Proves surjectivity of certain smooth maps with non-properness sets.
problem Surjectivity of linear operators and global diffeomorphisms of semialgebraic maps.
method Analytic proof involving C∞ semialgebraic local diffeomorphisms and linear partial differential operators. result A new analytic conjecture for polynomial local diffeomorphisms of Rn implies known results. Injective and surjective neural operators for function spaces.
problem Tackles injective and surjective neural operators in function spaces.
method Combines prior work in ReLU and operator learning, uses Fredholm theory and Leray-Schauder degree theory.
result Injective and surjective neural operators are universal approximators and maintain their properties in finite-rank implementations.
In an earlier paper, we showed that the moduli space of deformations of a smooth, compact, orientable special Lagrangian submanifold L in a symplectic manifold X with a non-integrable almost complex structure is a smooth manifold of dimension H^1(L), the space of harmonic 1-forms on L. We proved this first by showing t…
We exhibit a knot P in the solid torus, representing a generator of first homology, such that for any knot K in the 3-sphere, the satellite knot with pattern P and companion K is not smoothly slice in any homology 4-ball. As a consequence, we obtain a knot in a homology 3-sphere that does not bound a piecewise-…
Smooth approximations without critical points for continuous mappings between Banach spaces.
problem Approximating continuous mappings between Banach spaces with smooth mappings.
method Constructing a C∞ mapping that approximates a continuous mapping and has surjective linear derivative. result Every continuous mapping between certain Banach spaces can be uniformly approximated by smooth open mappings.
The paper proves neural networks are almost always surjective, impacting model safety.
problem Ensuring neural networks can generate any output, including harmful content.
method Analyzing fundamental neural architectures and generative models.
result Many neural architectures are almost always surjective, allowing for arbitrary outputs.
Study surjectivity of Kirwan map for generalized hyperkähler reduction.
problem Establishing surjectivity of Kirwan map for a specific class of Hamiltonian manifolds.
method Defined a close analogue of hyperkähler reduction for manifolds with equivariant functions under semi-linear G-actions. result Surjectivity of Kirwan map proved for the defined class of manifolds.
New unitary representations for Thompson's group V and virtual links.
problem Constructing unitary representations for Thompson's group V and virtual links.
method Defining a surjection from V to virtual equivalence classes of CC links, constructing unitary representations from kei and operator quandle colorings.
result Realizes all oriented almost classical virtual links.
Study Bernstein-Gelfand-Gelfand complexes on Lipschitz domains, computing cohomology and applying to elasticity models.
problem Cohomology of BGG complexes on bounded Lipschitz domains.
method Computes cohomology of conformal deformation and Hessian complexes in Sobolev spaces, allowing multiple input complexes.
result Establishes conformal Korn inequality and proposes generalizations of continuum models with microstructures.
The study generalizes cohomology results for hyperbolic groups.
problem Understanding cohomology of hyperbolic groups and their subgroups.
method Generalizing Gersten's theorem on ℓ∞-cohomology and applying it to hyperbolic groups. result Obtained hyperbolicity criteria for specific groups.
We consider localized deformation for initial data sets of the Einstein field equations with the dominant energy condition. Deformation results with the weak inequality need to be handled delicately. We introduce a modified constraint operator to absorb the first order change of the metric in the dominant energy condit…
SurVAE Flows combine VAEs and flows using surjective transformations.
problem Combining the strengths of VAEs and flows to model complex densities.
method Modular framework of composable deterministic and stochastic transformations.
result Exact likelihood computation and lower bound on likelihood.
The paper calculates Alexander polynomials for knots using finite group representations.
problem Calculating Alexander polynomials for knots using specific group representations.
method Defined twisted Alexander polynomials associated with regular representations of finite groups.
result Several formulas for the twisted Alexander polynomial are provided.
The symplectic representation of mapping classes is not surjective for certain types of mapping classes.
problem The surjectivity of the symplectic representation of mapping classes, particularly pseudo-Anosov ones, is not always preserved.
method Explicit construction of symplectic matrices with a bi-Perron leading eigenvalue that cannot be represented by orientable pseudo-Anosov mapping classes.
result The symplectic representation of orientable pseudo-Anosov mapping classes is not surjective.
Extends Jones' construction to map Thompson group to pointed links.
problem Constructing a map from Thompson group to pointed links.
method Extended Jones' construction, introduced operations, defined new monoid.
result Surjective map from central monoid to linked monoid.
Localized deformation of scalar curvature and mean curvature on manifolds.
problem Deforming scalar curvature and mean curvature on compact manifolds with boundary.
method Proving localized surjection of scalar curvature and mean curvature map, handling non-variational linearized problem.
result Localized deformations of scalar curvature and mean curvature on compact manifolds are possible.
The paper introduces two-tone colorings for links and shows conditions for surjective dihedral representations.
problem The challenge is to find conditions for links to admit surjective dihedral representations.
method The method involves introducing two-tone colorings and providing conditions for the link groups to admit such representations.
result Any link with at least 3 components admits a surjective homomorphism to the dihedral group of arbitrary degree.
Simplified proof of K3 surface period map surjectivity.
problem Surjectivity of period map on K3 surfaces.
method Utilizes hyperkähler geometry and collapsing techniques.
result Simple proof of Todorov's result on K3 surfaces.
The Hilbert map's image is discussed, showing when it's surjective.
problem Understanding when the Hilbert map is surjective.
method Analyzing the Hilbert map's properties to determine surjectivity.
result Necessary and sufficient conditions for the Hilbert map to be surjective.
If phi: G-->G' is a surjective homomorphism, we prove that the twisted Alexander polynomial of G is divisible by the twisted Alexander polynomial of G'. As an application, we show non-existence of surjective homomorphism between certain knot groups.
New insights into X-ray transform on hyperbolic disk, with functional relations and range characterizations.
problem Understanding the X-ray transform on hyperbolic geometry.
method Derived new singular value decompositions, range characterizations, and intertwining relations with wedge-type differential operators.
result Sharp understanding of boundary behavior and invertibility settings for the X-ray transform.
Proves generic surjectivity of vector bundles via degeneration.
problem Understanding generic surjectivity of vector bundles.
method Uses degeneration argument, Berndtsson's theorem, and Lempert's proof.
result Generalizes previous work on L2 division theorem. Research characterizes traces of Sobolev mappings into manifolds, identifying analytical obstructions and proving surjectivity under certain conditions.
problem Characterizing traces of Sobolev mappings into compact manifolds.
method Analytical and topological methods, including homotopy and fundamental groups.
result Identification of an analytical obstruction when p<m is an integer and πp(N) is non-trivial, and proof of surjectivity under specific conditions. Study differential operators on non-compact harmonic manifolds, finding conditions for radial fundamental solutions and dense heat-semigroups.
problem Conditions for differential operators on non-compact harmonic manifolds to have specific properties.
method Analyzing the algebra of differential operators, their commutation properties, and using geometric averages.
result Algebra of differential operators on non-compact harmonic manifolds has specific properties related to radial fundamental solutions and dense heat-semigroups.
Period maps surjective for certain gravitational instantons.
problem Understanding the uniformization of gravitational instantons.
method Uniformization theorems by Chen--Chen, Chen--Viaclovsky, Collins--Jacob--Lin, and Hein--Sun--Viaclovsky--Zhang.
result Period maps surjective for ALH*, ALG, and ALG* gravitational instantons.
Classifies non-linear Fredholm maps linking to stable homotopy groups of spheres.
problem Classifying non-linear proper Fredholm maps between Hilbert spaces.
method Using stable homotopy groups of spheres to classify maps up to proper homotopy.
result Determines the non-trivial kernel of the map from stable homotopy groups to non-linear proper Fredholm maps.
VAEs struggle with surjective multimodal data, especially class labels describing images.
problem VAEs struggle to capture variability in surjective multimodal data.
method Theoretical and empirical demonstration of VAEs with a mixture of experts posterior.
result VAEs with a mixture of experts posterior can disregard variation in surjective multimodal data.
The paper studies symmetries in quandles and their relative versions.
problem Understanding symmetries in quandle structures and their transformations.
method Introducing relative versions of inner automorphism and transvection groups, and using them to characterize and classify surjective homomorphisms.
result Characterization of connected homomorphisms and classification of quandle structures under certain symmetry assumptions.
Study on cohomology of special linear groups over Euclidean number rings.
problem Understanding cohomology of principal congruence subgroups of special linear groups.
method Analyzing cohomology groups and using properties of Tits buildings.
result Natural map between cohomology and homology is surjective and an isomorphism under certain conditions.
Eisenhart's theorem extended to sub-Riemannian metrics on specific Lie algebras.
problem Extending Eisenhart's theorem to sub-Riemannian metrics on step 2 distributions.
method Introducing ad-surjective step 2 nilpotent Lie algebras and extending Eisenhart's theorem.
result The theorem holds for sub-Riemannian metrics on ad-surjective step 2 distributions.
New calculus solves boundary value problems for elliptic operators.
problem Boundary value problems for 0-elliptic operators.
method Developed a new calculus called symbolic 0-calculus to handle boundary value problems.
result Construct left and right parametrices for 0-elliptic operators with boundary conditions.
Let G be a compact Lie group, and let LG denote the corresponding loop group. Let (X,ω) be a weakly symplectic Banach manifold. Consider a Hamiltonian action of LG on (X,ω), and assume that the moment map $μ: X \to L\fg^*$ is proper. We consider the function ∣μ∣2:X→R, and use a version of Morse theor…
Teichmüller space rigidity proven for Thurston metric.
problem Understanding isometries in Teichmüller space with Thurston metric.
method Analyzing R-linear surjective isometries between cotangent spaces. result Every isometry between hyperbolic surfaces induces an isometry in Teichmüller space.
Study surjections between braid groups on surfaces, focusing on lower central series.
problem Determine conditions for surjections between braid groups on surfaces.
method Utilizes properties of lower central series and combinatorial methods.
result Identifies specific values of m and n for which surjections exist between braid groups.
We study isomorphism classes of symplectic dual pairs P <- S -> P-, where P is an integrable Poisson manifold, S is symplectic, and the two maps are complete, surjective Poisson submersions with connected and simply-connected fibres. For fixed P, these Morita self-equivalences of P form a group Pic(P) under a natural `…
This paper is motivated by a relatively recent work by Joyce in special Lagrangian geometry, but the basic idea of the present paper goes back to an earlier pioneering work of Donaldson in Yang--Mills gauge theory; Donaldson discovered a global structure of a (compactified) moduli space of Yang--Mills instantons, and a…
For smooth compact connected manifolds with strictly convex boundary, no conjugate points and a hyperbolic trapped set, we prove an equivalence principle concerning the injectivity of the X-ray transform Im on symmetric solenoidal tensors and the surjectivity of an operator πm∗ on the set of solenoidal tensors…
The paper proves that certain surface mapping class groups do not virtually surject to the integers.
problem Proving that certain mapping class groups do not virtually surject to the integers.
method Analyzing the structure of finite-index subgroups of mapping class groups of surfaces of genus ≥ 3.
result Finite-index subgroups of mapping class groups of surfaces of genus ≥ 3 do not virtually surject to the integers.
We use Klyachko's methods to prove that the natural map G to G-hat, where G is a torsion-free group and G-hat is obtained by adding a new generator t and a new relator w, is surjective only if w is conjugate to gt or gt^{-1} for some g in G. This solves a special case of the surjectivity problem for group extensions, r…
For Anosov flows preserving a smooth measure on a closed manifold M, we define a natural self-adjoint operator Π which maps into the space of invariant distributions in ∩u<0Hu(M) and whose kernel is made of coboundaries in ∪s>0Hs(M). We describe relations to Liv…
For a Coxeter group W we have an associating bi-linear form B on a real vector space. We assume that B has the signature (n−1,1). In this case we have the Cannon-Thurston map for W, that is, a W-equivariant continuous surjection from the Gromov boundary of W to the limit set of W. We focus on the case w…
Kirwan surjectivity proven for equivariant Dolbeault cohomology.
problem Understanding cohomology of Kähler quotients for Hamiltonian actions.
method Using Cartan-Chern-Weil theory to define and prove a Kirwan map.
result A natural surjective Kirwan map established between equivariant Dolbeault cohomologies.
We prove an analogue of Kirwan surjectivity in the setting of equivariant basic cohomology of K-contact manifolds. If the Reeb vector field induces a free S1-action, the S1-quotient is a symplectic manifold and our result reproduces Kirwan's surjectivity for these symplectic manifolds. We further prove a Tolman-W…
Surjectivity of Cannon-Thurston map proven for metric graph bundles.
problem Proving surjectivity of Cannon-Thurston map in metric graph bundles.
method Generalized Mj-Sardar's result to include more types of fibers.
result Continuous extension map between boundaries is surjective.
Any knot in a solid torus, called a pattern or satellite operator, acts on knots in the 3-sphere via the satellite construction. We introduce a generalization of satellite operators which form a group (unlike traditional satellite operators), modulo a generalization of concordance. This group has an action on the set o…
We consider a class of topological objects in the 3-sphere S3 which will be called n-punctured ball tangles. Using the Kauffman bracket at A=eiπ/4, an invariant for a special type of n-punctured ball tangles is defined. The invariant Fn takes values in PM2×2n(Z), that is the set of $2…
P. Buser and P. Sarnak showed in 1994 that the maximum, over the moduli space of Riemann surfaces of genus s, of the least conformal length of a nonseparating loop, is logarithmic in s. We present an application of (polynomially) dense Euclidean packings, to estimates for an analogous 2-dimensional conformal systolic i…
We study the group of symplectic birational transformations of the plane. It is proved that this group is generated by SL(2,Z), the torus and a special map of order 5, as it was conjectured by A. Usnich. Then we consider a special subgroup H, of finite type, defined over any field which admits a…