Develops analysis of Hölder continuous mappings on Heisenberg groups.
problem Analyzing Hölder continuous mappings on Heisenberg groups.
method Theory of distributional Jacobians and pullbacks of differential forms.
result Simple proof of a generalization of the Gromov non-embedding theorem and new results about Hölder homotopy groups.
Classifies surfaces for pure mapping class groups with automatic continuity.
problem Determining surfaces for which pure mapping class groups have automatic continuity.
method Completely classified orientable infinite-type surfaces and specific cases of surfaces with finite ends.
result Classification of surfaces for automatic continuity of pure mapping class groups.
By a fixed continuous map from a 3-space to itself, a knot in the 3-space may be mapped to another knot in the 3-space. We analyze possible knot types of them. Then we map a knot repeatedly by a fixed continuous map and analyze possible infinite sequences of knot types.
Proves rigidity of maps between balls with Hölder boundary continuity.
problem Rigidity of proper holomorphic maps between unit balls with Hölder boundary continuity.
method Proves rigidity for maps with symmetries and Hölder boundary continuity.
result Proves rigidity for maps with Hölder exponent > 1/2 on the boundary.
Strict type-II blowup in harmonic map flow is proven to have Hölder continuous body map.
problem Finite-time singularity of harmonic map flow.
method Analysis of outer energy scale and Hölder continuity proof.
result Strictly type-II blowup body map is Hölder continuous.
Continuous epimorphisms between certain mapping class groups are induced by homeomorphisms.
problem Understanding continuous epimorphisms between specific mapping class groups.
method Analyzing subgroups of mapping class groups of infinite-genus 2-manifolds with no planar ends.
result Continuous epimorphisms are induced by homeomorphisms for the specified subgroups.
We consider harmonic maps into pseudo-Riemannian manifolds. We show the removability of isolated singularities for continuous maps, i.e. that any continuous map from an open subset of R^m into a pseudo-Riemannian manifold which is two times continuously differentiable and harmonic everywhere outside an isolated point i…
Proves properties of sub-Riemannian exponential map, showing it's not injective.
problem Regularity and continuity of sub-Riemannian exponential map.
method Used sub-Riemannian Jacobi fields and Maslov index of Jacobi curves.
result Exponential map of 3D Heisenberg group is not injective near conjugate vectors.
Continuous time analysis of bubble formation in harmonic maps.
problem Understanding bubble formation in harmonic map heat flow.
method Continuous time approach to analyze bubbling sequences.
result Solutions approach multi-bubble configurations in continuous time.
We study continuous maps between differential manifolds from a microlocal point of view. In particular, we characterize the Lipschitz continuity of these maps in terms of the microsupport of the constant sheaf on their graph. Furthermore, we give lower and upper bounds on the microsupport of the graph of a continuous m…
The paper explores unique continuation properties for polyharmonic maps between Riemannian manifolds.
problem Investigating unique continuation principles for polyharmonic maps.
method Analyzing critical points of higher order functionals to prove extensions of known results in harmonic and biharmonic cases.
result Proving extensions of unique continuation principles for k-harmonic maps.
We prove several unique continuation results for biharmonic maps between Riemannian manifolds.
The paper shows dense and residual sets of continuous maps with positive metric mean dimension.
problem Understanding the genericity of continuous maps with positive metric mean dimension.
method Analyzing continuous maps on compact Riemannian manifolds and Cantor sets.
result The set of continuous maps with metric mean dimension equal to a given value is dense and, for the dimension, residual in the space of continuous maps.
We solve the differentiability problem for the evolution map in Milnor's infinite dimensional setting. We first show that the evolution map of each Ck-semiregular Lie group G (for k∈N⊔{lip,∞}) admits a particular kind of sequentially continuity − called Mackey k-continuity. We …
The paper explores continuous limits of pentagram maps and their relation to KdV equations.
problem Understanding the continuous limits of pentagram maps and their associated KdV equations.
method Quantum calculus and geometric constructions to derive continuous limits and Lax representations.
result Continuous limits of pentagram maps yield specific KdV equations, providing a geometric interpretation.
Topological degrees of continuous mappings between manifolds of even dimension are studied in terms of index theory of pseudo-differential operators. The index formalism of non-commutative geometry is used to derive analytic integral formulas for the index of a 0:th order pseudo-differential operator twisted by a Hölde…
Harmonic map flow preserves almost-holomorphic maps without singularities.
problem Preserving almost-holomorphic maps without singularities.
method Harmonic map flow applied to almost-holomorphic maps.
result No singularities or necks appear in the limit at singular times.
Automatic continuity of polynomial maps and cocycles proved.
problem Proving continuity of polynomial maps and cocycles.
method The approach involves proving continuity of polynomial maps and cocycles.
result Automatic continuity of polynomial maps and cocycles.
Paper proves equivalence of derivatives for maps between Carnot groups.
problem Maps between Carnot groups and their derivatives.
method Elementary proof using Euclidean arguments and mean value estimates.
result Maps preserving horizontal curves are continuously Pansu differentiable.
The paper studies constraint maps with singularities and free boundaries, proving continuity near singularities and optimality.
problem Analyzing the structure of constraint maps with singularities and free boundaries.
method Establish continuity near singularities using a new quantitative unique continuation principle, and investigate the presence of branch points leading to new singularities.
result Topological singularities can only lie in the interior of the contact set in the uniformly convex setting, and the optimality of this result is proven.
Proves continuity and singular set dimension for 2D maps with Q values.
problem Interior regularity of 2D Q-valued maps. method Strong concentration-compactness theorem for equicontinuous maps.
result 2D Q-valued maps are Hölder continuous with singular set dimension ≤1. We prove local Holder continuity of quasi-n-harmonic mappings from Euclidean domains into metric spaces with non-positive curvature in the sense of Alexandrov. We also obtain global Holder continuity of such mappings from bounded Lipschitz domains.
Study continuation maps for Morse fundamental group properties.
problem Properties of continuation maps for Morse fundamental group.
method Analysis of continuation maps for Morse fundamental group, functoriality, and isomorphism to relative fundamental group.
result Continuation maps are isomorphic to relative fundamental groups.
In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally Hölder continuous. In [39], F. H. Lin proposed a challenge problem: Can the Hölder continuity …
Study on infinite-type surfaces shows stable commutator length is continuous and defines open subgroups.
problem Understanding stable commutator length on infinite-type surfaces.
method Analyzing mapping class groups of infinite-type surfaces, showing continuity and openness of commutator subgroups.
result Stable commutator length defines a continuous function on commutator subgroups of infinite-type mapping class groups.
CAM-GAN improves GANs for continual learning with efficient feature map transformations.
problem Efficient continual learning for GANs with reduced parameter growth.
method Designing and leveraging parameter-efficient feature map transformations, including global and task-specific parameters, residual bias, and Fisher information matrix.
result Significantly improved model performance and high-quality samples with fewer parameters.
Earth observation embeddings can convert discrete biome maps into continuous representations that better capture ecological variation.
problem Biome maps impose categorical boundaries that compress continuous variation in biotic communities.
method Fit a linear classifier on Earth observation embeddings to predict biome labels.
result Continuous biome representation outperforms discrete biome labels for predicting species occurrence.
Continuity of roots of hyperbolic polynomials with smooth coefficients.
problem Continuity of the solution map for hyperbolic polynomials.
method Proving continuity of the solution map from hyperbolic polynomials of degree d with C^d coefficients to their increasingly ordered roots.
result Continuity of the solution map for hyperbolic polynomials with C^d coefficients.
There is a well-known correspondence between infinite trees and ultrametric spaces which can be interpreted as an equivalence of categories and comes from considering the end space of the tree. In this equivalence, uniformly continuous maps between the end spaces are translated to some classes of coarse maps (or even c…
In this note we prove that reconstruction from magnitudes of frame coefficients (the so called "phase retrieval problem") can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map α:H→Rm is injective, with (α(x))k=∣<x,fk>∣2, where $…
Develops Lefschetz theory for noncompact manifolds.
problem Lefschetz fixed-point theory for noncompact manifolds.
method Introduces uniform bounded cohomology and develops obstruction theory.
result Uniform Lefschetz class vanishes if and only if map is homotopic to a strongly fixed-point free map.
A map f:X→Y between topological spaces is defined to be {\em scatteredly continuous} if for each subspace A⊂X the restriction f∣A has a point of continuity. We show that for a function f:X→Y from a perfectly paracompact hereditarily Baire Preiss-Simon space X into a regular space Y the scattere…
For a bounded domain equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from (Ω,g) to a compact Riemannian manifold (N,h)⊂Rk without boundary. We generalize the notion of stationary harmonic map and prove the partial regularity. We also discuss the global Li…
Classifies Tk-manifolds with continuous sections.
problem Classifying Tk-manifolds with specific properties. method Uses continuous sections to orbit map for classification.
result Classifies Tk-manifolds up to equivariant diffeomorphism. Classifies when homeomorphism groups of stable surfaces have automatic continuity.
problem Determining when homeomorphism groups of stable surfaces are continuous.
method Developed a general framework to prove automatic continuity for homeomorphism groups, applied to stable surfaces and Stone spaces.
result Classification of stable surfaces with respect to automatic continuity of their homeomorphism groups.
Constructs a map with prescribed local Lipschitz constants on a subset of a manifold.
problem Creating a Lipschitz map with specific local Lipschitz constants on a subset of a manifold.
method Constructs a Lipschitz map that matches a given map on a subset and has a local Lipschitz constant defined by a continuous function.
result A Lipschitz map can be constructed with a local Lipschitz constant prescribed by a continuous function.
A smooth map having only fold singularities is called a fold-map. We will give effective conditions for a continuous map to be homotopic to a fold-map from the viewpoint of the homotopy principle.
We consider a continuous map f:M→N between two manifolds and try to estimate its multiplicity from below, i.e. find a q-tuple of pairwise distinct points x1,...,xq∈M such that f(x1)=f(x2)=...=f(xq). We show that there are certain characteristic classes of vector bundle f∗TN−TM that guarant…
Continuity of earthquake flow map transfers Teichmüller dynamics results.
problem Transfer results from Teichmüller dynamics to earthquake flow.
method Analyze continuity of earthquake flow map and its inverse.
result Transfer results from Teichmüller dynamics to earthquake flow.
The definition of the grafting operation for quasifuchsian groups is extended by Bromberg to all b-groups. Although the grafting maps are not necessarily continuous at boundary groups, in this paper, we show that the grafting maps take every "standard" convergent sequence to a convergent sequence. As a consequence of…
Local Lipschitz continuity of sub-elliptic harmonic maps into CAT(0) spaces proved.
problem Proving Lipschitz continuity of sub-elliptic harmonic maps between singular spaces.
method Analyzing sub-elliptic harmonic maps from the Heisenberg group into CAT(0) spaces.
result Local Lipschitz continuity established for sub-elliptic harmonic maps.
We focus on the challenge of finding a diverse collection of quality solutions on complex continuous domains. While quality diver-sity (QD) algorithms like Novelty Search with Local Competition (NSLC) and MAP-Elites are designed to generate a diverse range of solutions, these algorithms require a large number of evalua…
The paper shows how Sobolev maps affect currents in metric spaces.
problem Understanding how Sobolev maps affect currents in metric spaces.
method Proving that a Sobolev map pushes almost every compactly supported integral current to an Ambrosio-Kirchheim integral current.
result The paper proves an isoperimetric inequality for Sobolev mappings relative to bounded, closed, and additive cochains.
Extends holomorphic functions on complex manifolds to larger spaces.
problem Extending holomorphic functions on complex manifolds.
method Proving the existence of a larger space B(S,X) for continuous maps that allows holomorphic continuation. result Bounded holomorphic functions on C(S,X) can be extended to holomorphic functions on B(S,X). The bending map of a hyperbolic 3-manifold maps a convex cocompact hyperbolic metric on a hyperbolic 3-manifold with boundary to its bending measured geodesic lamination. In the present paper we study the extension of this map to the space of geometrically finite hyperbolic metrics. We introduce a relationship on the s…
This survey covers in our opinion the most important results in the theory of continuous selections of multivalued mappings (approximately) from 2002 through 2012. It extends and continues our previous such survey which appeared in Recent Progress in General Topology, II, which was published in 2002. In comparison, our…
Develops a new method for learning discrete distributions without embedding them in a continuous space.
problem Challenges in learning discrete distributions using current methodologies.
method Introduces a MAD invertible map and a mixed variational flow (MAD Mix) for discrete distributions.
result MAD Mix produces more reliable approximations than continuous-embedding flows.
New approach shows continuity and compactness of martingale measures.
problem Stability of martingale optimal transport problem.
method Set-valued map theory and lower-upper hemicontinuity.
result Lower and upper hemicontinuity of the set of martingale measures.