Extends tangent functor to microformal morphisms, creating non-linear pullbacks for forms and cohomology.
problem Generalizing smooth maps to microformal morphisms for new types of mappings.
method Introduces microformal morphisms and shows how they act on functions and forms via non-linear pullbacks.
result Non-linear pullbacks of forms respect de Rham differentials and induce transformations of cohomology.
This work develops methods to analyze data on curved spaces using deep learning.
problem Analyzing data in non-linear, curved spaces.
method Pullback Riemannian geometry through diffeomorphisms.
result Diffeomorphisms need to map data into geodesic subspaces to ensure proper data analysis.
Develops a Riemannian archetypal analysis for interpretable non-linear data.
problem Limited performance of classical archetypal analysis on non-linear data.
method Riemannian geometry for data-driven pullback, geodesic convex combinations, convex relaxation followed by non-convex refinement.
result Combines interpretability of classical archetypal analysis with expressive power of modern non-linear models.
Defines simplicity of non-linear mappings using information geometry.
problem Finding a simple yet effective non-linear mapping from latent to observation space.
method Formalizes simplicity through information geometry, independent of empirical data.
result Proves basic properties of the defined simplicity measure.
Study differential operators over maps and their applications in supermanifolds.
problem Understanding differential operators over smooth maps and their applications.
method Recall and study differential operators, formal ℏ-differential operators, pullbacks by thick morphisms, and quantization of symplectic micromorphisms. result Developed constructions and examples of differential operators over maps.
Study hermitian Yang-Mills connections on pullback bundles for holomorphic submersions.
problem Investigate hermitian Yang-Mills connections on pullback bundles for holomorphic submersions.
method Obtain a criterion for the existence of hermitian Yang-Mills connections on pullback bundles, using intersection numbers on the base.
result Determine conditions under which pullback bundles of stable or unstable bundles remain stable or unstable for adiabatic classes.
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.
New approach connects Finsler geometry's metric and connections.
problem Deriving Finsler geometry's metric and connections from compatibility axioms.
method Compatibility axioms between metric and Finsler connection.
result Metrical formulation of Finsler geometry for field theory.
Study the pullbacks and blowups of Lie algebroids and related structures.
problem Understanding the relationship between Lie algebroids, singular foliations, and Dirac structures under maps.
method Examine pullbacks and blowups of Lie algebroids and related structures under maps with constant rank or transversality assumptions.
result Establish the relation between the blowup of a Lie algebroid and its singular foliation.
Direct proof of cohomology isomorphism for pullback Lie algebroids.
problem Cohomology of pullback Lie algebroids under surjective submersions.
method Differential-geometric proof, leafwise cohomology vanishing, exhaustion-and-patching argument.
result Cohomology isomorphism and injectivity properties for pullback Lie algebroids.
Spectral sequence analysis for Sobolev mappings in Carnot groups.
problem Analyzing spectral sequences for Sobolev mappings in Carnot groups.
method Showed Pansu pullback induces a spectral sequence mapping.
result Pansu pullback induces a spectral sequence mapping.
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 geometric non-linear Schrodinger equation (GNLS) on the complex Grassmannian manifold M is the Hamiltonian equation for the energy functional on C(R,M) with respect to the symplectic form induced from the Kahler form on M. It has a Lax pair that is gauge equivalent to the Lax pair of the matrix non-linear Schroding…
We consider the group of sense-preserving diffeomorphisms $\Diff S^1$ of the unit circle and its central extension, the Virasoro-Bott group, with their respective horizontal distributions chosen to be Ehresmann connections with respect to a projection to the smooth universal Teichmüller space and the universal Teichmül…
This work improves manifold learning for multi-modal data.
problem Distortions and modeling errors in multi-modal data.
method Isometrizing learned Riemannian structure and balancing regularity and expressivity.
result The synergy of proposed approaches enhances manifold learning.
Exposes graded and microformal geometry, focusing on Q-manifolds.
problem Describes new geometric structures and their applications.
method Introduces Q-manifolds and microformal geometry. result Establishes connections between Q-manifolds and Lie algebras. Explains linearizing a nonlinear connection on a pullback bundle.
problem Clarifying the geometric meaning of linearized connections.
method Fiberwise linear approximation of a vector bundle connection.
result Clarifies the geometric meaning of linearized connections.
Study Liouville-type results for stationary maps related to pullback metrics.
problem Analyzing stationary maps and their properties related to pullback metrics.
method Derive first variation formula, stress-energy tensor, and monotonicity formula.
result Derive Liouville-type results and investigate boundary value problems.
Sobolev mappings preserve the Rumin complex on contact manifolds.
problem Preserving the Rumin complex under Sobolev mappings on contact manifolds.
method Using the Pullback Theorem, Pansu pullback is shown to induce chain mappings between Rumin complexes and de Rham complexes.
result The Rumin flat complex is bilipschitz invariant under Sobolev mappings between contact manifolds.
This paper improves probabilistic latent models on hyperbolic spaces.
problem Uncertainty in predictions due to geodesics crossing low-data regions.
method Augmenting hyperbolic manifold with a pullback metric for probabilistic pullback metrics.
result Geodesics on pullback metric respect both geometry and data distribution, reducing uncertainty.
PFM generates novel samples on data manifolds using pullback geometry.
problem Generating novel samples on complex data manifolds.
method Pullback Flow Matching framework leveraging pullback geometry and isometric learning.
result PFM achieves improved manifold learning and generative performance.
This paper introduces tangent display maps to simplify tangent category theory.
problem The category of smooth manifolds does not admit all pullbacks, complicating tangent category theory.
method Develops tangent display maps as a special class of maps well-behaved with respect to pullbacks.
result Tangent display maps simplify previous work in tangent categories and provide a new way to define open subobjects.
Adopting the pullback approach to global Finsler geometry, the aim of the present paper is to provide new intrinsic (coordinate-free) proofs of intrinsic versions of the existence and uniqueness theorems for the Cartan and Berwald connections on a Finsler manifold. To accomplish this, the notions of semispray and nonli…
The paper introduces new functors for cohomology groups of manifolds.
problem Behavior of cohomology groups under uniform maps.
method Introducing contravariant functors between manifold categories and vector space categories.
result Uniform homotopy invariance of cohomology groups.
We introduce mappings between spaces of functions on (super)manifolds that generalize pullbacks with respect to smooth maps but are, in general, nonlinear (actually, formal). The construction is based on canonical relations and generating functions. (The underlying structure is a formal category, which is a "thickening…
Extends symphonic maps to bi-symphonic maps between Riemannian manifolds.
problem Defining and exploring new types of maps between Riemannian manifolds.
method Introduces bi-symphonic maps by analyzing the bi-energy functional.
result New types of maps (bi-symphonic) with associated bi-energy functional.
Study of permutational wreath pullbacks and their properties.
problem Structural study of permutational wreath pullbacks and their properties.
method Systematic structural study of permutational wreath pullbacks, focusing on center, abelianization, and functorial behavior.
result Established a criterion for the abelian kernel to be characteristic and for the wreath product to inherit the R-infinity property.
Study on mappings between nonrigid Carnot groups, proving quasisymmetric rigidity.
problem Quasisymmetric homeomorphisms in nonrigid Carnot groups.
method Use pullback theorem from previous work to show reducibility and rigidity.
result Quasisymmetric homeomorphisms are reducible in nonrigid Carnot groups, except for specific cases.
Proposes a scalable framework for extracting data manifold geometry.
problem Efficiently mapping and learning data manifold geometry.
method Score-based pullback Riemannian geometry integrating pullback Riemannian geometry and generative models.
result High-quality geodesics and reliable intrinsic dimension estimation.
Symplectic forms match on circle pattern space.
problem Matching symplectic forms on circle pattern space.
method Pullback of symplectic forms to circle pattern space.
result Symplectic forms on circle pattern space coincide.
We present a family of complexes playing the same role, for homogeneous variational problems, that the horizontal parts of the variational bicomplex play for variational problems on a fibred manifold. We show that, modulo certain pullbacks, each of these complexes (apart from the first one) is globally exact. All the c…
Associated to a Thurston map f:S2→S2 with postcritical set P are several different invariants obtained via pullback: a relation on the set of free homotopy classes of curves in S2−P, a linear operator on the free R-module generated by these homotopy classes of curves, a virtual endomorphism on the pur…
New construction of Atiyah and Todd classes for Lie pair pullback dg Lie algebroids.
problem Constructing Atiyah and Todd classes for Lie pair pullback dg Lie algebroids.
method Using homological perturbation lemma and contraction, constructing isomorphisms between cochain complexes and Chevalley-Eilenberg cohomologies.
result Identifies Atiyah and Todd classes of Lie pair pullback dg Lie algebroids with those of the Lie pair.
New distances for comparing multivariate normal distributions.
problem Comparing multivariate normal distributions efficiently and accurately.
method Approximated Fisher-Rao distance and pullback SPD cone distances.
result Efficient computation of distances between normal distributions.
Proves singularities of codimension one objects under finite holomorphic maps.
problem Analyzing singularities of codimension one objects under finite holomorphic maps.
method Generalizes previous results by proving the singularity of pullbacks of singular codimension one holomorphic foliations.
result The preimage of a germ of a singular analytic hypersurface under a germ of a finite holomorphic map is again singular.
Lean 4 formalizes Stokes' theorem for smooth singular cubes.
problem Formalizing Stokes' theorem for singular cubes in arbitrary dimensions.
method Using true differential-form pullback via Frechet derivative, bridging to mathlib4's extDeriv.
result d^2=0 for singular cubical chains, chain-level Stokes extended.
Study on hermitian Yang-Mills connections on blown-up manifolds.
problem Analyzing hermitian Yang-Mills connections on blown-up Kähler manifolds.
method Investigates connections for pullback vector bundles under specific conditions.
result Provides numerical criterion for convergence of hermitian Yang-Mills connections.
Study shows essential properties of Z2n-manifolds.
problem Existence of categorical products in Z2n-manifolds. method Detailed analysis of function sheaves and morphisms.
result Existence of categorical products in Z2n-manifolds. The paper defines and analyzes n-fold vector bundles and their decompositions.
problem Understanding and decomposing n-fold vector bundles. method Introducing n-fold vector bundles as functors and studying their cores and decompositions. result Any n-fold vector bundle admits a non-canonical isomorphism to a decomposed n-fold vector bundle. To a closed Riemannian manifold, we associate a set of (special values of) a family of Dirichlet series, indexed by functions on the manifold. We study the meaning of equality of two such families of spectral Dirichlet series under pullback along a map. This allows us to give a spectral characterization of when a smoot…
We extend the category of (super)manifolds and their smooth mappings by introducing a notion of microformal or "thick" morphisms. They are formal canonical relations of a special form, constructed with the help of formal power expansions in cotangent directions. The result is a formal category so that its composition l…
We generalize the concept of sub-Riemannian geometry to infinite-dimensional manifolds modeled on convenient vector spaces. On a sub-Riemannian manifold M, the metric is defined only on a sub-bundle $\calH$ of the tangent bundle TM, called the horizontal distribution. Similarly to the finite-dimensional case, we ar…
Adjusting conventional Chern-Simons theory to G2-manifolds, one describes G2-instantons on bundles over a certain class of 7-dimensional flat tori which fiber non-trivially over T4, by a pullback argument. Moreover, if c2=0, any (generic) deformation of the G2-structure away from s…
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.
A new connection in Finsler geometry unifies various types of connections.
problem Introducing a unified connection in Finsler geometry.
method Using the pullback formalism, a new linear connection is introduced and investigated.
result The existence and uniqueness of the new connection are proved intrinsically.
Generalizes Kawai theorem for orbifold Riemann surfaces.
problem Proving a generalization of Kawai theorem for orbifold Riemann surfaces.
method Using a formula for the differential of a holomorphic map from the cotangent bundle of the Teichmüller space to the PSL(2,C)-character variety, and evaluating the pullback of Goldman symplectic form. result Generalization of Kawai theorem for orbifold Riemann surfaces and Goldman's theorem.
Article studies symmetry in smooth vector bundles using advanced operations.
problem Symmetry phenomena in smooth vector bundles after two iterations of the normal functor.
method Developed theory of pullback and quotient for double vector bundles and morphisms, focusing on naturality of the normal functor.
result Expected symmetry is obtained through universal behavior and compatibility of operations.
Study compactness and continuity in Sobolev wave front set spaces for smooth vector bundles.
problem Compactness and continuity in Sobolev wave front set spaces for smooth vector bundles.
method Introduced a locally convex topology, extended compactness theorem, studied pseudo-differential operators, and applied to microlocal defect measures.
result Extended microlocal defect measures and compensated compactness theorem to Sobolev wave front set spaces.