arXiv research

In this article we introduce conformal Riemannian morphisms. The idea of conformal Riemannian morphism generalizes the notions of an isometric immersion, a Riemannian submersion, an isometry, a Riemannian map and a conformal Riemannian map. We show that every injective conformal Riemannian morphism is an injective conf…

We use assembly maps to study $\mathbf{TC}(\mathbb{A}[G];p)$, the topological cyclic homology at a prime $p$ of the group algebra of a discrete group $G$ with coefficients in a connective ring spectrum $\mathbb{A}$. For any finite group, we prove that the assembly map for the family of cyclic subgroups is an isomorphis…

Injective and surjective neural operators for function spaces.

Proves surjectivity of certain smooth maps with non-properness sets.

The paper studies cohomology on incomplete manifolds and stratified spaces.

Study big mapping class groups and their co-Hopfian property, finding new examples and proving injective homomorphisms results.

In this paper, the first of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs is an expander and the girth of the graphs tends to infinity, then the coarse Baum-Connes assembly map is injective, but not surjective, for the associated metric space $X$. Exp…

In this paper, we consider a natural map from the Kahler cone to the balanced cone of a Kahler manifold. We study its injectivity and surjecticity. We also give an analytic characterization theorem on a nef class being Kahler.

We study a model situation in which direct limit ($\text{colim}$) and inverse limit ($\lim$) do not commute, and offer some computations of their "commutator". The homology of a separable metrizable space $X$ has two well-known approximants: $qH_n(X)$ ("Čech homology") and $pH_n(X)$ ("Čech homology with compact support…

Researchers study metrics with maximal Ricci curvature on homogeneous spaces.

This paper generalizes monodromy maps for projective structures with poles.

For Anosov flows preserving a smooth measure on a closed manifold $\mathcal{M}$, we define a natural self-adjoint operator $Π$ which maps into the space of invariant distributions in $\cap_{u<0} H^{u}(\mathcal{M})$ and whose kernel is made of coboundaries in $\cup_{s>0} H^{s}(\mathcal{M})$. We describe relations to Liv…

The symplectic representation of mapping classes is not surjective for certain types of mapping classes.

The Hilbert map's image is discussed, showing when it's surjective.

Simplified proof of K3 surface period map surjectivity.

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 $I_m$ on symmetric solenoidal tensors and the surjectivity of an operator ${π_m}_*$ on the set of solenoidal tensors…

The paper studies Morse theory for Lie algebra actions on Riemannian foliations.

Study of section conjecture analogues over complex numbers and Kodaira fibrations.

S. Satoh has defined a construction to obtain a ribbon torus knot given a welded knot. This construction is known to be surjective. We show that it is not injective. Using the invariant of the peripheral structure, it is possible to provide a restriction on this failure of injectivity. In particular we also provide an …

In this paper we consider the Gaussian thermostat ray transform on both closed Riemannian surfaces and compact Riemannian surfaces with boundary. We establish certain results on the injectivity of the thermostat ray transform and the surjectivity of its adjoint.

This paper is a systematic approach to the construction of coronas (i.e. Higson dominated boundaries at infinity) of combable spaces. We introduce three additional properties for combings: properness, coherence and expandingness. Properness is the condition under which our construction of the corona works. Under the as…

Period maps surjective for certain gravitational instantons.

Consider the Hamiltonian action of a torus on a compact twisted generalized complex manifold $M$. We first observe that Kirwan injectivity and surjectivity hold for ordinary equivariant cohomology in this setting. Then we prove that these two results hold for the twisted equivariant cohomology as well.

Study surjectivity of Kirwan map for generalized hyperkähler reduction.

The paper proves that certain surface mapping class groups do not virtually surject to the integers.

New mappings in Minkowski spacetime classified under mild 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 $|μ|^2: X \to \R$, and use a version of Morse theor…

Maps between classifying spaces for certain groups are studied, with rational cohomology results.

This article considers inverse problems on closed Riemannian surfaces whose geodesic flow is Anosov. We prove spectral rigidity for any Anosov surface and injectivity of the geodesic ray transform on solenoidal 2-tensors. We also establish surjectivity results for the adjoint of the geodesic ray transform on solenoidal…

Surjectivity of Cannon-Thurston map proven for metric graph bundles.

Let $f:M^m\to N^n$ be a smooth map between two differential manifolds with $N$ connected, $f(M)$ closed and $f(M)\neq N$. In this short note, we show that either all the points of $M$ are critical points of $f$ or the dimension the collection of all critical points of $f$ is not less than $n-1$. Some consequences of th…

Classifies $π_1$-injective maps between non-compact surfaces.

For a Liouville domain $W$ satisfying $c_1(W)=0$, we propose in this note two versions of symplectic Tate homology $\underrightarrow{H}\underleftarrow{T}(W)$ and $\underleftarrow{H}\underrightarrow{T}(W)$ which are related by a canonical map $κ\colon \underrightarrow{H}\underleftarrow{T}(W) \to \underleftarrow{H}\under…

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 $S^1$-action, the $S^1$-quotient is a symplectic manifold and our result reproduces Kirwan's surjectivity for these symplectic manifolds. We further prove a Tolman-W…

For a complex polynomial in two variables we study the morphism induced in homology by the embedding of an irregular fiber in a regular neighborhood of it. We give necessary and sufficient conditions for this morphism to be injective, surjective. Particularly this morphism is an isomorphism if and only if the correspon…

We study spaces with a cuspidal (or horn-like) singularity embedded in a smooth Riemannian manifold and analyze the geodesics in these spaces which start at the singularity. This provides a basis for understanding the intrinsic geometry of such spaces near the singularity. We show that these geodesics combine to natura…

The paper proves neural networks are almost always surjective, impacting model safety.

Proves polynomial injectivity of Fubini-Study map for ample line bundles.

Assume (M,g,Ω) is a closed, oriented Riemannian surface equipped with an Anosov magnetic flow. We establish certain results on the surjectivity of the adjoint of the magnetic ray transform, and use these to prove the injectivity of the magnetic ray transform on sums of tensors of degree at most two. In the final sectio…

Let G be a finite group. To every smooth G-action on a compact, connected and oriented surface we can associate its data of singular orbits. The set of such data becomes an Abelian group B_G under the G-equivariant connected sum. We will show that the map which sends G to B_G is functorial and carries many features of …

The paper studies symmetries in quandles and their relative versions.

Paper proves DN map determination for simple surfaces with low regularity metrics.

This paper proves a map from flow-spines to contact structures is surjective.

The period map for 4-manifolds is dense and surjective under certain conditions.

The exponential map fails to be injective near critical points in sub-Riemannian geometry.

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…

New rigidity theorem on static manifolds with boundary.

A cone spherical metric is called irreducible if any developing map of the metric does not have monodromy in ${\rm U(1)}$. By using the theory of indigenous bundles, we construct on a compact Riemann surface $X$ of genus $g_X \geq 1$ a canonical surjective map from the moduli space of stable extensions of two line bund…