arXiv research

New hyperbolic groups found with specific subgroup properties.

Researchers classify and characterize Bäcklund transformations for hyperbolic Monge-Ampère systems.

The paper refines 2-factor homology to a stable homotopy type for planar trivalent graphs with perfect matchings.

Natural coordinates for SL3-webs on surfaces are shown to be consistent under triangulation changes.

We study finite type invariants of nullhomologous knots in a closed 3-manifold $M$ defined in terms of certain descending filtration $\{\mathscr{K}_n(M)\}_{n\geq 0}$ of the vector space $\mathscr{K}(M)$ spanned by isotopy classes of nullhomologous knots in $M$. The filtration $\{\mathscr{K}_n(M)\}_{n \geq 0}$ is define…

Unified treatment of two extension problems using heat equation in Heisenberg group.

This paper improves the approximation of machine learning models by transforming them to better fit locally $p$-integrable functions.

The Liouville theorem and $C^α$-estimate for Calabi-Yau cones establish uniqueness and asymptotic behavior of metrics.

Cluster algebras match for specific Lie algebras and surfaces.

Defined a new graph type for compact surfaces, proving its connectedness and infinite diameter.

Let $M$ be a connected open Riemann surface. We prove that the space $\mathscr L(M,\mathbb C^{2n+1})$ of all holomorphic Legendrian immersions of $M$ into $\mathbb C^{2n+1}$, $n\geq 1$, endowed with the standard holomorphic contact structure, is weakly homotopy equivalent to the space $\mathscr C(M,\mathbb S^{4n-1})$ o…

In this work we prove a Baum-Bott type formula for non-compact complex manifold of the form $\tilde{X}=X- \mathcal{D}$, where $X$ is a complex compact manifold and $\mathcal{D}$ is a normal crossing divisor on $X$. As applications, we provide a Poincaré-Hopf type Theorem and an optimal description for a smooth hypersur…

Establishes scattering theory for de Sitter vacuum solutions in even dimensions.

The paper studies the geometry of $G_2$-moduli spaces and their embeddings.

The paper develops a Feynman-Kac formula for perturbations of order ≤ 1 in noncommutative geometry.

Let $Ω$ be a smooth compact oriented 3-dimensional Riemannian manifold with boundary. A quaternion field is a pair $q=\{α,u\}$ of a function $α$ and a vector field $u$ on $Ω$. A field $q$ is {\it harmonic} if $α, u$ are continuous in $Ω$ and $\nablaα={\rm rot\,}u,\,{\rm div\,}u=0$ holds into $Ω$. The space ${\mathscr Q…

Study complex hyperbolic lattices and their subgroups, proving new finiteness properties.

Constructs currents representing Baum-Bott residues for foliations.

The paper establishes Pogorelov type estimates for solutions to Hessian quotient equations in hyperbolic space.

We study the Ozsváth-Szabó-Thurston transverse invariant in combinatorial link Floer homology for certain transverse cables $\mathscr{L}_{p,q}$ of transverse link $L$ in $S^3$. Transverse cables $\mathscr{L}_{p,q}$ are constructed from the grid diagram of $L$. The main result is $\hatθ(\mathscr{L}_{p,q})=0$ if and only…

New hyperbolic groups exhibit unusual finiteness properties.

Let $M$ be a smooth closed orientable surface and $F=F_{p,q,r}$ be the space of Morse functions on $M$ having exactly $p$ critical points of local minima, $q\ge1$ saddle critical points, and $r$ critical points of local maxima, moreover all the points are fixed. Let $F_f$ be the connected component of a function $f\in …

Optimizes quickest change detection with bounded means under ARL constraint.

Study of J-Hermitian matrices and geometric mean definition.

The paper studies a semigroup generated by finite intervals and characterizes its properties.

New quantum knot invariants derived from Verma modules.

Classifies algebraic concordance for almost classical knots.

We propose a general strategy to derive null-homotopy operators for differential complexes based on the Bernstein-Gelfand-Gelfand (BGG) construction and properties of the de Rham complex. Focusing on the elasticity complex, we derive path integral operators $\mathscr{P}$ for elasticity satisfying $\mathscr{D}\mathscr{P…

Constructs diffeological moduli stacks for Higgs and flat bundles on Kähler manifolds

Compact foliations preserve entropy if leaves are strictly convex projective.

Study K-theory of Etesi $C^*$-algebras to understand smooth manifolds.

Let $Y\subset{\mathbb R}^n$ be a triangulable set and let $r$ be either a positive integer or $r=\infty$. We say that $Y$ is a $\mathscr{C}^r$-approximation target space, or a $\mathscr{C}^r\text{-}\mathtt{ats}$ for short, if it has the following universal approximation property: For each $m\in{\mathbb N}$ and each loc…

We establish the following Hadamard--Stoker type theorem: Let $f:M^n\rightarrow\mathscr{H}^n\times\mathbb R$ be a complete connected hypersurface with positive definite second fundamental form, where $\mathscr H^n$ is a Hadamard manifold. If the height function of $f$ has a critical point, then it is an embedding and $…

The paper finds the optimal wealth growth rate in betting games.

The paper improves the smoothness of vector fields on manifolds.

The paper explores the geometry and dynamics of free splitting and free factor complexes for groups.

This paper shows how to construct sequential tests with power one against weakly compact sets in Polish spaces.

A new DL framework preserves geometric structures for causal predictions.

Double vector bundles may be dualized in two distinct ways and these duals are themselves dual. These two dualizations generate a group, denoted $\mathscr{D}\mathscr{F}_2$, which is the symmetric group $S_3$ on three symbols. In the case of triple vector bundles the authors proved in a previous paper that the correspon…

New $3$-manifolds created from $4$-regular graphs with unique Eulerian cycles.

This paper extends Euclidean theorems to anisotropic settings for varifolds.

We apply Lescop's construction of $\mathbb{Z}$-equivariant perturbative invariant of knots and 3-manifolds to the explicit equivariant propagator of "AL-paths" given in arXiv:1403.8030. We obtain an invariant $\hat{Z}_n$ of certain equivalence classes of fiberwise Morse functions on a 3-manifold fibered over $S^1$, whi…

We consider a class $\mathscr{X}$ of continuous functions on $[0,1]$ that is of interest from two different perspectives. First, it is closely related to sets of functions that have been studied as generalizations of the Takagi function. Second, each function in $\mathscr{X}$ admits a linear pathwise quadratic variatio…

In this paper, we introduce a new energy density function $\mathscr Y$ on the projective bundle $\mathbb{P}(T_M)\>M$ for a smooth map $f:(M,h)\>(N,g)$ between Riemannian manifolds $\mathscr Y=g_{ij}f^i_αf^j_β\frac{W^αW^β}{\sum h_{γδ} W^γW^δ}.$ We get new Hessian estimates to this energy density and obtain various new…

The paper constructs bases for cluster varieties using ${ m SL}_3$-webs and laminations.

Classifies embeddings of surfaces in 3D space with product structure.

Operads help quantify polygon spaces, proving dimensions equal.

In response to a 1997 problem of M. Vidyasagar, we state a criterion for PAC learnability of a concept class $\mathscr C$ under the family of all non-atomic (diffuse) measures on the domain $Ω$. The uniform Glivenko--Cantelli property with respect to non-atomic measures is no longer a necessary condition, and consisten…