minimizing submanifolds.
method Proved non-smoothness by contradiction and established Hausdorff dimension bounds.
result Area-minimizing submanifolds are not generically smooth, resolving a conjecture.
Smooth minimizing hypersurfaces in 11D are generic, with singularities in higher dimensions.
problem Finding smooth minimizing hypersurfaces in high dimensions.
method Analyzing the Plateau problem and area minimization in integral homology.
result Smooth minimizing hypersurfaces are generic in 11D, with singularities in higher dimensions.
This paper determines a minimal generating set and abelianization of a specific subgroup of SL(n,Z).
problem Understanding the structure of the level d principal congruence subgroup of SL(n,Z).
method Direct computation and theorems about the subgroup without relying on previous results.
result A minimal generating set and abelianization of the level d principal congruence subgroup of SL(n,Z).
Minimal sets of moves for rotational Reidemeister diagrams are identified.
problem Understanding the minimal sets of moves for rotational Reidemeister diagrams.
method Detailed description and proof of minimal generating sets for rotational Reidemeister moves.
result Minimal generating sets for oriented, framed links contain 5 moves.
The paper proves prevalent existence and partially determines moduli space of area-minimizing surfaces with fractal singular sets.
problem Existence and moduli space of area-minimizing surfaces with fractal singular sets.
method Proof of prevalent existence, determination of moduli space, refinement of strata.
result Sharp results on moduli space and refinement of strata, showing fractal singularities do not completely dissolve under generic perturbations.
The study finds generic regularity of minimal hypersurfaces in Riemannian manifolds.
problem Finding regularity of minimal hypersurfaces in Riemannian manifolds.
method Estimate for a one-parameter min-max minimal hypersurface.
result Generic regularity of minimal hypersurfaces in 8-dimensional Riemannian manifolds with positive Ricci curvature.
Minimal moves for surfaces in 4D discovered, linking planar and spatial moves.
problem Finding the minimal set of moves for surfaces in 4D.
method Derived minimal generating set of spatial moves, translated into planar moves.
result Minimal generating set of spatial moves for surfaces in 4D.
Perturbs area-minimizing hypersurfaces to reduce singular set's dimension.
problem Reduces the dimension of the singular set of area-minimizing hypersurfaces.
method Perturbs a smooth hypersurface to minimize the Minkowski dimension of the singular set.
result The singular set of the perturbed minimizing current has Minkowski dimension less than n-9.
We construct a minimal generating set of the level 2 mapping class group of a nonorientable surface of genus g, and determine its abelianization for g≥4.
Minimal generating sets found for surface mapping groups.
problem Finding the smallest sets of elements needed to generate mapping class groups of surfaces.
method Analyzing surfaces with different genera and punctures to find minimal generating sets.
result Minimal generating sets found for Mg,p and Mg,p± with g≥3 and p≥0. New method improves smoothness of minimizing currents near singular points.
problem Improving smoothness of minimizing currents near singular points.
method New method to estimate the full singular set of the foliation by minimizers and proof of superlinear decay of closeness.
result Generic smoothness of minimizers improved to n−9−εn for n≥11. We study the centralizer of a braid from the point of view of Garside theory, showing that generically a minimal set of generators can be computed very efficiently, as the ultra summit set of a generic braid has a very particular structure. We present an algorithm to compute the centralizer of a braid whose generic-cas…
Smoothness of collapsed regions in soap films is proven, indicating wetted singularities.
problem Smoothness of collapsed regions in soap films.
method Study of generalized minimizers in capillarity model.
result Collapsed regions are smooth outside of dimensionally small singular sets.
New formulas limit minimal submanifolds' area in curved spaces.
problem Bounding minimal submanifolds' area in curved spaces.
method Developed new monotonicity formulae involving energy-like integrals over non-geodesic sets.
result Imply sharp area bounds for minimal submanifolds through a prescribed point.
Survey of minimal generating sets for nonorientable mapping class groups.
problem Challenges in generating minimal sets for nonorientable surfaces.
method Detailed analysis of various generating sets, including torsions, involutions, and commutators.
result For large genus, both Mod(Ng) and Tg are generated by two elements. Optimal regularity theory for stable minimal hypersurfaces with small singular set.
problem Optimal regularity of stable minimal hypersurfaces with small singular set.
method Analysis of stable minimal hypersurfaces in a specific domain with small singular set.
result Optimal size assumption on the non-immersed singular set guarantees optimal regularity.
Study of logarithms in SVD-closed subgroups of unitary group.
problem Understanding logarithms in SVD-closed subgroups of unitary groups.
method Analysis of generalized principal logarithms and minimizing geodesics.
result Set of generalized principal logarithms is a disjoint union of diffeomorphic subsets.
This study provides a new mathematical structure for Koopman eigenfunctions.
problem Understanding and representing nonlinear dynamics as linear.
method Theoretical, analytical, and numerical approaches to Koopman eigenfunction space.
result Equivalence of minimal generating set and maximal independent set, defining conditions for independence.
New examples show flat singular sets can be arbitrarily complex.
problem Understanding the structure of singular sets in almost-minimizing currents.
method Construction of specific examples of area almost-minimizing currents.
result Flat singular sets can contain any closed empty interior subset of a plane.
This chapter surveys minimal generating sets for mapping class groups of orientable surfaces.
problem Determining minimal generating sets for mapping class groups.
method Exploration of classical and recent results, including new findings for specific cases.
result For even number of punctures p≥8, the group Mod(Σ13,p) is generated by three involutions. Minimal involutions generate a subgroup of nonorientable surfaces.
problem Generating a minimal set of involutions for a specific subgroup.
method Obtained a minimal generating set of involutions.
result Minimal involutions for the level 2 subgroup of a nonorientable surface.
The paper explores generic properties of minimal surfaces in high dimensions.
problem Understanding the behavior of minimal surfaces in high-dimensional spaces.
method Analyzing the space of conformal minimal immersions using Baire category theory.
result A generic conformal minimal immersion is chaotic in various ways, such as being non-proper, almost proper, and g-complete. Paper proves generalizations of Bernstein's theorem in higher dimensions.
problem Proving theorems about sets in higher-dimensional spaces.
method Analyzes properties of sets with monotonicity formula and uses geometric and analytic techniques.
result Generalizations of Bernstein's theorem in higher dimensions.
The study bounds Hausdorff measure of flat singular points in area-minimizing currents.
problem Bounding Hausdorff measure of flat singular points in area-minimizing currents.
method Proving locally finite (m−2)-dimensional Hausdorff measure and Minkowski content bounds. result The set of flat singular points has locally finite (m−2)-dimensional Hausdorff measure. We show that directed minimal cones in (n+1)-dimensional Euclidean space which have at most one singularity are - besides the trivial cases: empty set, whole space - half spaces. Using blow-up techniques, this result can be used to get C^{1,lambda}-regularity for the measure-theoretic boundary of almost minimal Cacciop…
New algorithm for online convex minimization over integer lattice.
problem Online decision-making with nonlinear combinatorial objectives.
method Introduces online Latural-convex minimization and proposes efficient algorithms. result Tight regret bound for full information setting algorithm.
Study minimizers of quasi-perimeters in RCD spaces with volume constraints.
problem Regularity and topological properties of volume constrained minimizers in RCD spaces.
method New Deformation Lemma and study of interior and exterior points.
result Volume constrained minimizers are open bounded sets with Ahlfors regular boundary.
This paper deals with continuity preservation when minimizing generalized total variation with a L2 fidelity term or a Dirichlet boundary condition. We extend several recent results in the two cases, mainly by showing comparison principles for the prescribed mean curvature problem satisfied by the level-sets of such…
Paper proves existence of area-minimizing submanifolds on almost any manifold with fractal singular sets.
problem Existence of area-minimizing submanifolds with fractal singular sets.
method Constructing and proving existence on almost any smooth manifold.
result Existence of area-minimizing submanifolds with fractal singular sets on almost any smooth manifold.
Improved flatness in annuli using PDE methods.
problem Flatness improvement in annuli.
method PDE-based approach adapted to exterior domains.
result Alternative proof of minimal surface end-structure and asymptotics.
For any smooth Riemannian metric on an (n+1)-dimensional compact manifold with boundary (M,∂M) where 3≤(n+1)≤7, we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min-max theory in the Almgren-Pitts setting. We apply our Morse index estimates t…
Minimal surfaces with complex branching structures constructed using various methods.
problem Construct minimal surfaces with specific branching sets.
method Three constructions using perturbation, barrier methods, and bifurcation theory.
result Minimal surfaces with stratified branching sets constructed as graphs of two-valued functions.
We develop a theory of "minimal θ-graphs" and characterize the behavior of limit laminations of such surfaces, including an understanding of their limit leaves and their curvature blow-up sets. We use this to prove that it is possible to realize families of catenoids in euclidean space as limit leaves of sequences of…
Minimal graph level sets are concave if boundary is concave.
problem Understanding curvature of minimal graph level sets.
method Proved an inequality and showed geometric properties.
result Level sets of minimal graphs are concave if boundary is concave.
Paper extends SMM to weakly convex and multi-convex surrogates for non-convex optimization.
problem Non-convex optimization with weakly convex or multi-convex surrogates.
method Stochastic majorization-minimization with proximal regularization or block-minimization.
result Convergence rates for empirical and expected losses under non-i.i.d. data.
We prove that a theorem of Pawlucki, showing that Whitney regularity for a subanalytic set with a smooth singular locus of codimension one implies the set is a finite union of differentiable manifolds with boundary, applies to definable sets in polynomially bounded o-minimal structures. We give a refined version of Paw…
Proves a function's locally least gradient property if its level sets are minimal laminations.
problem Understanding the relationship between 1-harmonic functions and minimal laminations.
method Analyzes minimal laminations and their convergence properties, then applies to 1-harmonic functions.
result Proves a function is 1-harmonic if its level sets are minimal laminations.
Most known examples of doubly periodic minimal surfaces in R3 with parallel ends limit as a foliation of R3 by horizontal noded planes, with the location of the nodes satisfying a set of balance equations. Conversely, for each set of points providing a balanced configuration, there is a correspo…
In \cite{CHMY04}, we studied p-mean curvature and the associated p-minimal surfaces in the Heisenberg group from the viewpoint of PDE and differential geometry. In this paper, we look into the problem through the variational formulation. We study a generalized p-area and associated (p-) minimizers in general di…