arXiv research

Generalizes Bestvina's $\mathcal{Z}$-boundaries to coarse $\mathcal{Z}$-boundaries.

Constructs hyper-Kähler models using Riemann-Hilbert problems.

The paper connects Bergman geometry with information geometry.

We extend Thurston's metric to projective filling currents, embedding Teichmüller space into the larger space.

A new approach to Riemannian geometry using embedded and submersion structures.

Study geometric and representation theory of statistical transformation models.

Study local invariants and geometry of sub-Laplacian on H-type foliations.

Researchers mapped the moduli space of a specific group in 3D complex hyperbolic geometry.

Symplectic coordinates found on projective structures on orbifolds.

High-dimensional models trained on smooth manifolds achieve optimal rates in Wasserstein metrics.

We introduce a category of rigid geometries on singular spaces which are leaf spaces of foliations and are considered as leaf manifolds. We single out a special category $\mathfrak F_0$ of leaf manifolds containing the orbifold category as a full subcategory. Objects of $\mathfrak F_0$ may have non-Hausdorff topology u…

The study constructs optimal tori on Fano manifolds and confirms mirror symmetry.

Construct spectral triples on C*-algebras with group actions.

Motivated by generalized geometry (à la Hitchin), we discuss the integrability conditions for four natural almost complex structures on the product bundle ${\mathcal Z}\times {\mathcal Z}\to M$, where ${\mathcal Z}$ is the twistor space of a Riemannian 4-manifold $M$ endowed with a metric connection $D$ with skew-symme…

We study the geometry of multidimensional scalar $2^{nd}$ order PDEs (i.e. PDEs with $n$ independent variables) with one unknown function, viewed as hypersurfaces $\mathcal{E}$ in the Lagrangian Grassmann bundle $M^{(1)}$ over a $(2n+1)$-dimensional contact manifold $(M,\mathcal{C})$. We develop the theory of character…

We consider two Riemannian geometries for the manifold $\mathcal{M}(p,m\times n)$ of all $m\times n$ matrices of rank $p$. The geometries are induced on $\mathcal{M}(p,m\times n)$ by viewing it as the base manifold of the submersion $π:(M,N)\mapsto MN^T$, selecting an adequate Riemannian metric on the total space, and …

The paper explores isomorphisms on isoparametric hypersurfaces in spheres, leading to new geometric structures.

Study on harmonic Higgs bundles with vanishing endormorphism and eigenvalues of Q.

The present paper is a continuation of Le Anh Vu's ones [13], [14], [15]. Specifically, the paper is concerned with the subclass of connected and simply connected MD5-groups such that their MD5-algebras $\mathcal{G}$ have the derived ideal ${\mathcal{G}}^{1} : = [ \mathcal{G},\mathcal{G} ]\equiv$ ${\bf{R}}^{3}$. We sha…

We develop various properties of symmetric generalized complex structures (in connection with their holomorphic space and B-field transformations), which are analogous to the well-known results of Gualtieri on skew-symmetric generalized complex structures. Given a symmetric or skew-symmetric generalized complex structu…

This paper explains a 2-form on moduli spaces using quasi-Hamiltonian and Dirac geometry.

The paper studies gradients of geodesic-length functions and systoles on Teichmüller spaces.

Jordan algebras in information geometry linked to metrics on probability distributions.

The paper explores anomalous subvarieties in hyperbolic 3-manifolds and their geometric implications.

The paper studies formal geometry of dg manifolds and proves isomorphism of their calculi.

The purpose of this paper is to study $\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifolds of a paracosymplectic manifold $\widetilde{M}$. We prove that the distributions associated with the definition of $\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifold $M$ are always integrable. A nece…

New spectral sequence for $\mathcal{K}$-manifolds, computing cohomology and harmonic forms.

We analyze the asymptotic cones of Teichmüller space with the Teichmüller metric, $(\mathcal{T}(S),d_T)$. We give a new proof of a theorem of Eskin-Masur-Rafi which bounds the dimension of quasiisometrically embedded flats in $(\mathcal{T}(S),d_T)$. Our approach is an application of the ideas of Behrstock and Behrstock…

We propose that a simple, Lagrangian 2d $\mathcal{N}=(0, 2)$ duality interface between the 3d $\mathcal{N}=2$ XYZ model and 3d $\mathcal{N}=2$ SQED can be associated to the simplest triangulated 4-manifold: the 4-simplex. We then begin to flesh out a dictionary between more general triangulated 4-manifolds with boundar…

Study reveals limits of detecting local geometry in random graphs.

Let $T$ be a circle and $LT$ be its loop group. Let $\mathcal{M}$ be an infinite dimensional manifold equipped with a nice $LT$-action. We construct an analytic $LT$-equivariant index for $\mathcal{M}$, and justify it in terms of noncommutative geometry. More precisely, we construct a Hilbert space $\mathcal{H}$ consis…

Suppose $(X,ω)$ is a compact Kähler manifold. We introduce and explore the metric geometry of the $L^{p,q}$-Calabi Finsler structure on the space of Kähler metrics $\mathcal H$. After noticing that the $L^{p,q}$-Calabi and $L^{p'}$-Mabuchi path length topologies on $\mathcal H$ do not typically dominate each other, we …

The paper explores cone structures and their connections to parabolic geometries in complex manifolds.

New spinorial field equation reveals geometric properties of Sasaki manifolds.

Study on a specific obstruction in four-dimensional geometry.

Constructs graded jet bundles for Z-graded manifolds and vector bundles.

We describe a bicategory $(\mathcal{R}ed\,\mathcal{O}rb)$ of reduced orbifolds in the framework of classical differential geometry (i.e. without any explicit reference to notions of Lie groupoids or differentiable stacks, but only using orbifold atlases, local lifts and changes of charts). In order to construct such a …

Let $\mathcal M_g$ denote the moduli space of compact Riemann surfaces of genus $g$ and let $\mathcal A_g$ be the space of principally polarized abelian varieties of (complex) dimension $g$. Let $J:\mathcal M_g\longrightarrow \mathcal A_g$ be the map which associates to a Riemann surface its Jacobian. The map $J$ is in…

The Weyl tube theorem is extended to Kähler manifolds.

New invariant defined for Weinstein domains, related to Kirby-Thompson's invariant.

Let $\mathcal M= (M,\mathcal O_\mathcal M)$ be a smooth supermanifold with connection $\nabla$ and Batchelor model $\mathcal O_\mathcal M\congΓ_{ΛE^\ast}$. From $(\mathcal M,\nabla)$ we construct a connection on the total space of the vector bundle $E\to{M}$. This reduction of $\nabla$ is well-defined independently of …

Let $\{{\cdot},{\cdot}\}_{\boldsymbol{\mathcal{P}}}$ be a variational Poisson bracket in a field model on an affine bundle $π$ over an affine base manifold $M^m$. Denote by $\times$ the commutative associative multiplication in the Poisson algebra $\boldsymbol{\mathcal{A}}$ of local functionals $Γ(π)\to\Bbbk$ that take…

In 1931 Elie Cartan constructed a geometry which was rarely considered. Cartan proposed a way to define an infinitesimal metric $ds$ starting from a variational problem on hypersurfaces in an $n$-dimensional manifold $\mathcal{M}$. This distance depends not only of the point $\textsc{m}\in\mathcal{M}$ but on the orient…

Suppose $S$ is a closed, oriented surface of genus at least two. This paper investigates the geometry of the homology multicurve complex, $\mathcal{HC}(S,α)$, of $S$; a complex closely related to complexes studied by Bestvina-Bux-Margalit and Hatcher. A path in $\mathcal{HC}(S,α)$ corresponds to a homotopy class of imm…

The paper studies the geometry and topology of a specific foliation on a complex surface.

Extended Regge complex for linearized Riemann-Cartan geometry and cohomology.

In category theory, logic and geometry cooperate with each other producing what is known under the name Synthetic Differential Geometry (SDG). The main difference between SDG and standard differential geometry is that the intuitionistic logic of SDG enforces the existence of infinitesimal objects which essentially modi…

Develops an L^p theory for Dolbeault-Dirac operators on compact Kähler manifolds.