Maps can be embedded in higher dimensions if they lift to embeddings in product spaces.
problem Embedding maps in higher dimensions without self-intersections.
method Lifting maps to embeddings in product spaces.
result Maps can be embedded in higher dimensions if they lift to embeddings in product spaces.
The paper generalizes Reeb spaces for special generic maps and lifts smooth functions.
problem Constructing lifts of smooth maps, especially Morse functions.
method Defining and generalizing quotient maps onto Reeb spaces of special generic maps and constructing lifts.
result Lifts of Morse functions can be constructed using the generalized maps.
Study lifts plane mappings to Heisenberg group.
problem Contact quasiconformal mappings in hyperbolic Heisenberg group.
method Lifting Theorem for symplectic mappings.
result Symplectic mappings lifted to Heisenberg group.
Existence and rigidity results for lifts in Carnot groups.
problem Existence and properties of lifts for maps between Carnot groups.
method Use central extensions to define lifts and prove existence and rigidity results for Lipschitz, Sobolev, and quasiconformal maps.
result Quasiconformal maps admit contact lifts that are bi-Lipschitz.
The paper lifts spherical Morse functions to immersions and embeddings.
problem Lifting spherical Morse functions to other maps.
method New methods to lift to special generic maps with non-positive codimensions.
result Constructs most lifts to special generic maps.
Isothermic nets created from special maps for smooth surfaces.
problem Creating discrete curvature lines on surfaces.
method Special discrete holomorphic maps and lifted-folding.
result Isothermic nets with spherical parameter lines constructed efficiently.
Criterion for lifting smooth contact maps between Carnot groups to central extensions.
problem Existence of smooth contact map lifts between Carnot groups and their central extensions.
method Criterion using pullbacks and Lie algebra cohomology classes.
result Necessary and sufficient conditions for lifting are formulated.
The paper explores conditions for lifting maps between graphs to embeddings.
problem Conditions for embedding maps between graphs.
method Combinatorial techniques and satisfiability of 3-CNF formulas.
result Established necessary and sufficient conditions for lifting.
We deal here with the geometry of the twistor fibration $\mathcal{Z} \to \bb{S}^3_1$ over the De Sitter 3-space. The total space Z is a five dimensional reductive homogeneous space with two canonical invariant almost CR structures. Fixed the normal metric on Z we study the harmonic map equation …
Given a generic PL map or a generic smooth fold map f:Nn→Mm, where m≥n and 2(m+k)≥3(n+1), we prove that f lifts to a PL or smooth embedding N→M×Rk if and only if its double point locus {(x,y)∈N×N∣f(x)=f(y),x=y} admits an equivariant map to Sk−1. As a coro…
Develops a lifting theory for exponential maps in semi-Riemannian geometry.
problem Overcoming singularities in exponential maps to prove geodesic connectivity.
method Lifting theory for semi-Riemannian manifolds with path-continuation property.
result General path-lifting theorem extending globally under certain conditions.
We study subelliptic biharmonic maps, i.e. smooth maps from a compact strictly pseudoconvex CR manifold M into a Riemannian manifold N which are critical points of a certain bienergy functional. We show that a map is subelliptic biharmonic if and only if its vertical lift to the (total space of the) canonical circle bu…
A method improves Cryo-EM 3D map refinement by regularizing rotation estimation.
problem Noise-robustness vs. data-consistency in Cryo-EM 3D map reconstruction.
method Ellipsoidal support lifting (ESL) for regularizing and approximating the global minimizer over Riemannian manifolds.
result The induced bias due to regularizing effect of ESL estimates better rotations than global optimisation.
The paper solves the Nielsen realization problem for high degree del Pezzo surfaces.
problem Which finite subgroups of the mapping class group of a del Pezzo surface lift to the diffeomorphism group?
method Classification and partial answers for d≥7, equivariant connected sum for d=6. result Complete classification for d≥7, partial answer for d=6. Study of lifting maps in branched covers of 3-manifolds, showing non-injectivity.
problem Injectivity of lifting maps in branched covers of 3-manifolds.
method Analysis of branched covers and mapping class groups.
result Lifting map is generally not injective for most regular branched covers of 3-manifolds.
We show that one can lift locally real analytic curves from the orbit space of a compact Lie group representation, and that one can lift smooth curves even globally, but under an assumption.
For a generic degree d smooth map f: N^n -> M^n we introduce its "transverse fundamental group" π(f), which reduces to π_1(M) in the case where f is a covering, and in general admits a monodromy homomorphism π(f) -> S_{|d|}; nevertheless, we show that π(f) can be non-trivial already for rather simple degree 1 maps S^n …
The study proves unique path lifting properties and their implications on quotient spaces and covering maps.
problem Understanding unique path lifting properties and their implications on quotient spaces and covering maps.
method The study uses group actions on R-trees and path lifting properties to prove the main results. result Every map of manifolds with the unique path lifting property is a covering map.
Develops a new algebraic framework for differential geometry of infinite dimensional spaces.
problem Creating a differential geometry for infinite dimensional spaces without topology or local coordinates.
method Introduces a general algebraic framework for lifted geometry applicable to various infinite dimensional spaces.
result Stokes' Theorem appears as a form of differentiability in the lifted geometry of spaces of submanifolds.
Survey of twistor lifts of surfaces in 4-dimensional spaces.
problem Understanding the properties of surfaces in 4-dimensional Euclidean space.
method Definitions of Gauss maps and twistor lifts using orthogonal complex structures.
result Holomorphicity and isotropicity of minimal surfaces in E4. We discuss various lifting and reduction problems for bundles and gerbes in the context of a strict Lie 2-group. We obtain a geometrical formulation (and a new proof) for the exactness of Breen's long exact sequence in non-abelian cohomology. We use our geometrical formulation in order to define a transgression map in …
The present research work proposes a new fast fixed-point averaging algorithm on the compact Stiefel manifold based on a mixed retraction/lifting pair. Numerical comparisons between fixed-point algorithms based on the proposed non-associated retraction/lifting map pair and two associated retraction/lifting pairs confir…
We show that any pseudo-Anosov map that is a lift of pseudo-Anosov homeomorphism of a nonorientable surface has vanishing SAF invariant. We also provide a criterion to certify that a pseudo-Anosov map is not such a lift.
This paper extends braid lifting to coloured braid groupoids for all simple disc covers.
problem Lifting braids to homeomorphisms on branched covers of the disc.
method Defines a map from a coloured braid groupoid to a mapping class groupoid for all simple covers of the disc.
result Characterizes the lift of every coloured braid, recovering classical lifting on liftable braids.
New L∞ liftings derived from Chern-Simons classes for coherent sheaves.
problem Liftings of semiregularity maps for coherent sheaves on complex manifolds.
method Introducing Chern-Simons classes for curved DG-pairs and proving canonical liftings.
result Canonical L∞ liftings of Buchweitz-Flenner semiregularity maps. We construct an analytic multiplicative model of smooth K-theory. We further introduce the notion of a smooth K-orientation of a proper submersion and define the associated push-forward which satisfies functoriality, compatibility with pull-back diagrams, and projection and bordism formulas. We construct a multiplicati…
Introduces Hurewicz fibrations for embedding maps of orbifold charts.
problem No specific problem stated; focuses on new concept definition.
method Defines E-fibration embedding and studies its properties.
result Introduces and studies properties of E-fibration embedding.
We show that given a harmonic map φ from a Riemann surface to a classical compact simply connected inner symmetric space, there is a J2-holomorphic twistor lift of φ (or its negative) if and only if it is nilconformal. In the case of harmonic maps of finite uniton number, we give algebraic formulae i…
Criteria for lifting manifold diffeomorphisms to vector bundle automorphisms.
problem Lifting diffeomorphisms to vector bundles.
method Criteria for lifting diffeomorphisms to linear automorphisms of vector bundles.
result Criteria for lifting diffeomorphisms to linear automorphisms of vector bundles.
Commutes Pansu pullback with spectral complexes in Carnot groups.
problem Understanding the relationship between Pansu pullback and spectral complexes in Carnot groups.
method Proving commutativity between Pansu pullback and differentials in spectral complexes.
result Commutes Pansu pullback with spectral complexes in Carnot groups.
Proof of Mostow rigidity using harmonic maps.
problem Proving Mostow rigidity for hyperbolic manifolds.
method Equivariant harmonic lift with gradient estimate.
result Established harmonic-map proof of Mostow rigidity.
Lifts of maps to frame bundles studied for Riemannian manifolds.
problem Analyzing lifts of maps to frame bundles for Riemannian manifolds.
method Defining subbundles and considering lifts of submersions, studying conformality and harmonicity with Mok metrics.
result Properties of lifts of maps to frame bundles explored and studied.
The paper discusses Gauss maps for Möbius surfaces in spheres and their applications to Willmore surfaces.
problem Understanding Gauss maps and their relation to Willmore surfaces in spheres.
method Definition and study of Lorentzian 2-plane lifts for Möbius surfaces, and equivalence to Willmore condition.
result The conformal harmonicity of a Lorentzian 2-plane lift is equivalent to the Willmore condition for a surface.
We show that a harmonic map from a Riemann surface into the exceptional symmetric space G2/SO(4) has a J2-holomorphic twistor lift into one of the three flag manifolds of G2 if and only if it is `nilconformal', i.e., has nilpotent derivative. Then we find relationships with almost complex maps from a…
A conformal map from a Riemann surface to the Euclidean four-space is explained in terms of its twistor lift. A local factorization of a differential of a conformal map is obtained. As an application, the factorization of a differential provides an upper bound of the area of a super-conformal map around a branch point.
We study the notion of p-quasihomotopy in Newtonian classes of mappings and link it to questions concerning lifts of Newtonian maps, under the assumption that the target space is nonpositively curved. Using this connection we prove that every p-quasihomotopy class of Newtonian maps contains a minimizer of the p-e…
In the present paper, we study complete and vertical lifts of tensor fields from a smooth manifold M to its Weil bundle TAM defined by a Frobenius Weil algebra A. For a Poisson manifold (M,w), we show that the complete lift wC and the vertical lift wV of the Poisson tensor w are Poisson tensors on $T^…
This paper studies Poisson structures defined by divisor ideals.
problem Understanding Poisson structures with degeneracy captured by divisor ideals.
method Developed a framework using divisor ideals and Lie algebroids.
result Effective methods for studying Poisson structures of divisor-type.
The paper connects connections on sheaves to an L∞ morphism lifting semiregularity maps.
problem Understanding connections on sheaves and their relationship to semiregularity maps.
method Proves a canonical association of a connection of type (1,0) on a sheaf to an L∞ morphism. result Establishes a connection between connections on sheaves and an L∞ morphism lifting semiregularity maps. A stable smooth map f:N→M is called "k-realizable" if its composition with the inclusion M⊂M×Rk is C0-approximable by smooth embeddings; and a "k-prem" if the same composition is C∞-approximable by smooth embeddings, or equivalently if f lifts vertically to a smooth embedding $…
We introduce the complete lifts of maps between (real and complex) Euclidean spaces and study their properties concerning holomorphicity, harmonicity and horizontal weakly conformality. As applications, we are able to use this concept to characterize holomorphic maps $φ:{\Bbb C}^{m}\supset U\longrightarrow {\Bbb C}^{n}…
Solve arc diagrams on surfaces via branched covers.
problem Computing arc diagrams on surfaces via branched covers.
method Represent branched covers combinatorially and solve membership problem.
result Efficient solution for triangulated arc diagrams.
The abstract theorem extends a Lie group result to Lie groupoids.
problem Expressing functions on Lie groupoids as convolutions of two functions.
method Using a lemma from Dixmier-Malliavin, Lie algebroids, and exponential map.
result Every smooth, compactly-supported function on a Lie groupoid can be expressed as a finite sum of convolutions of two such functions.
We define compactifications of vector spaces which are functorial with respect to certain linear maps. These "many-body" compactifications are manifolds with corners, and the linear maps lift to b-maps in the sense of Melrose. We derive a simple criterion under which the lifted maps are in fact b-fibrations, and identi…
A new method lifts training of input-convex neural networks to avoid dead weights and plateaued loss.
problem Training input-convex neural networks with non-negative weights.
method Introduces a hypernetwork that emits non-negative weights from a summary of the input batch, adding stochasticity to soften the loss landscape.
result The lift method achieves lower test loss than projected gradient descent and direct softplus reparametrization.
New maps show some surfaces can't be sections of 4D spheres.
problem Finding sections for certain 4D sphere maps.
method Exhibited singular fibrations with high genus fibers.
result Some regular fibers cannot be sections.
The paper studies which branched covers can be lifted to braided embeddings.
problem Which branched covers lift to braided embeddings?
method Analyzes conditions for liftability using Hansen's criterion and examples in different dimensions.
result Not all branched covers lift to braided embeddings; examples are provided in various dimensions.
We deal with a Lie group G acting by isometries on a Riemannian manifold M, such that the quotient M/G is an orbifold, or, equivalently, all slice representations are polar. We show that any smooth orbifold symmetric 2-tensor on M/G lifts to a smooth G-invariant symmetric 2-tensor on M. The proof relies on a fact about…