arXiv research

A dense amalgam connects boundaries of groups split by finite subgroups.

Sharp boundaries for detecting dense subhypergraphs established.

We introduce and study the operation, called dense amalgam, which to any tuple X_1,...,X_k of non-empty compact metric spaces associates some disconnected perfect compact metric space, denoted $\widetilde\sqcup(X_1,...,X_k)$, in which there are many appropriately distributed copies of the spaces X_1,...,X_k. We then sh…

Detects dense subhypergraphs in heterogeneous random hypergraphs.

In this paper, we study dense subsets of boundaries of CAT(0) groups. Suppose that a group $G$ acts geometrically on a CAT(0) space $X$ and suppose that there exists an element $g_0\in G$ such that (1) $Z_{g_0}$ is finite, (2) $X\setminus F_{g_0}$ is not connnected, and (3) each component of $X\setminus F_{g_0}$ is con…

Maximal cusps are not dense on Teichmüller space for infinite-type surfaces.

Let M be a compact, orientable, mean convex 3-manifold with boundary. We show that the set of all simple closed curves in the boundary of M which bound unique area minimizing disks in M is dense in the space of simple closed curves in the boundary of M which are nullhomotopic in M. We also show that the set of all simp…

We show that the topological groups $Diff_{+}^{1}(I)$ and $Diff_{+}^{1}(\mathbb{S}^1)$ of orientation-preserving $C^1$-diffeomorphisms of the interval and the circle, respectively, admit finitely generated dense subgroups. We also investigate the question of genericity (in the sense of Baire category) of such finite to…

In this paper we construct a properly embedded holomorphic disc in the unit ball $\mathbb{B}^2$ of $\mathbb{C}^2$ having a surprising combination of properties: on the one hand, it has finite area and hence is the zero set of a bounded holomorphic function on $\mathbb{B}^2$; on the other hand, its boundary curve is eve…

Narasimhan and Ramadas showed that the restricted holonomy group of the Coulomb connection is dense in the connected component of the identity of the gauge group when one considers the product principal bundle $S^3 \times SU(2)\to S^3$. Instead of a base manifold S^3, we consider here a base manifold of dimension $n\ge…

Integral points are potentially dense in character varieties of quasi-projective varieties.

Generic density of equivariant min-max hypersurfaces in Riemannian manifolds.

Narasimhan and Ramadas showed that the Gribov ambiguity was maximal for the product SU(2) bundle over S^3. Specifically they showed that the holonomy group of the Coulomb connection is dense in the connected component of the identity of the gauge group. Instead of base manifold S^3, we consider here a base manifold wit…

We describe, under some additional technical assumptions, the Gromov boundary of the free product of several $G_i$'s amalgamated wrt. $H$, where $G_i$ are hyperbolic groups with boundary homeomorphic to a densely punctured $n$-sphere, and $H$ is their common subgroup corresponding to a peripheral sphere in each of the …

Study on horospheres in higher rank homogeneous spaces, proving density properties.

We begin by showing that commensurators of Zariski dense subgroups of isometry groups of symmetric spaces of non-compact type are discrete provided that the limit set on the Furstenberg boundary is not invariant under the action of a (virtual) simple factor. In particular for rank one or simple Lie groups, Zariski dens…

We define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type we show that there cannot be Zariski-dense maximal representations, and whenever the existenc…

Researchers describe the Gromov boundary of a graph related to surfaces.

In this paper, based on research on rank-one isometries by W.Ballmann and M.Brin and recent research on rank-one isometries of Coxeter groups by P.Caprace and K.Fujiwara, we study a topological fractal structure of boundaries of Coxeter groups. We also show that the limit-point set is dense in a boundary of a Coxeter g…

The paper explores the twisted Rokhlin property in mapping class groups of surfaces.

Image segmentation is widely used in a variety of computer vision tasks, such as object localization and recognition, boundary detection, and medical imaging. This thesis proposes deep learning architectures to improve automatic object localization and boundary delineation for salient object segmentation in natural ima…

In 1981 Masur proved the existence of a dense geodesic in the moduli space for a Teichmüller space. We prove an analogue theorem for reduced Outer Space endowed with the Lipschitz metric. We also prove two results possibly of independent interest: we show Brun's unordered algorithm weakly converges and from this prove …

In this paper, we show that the boundary $\partialΣ(W,S)$ of a right-angled Coxeter system $(W,S)$ is minimal if and only if $W_{\tilde{S}}$ is irreducible, where $W_{\tilde{S}}$ is the minimum parabolic subgroup of finite index in $W$. We also provide several applications and remarks. In particular, we obtain that for…

We study the TQFT mapping class group representations for surfaces with boundary associated with the $SU(2)$ gauge group, or equivalently the quantum group $U_q(\Sl(2))$. We show that at a prime root of unity, these representations are all irreducible. We also examine braid group representations for transcendental valu…

The square-peg problem is solved using configuration spaces and multijet transversality.

In this paper, we give several results on area minimizing surfaces in strictly mean convex 3-manifolds. First, we study the genus of absolutely area minimizing surfaces in a compact, orientable, strictly mean convex 3-manifold M bounded by a simple closed curve in the boundary of M. Our main result is that for any g>=0…

New static vacuum metrics confirmed for near Euclidean boundary data.

Study on well-posedness of vacuum Einstein equations with specific boundary conditions.

In the curve complex for a surface, a handlebody set is the set of loops that bound properly embedded disks in a given handlebody bounded by the surface. A boundary set is the set of non-separating loops in the curve complex that bound two-sided, properly embedded surfaces. For a Heegaard splitting, the distance betwee…

The study shows that certain metrics on spheres prevent stable tangent cones for area-minimizing boundaries.

The paper studies elliptic operators on manifolds with boundary.

For any smooth Riemannian metric on an $(n+1)$-dimensional compact manifold with boundary $(M,\partial M)$ where $3\leq (n+1)\leq 7$, we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min-max theory in the Almgren-Pitts setting. We apply our Morse index estimates t…

We study the boundary rigidity problem for compact Riemannian manifolds with boundary $(M,g)$: is the Riemannian metric $g$ uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function $ρ_g(x,y)$ known for all boundary points $x$ and $y$? We prove in this paper global uniqueness …

We study the $\mathrm{SL}_2(\mathbb{R})$-action on the moduli space of (triangulable) dilation tori with one boundary component. We prove that every orbit is either closed or dense, and that every orbit of the Teichmuller flow escapes to infinity.

Extends static vacuum metrics with specific boundary conditions.

Geodesic currents in strongly hyperbolic spaces are dense.

Study subgroups preserving proper domains in flag manifolds.

Study on magnetic field and potential systems to prove rigidity results.

Study on moduli spaces of Seiberg-Witten equations on manifolds with boundary.

The study finds discrete subgroups with full limit sets in higher rank Lie groups.

We study regularity properties of solutions to the Dirichlet problem for the complex Homogeneous Monge-Ampère equation. We show that for certain boundary data on $\mathbb P^1$ the solution $Φ$ to this Dirichlet problem is connected via a Legendre transform to an associated flow in the complex plane called the Hele-Shaw…

Bounds on conformal dimension for certain Coxeter group boundaries.

Study Dirichlet-to-Neumann maps on manifolds, focusing on covering and total spaces.

We announce a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle $(X,\F)$ with boundary; in particular, we define a Godbillon-Vey eta invariant on the boundary foliation, that is, a secondary invariant for longitudinal Dirac operators on type III foliations. Our theorem generalizes the cl…

Dense Associative Memories outperform classical networks in robustness and signal processing.

Here, a natural extension of Sobolev spaces is defined for a Finsler structure $F$ and it is shown that the set of all real $C^{\infty}$ functions with compact support on a forward geodesically complete Finsler manifold $(M, F)$, is dense in the extended Sobolev space $H_1^p (M)$. As a consequence, the weak solutions $…

We show that a small neighborhood of a closed symplectic submanifold in a geometrically bounded aspherical symplectic manifold has non-vanishing symplectic homology. As a consequence, we establish the existence of contractible closed characteristics on any thickening of the boundary of the neighborhood. When applied to…

Study shows singularity of stationary measure on Furstenberg boundary for certain random walks.