arXiv research

Representing 3D shape deformations by linear models in high-dimensional space has many applications in computer vision and medical imaging, such as shape-based interpolation or segmentation. Commonly, using Principal Components Analysis a low-dimensional (affine) subspace of the high-dimensional shape space is determin…

The aim of this paper is to establish an infinite dimensional generalization of the Schlesinger system -- a system of PDE's describing isomonodromic deformations of Fuchsian systems. This universal Schlesinger system first appeared in a paper by Korotkin and Samtleben in the finite dimensional case (i.e. when it reduce…

The aim of this paper is to give a new link between integrable systems and minimal surface theory. The dressing operation uses the associated family of flat connections of a harmonic map to construct new harmonic maps. Since a minimal surface in 3-space is a Willmore surface, its conformal Gauss map is harmonic and a d…

This paper constructs new Einstein metrics from old ones using specific deformation factors.

Study new involutivity theorems for Poisson quasi-Nijenhuis manifolds.

New methods for $Z$-transform inversion and Wiener-Hopf factorization.

We use Higgs bundles to answer the following question: When can a maximal Sp(4,R)-representation of a surface group be deformed to a representation which factors through a proper reductive subgroup of Sp(4,R)?

Factorization of DE coefficients is violated in antiparallel triple pretzels, but described elegantly.

Develops tools to construct Einstein 4-manifolds from conformal foliations.

This research classifies singular foliations and finds a universal deformation.

One of the key challenges of visual perception is to extract abstract models of 3D objects and object categories from visual measurements, which are affected by complex nuisance factors such as viewpoint, occlusion, motion, and deformations. Starting from the recent idea of viewpoint factorization, we propose a new app…

Study of symmetries in deformed q-map spaces reveals a complex group structure.

This paper extends previous work on genus two fibrations by studying and resolving singular fibers.

We study a notion of deformation for simplicial trees with group actions (G-trees). Here G is a fixed, arbitrary group. Two G-trees are related by a deformation if there is a finite sequence of collapse and expansion moves joining them. We show that this relation on the set of G-trees has several characterizations, in …

Given an ideal triangulation of a connected 3-manifold with non-empty boundary consisting of a disjoint union of tori, a point of the deformation variety is an assignment of complex numbers to the dihedral angles of the tetrahedra subject to Thurston's gluing equations. From this, one can recover a representation of th…

Learning automatically the structure of object categories remains an important open problem in computer vision. In this paper, we propose a novel unsupervised approach that can discover and learn landmarks in object categories, thus characterizing their structure. Our approach is based on factorizing image deformations…

The paper shows deep connections between exotic smoothings of small R^4, noncommutative algebras of foliations and quantization. At first, based on the close relation of foliations and noncommutative C*-algebras we show that cyclic cohomology invariants characterize some small exotic R^4. Certain exotic smooth R^4's de…

In this paper we prove certain Hurwitz equivalence properties in $B_n$. Our main result is that every two Artin's factorizations of $Δ_n ^2$ of the form $H_{i_1} ... H_{i_{n(n-1)}}, \quad F_{j_1} ... F_{j_{n(n-1)}}$ (with $i_k, j_k \in \{1,...,n-1 \}$), where $\{H_1,...,H_{n-1} \}, \{F_1,...,F_{n-1} \}$ are frames, are…

The Reshetikhin-Turaev sl(N) polynomial of links colored by wedge powers of the defining representation has been categorified via several different approaches. Here, we give a concise introduction to the categorification using matrix factorizations, which is a direct generalization of the Khovanov-Rozansky homology. Fu…

For a finitely generated group $G$, we introduce an asymmetric pseudometric on projectivized deformation spaces of $G$-trees, using stretching factors of $G$-equivariant Lipschitz maps, that generalizes the Lipschitz metric on Outer space and is an analogue of the Thurston metric on Teichmüller space. We show that in t…

In this paper we prove certain Hurwitz equivalence properties in the braid group. Our main result is that every two factorizations of $Δ_n ^2$ where the elements of the factorization are semi-frame are Hurwitz equivalent. The results of this paper are generalization of the results in \cite{B4}. We use a new presentatio…

For a perfect Lie algebra $\mathfrak{h}$ we classify all Lie algebras containing $\mathfrak{h}$ as a subalgebra of codimension $1$. The automorphism groups of such Lie algebras are fully determined as subgroups of the semidirect product $\mathfrak{h} \ltimes (k^* \times {\rm Aut}_{\rm Lie} (\mathfrak{h}))$. In the non-…

New invariants found for mappings between non-symmetric affine spaces.

New result on critical points of Bethe free energy under deformation retracts.

In \cite{Luo0}, Feng Luo conjectured that the discrete Yamabe flow will converge to the constant curvature PL-metric after finite number of surgeries on the triangulation. In this paper, we prove that the flow can always be extended (without surgeries) to a solution that converges exponentially fast to the constant cur…

Simplified KR polynomial for bipartite links reduces to tensor products of vector spaces.

Unsupervised mesh disentanglement separates identity and pose.

We consider invariant Riemannian metrics on compact homogeneous spaces G/H where an intermediate subgroup K between G and H exists, so that the homogeneous space G/H is the total space of a Riemannian submersion. We study the question as to whether enlarging the fibers of the submersion by a constant scaling factor ret…

Fast method developed for pricing barrier options and joint Lévy process distributions.

Study singular fibers in genus 2 algebraic fibrations and their monodromy factorizations.

In this paper we investigate the curvature of conformal deformations by noncommutative Weyl factors of a flat metric on a noncommutative 2-torus, by analyzing in the framework of spectral triples functionals associated to perturbed Dolbeault operators. The analogue of Gaussian curvature turns out to be a sum of two fun…

We give several equivalent characterizations of orthogonal subbundles of the generalized tangent bundle defined, up to B-field transform, by almost product and local product structures. We also introduce a pure spinor formalism for generalized CRF-structure and investigate the resulting decomposition of the de Rham ope…

We show the existence of a smooth solution for the flow deformed by the square root of the scalar curvature multiplied by a positive anisotropic factor $ψ$ given a strictly convex initial hypersurface in Euclidean space suitably pinched. We also prove the convergence of rescaled surfaces to a smooth limit manifold whic…

The paper defines a new metric and studies proper biharmonic maps on tangent bundles.

The paper studies Jacobi fields and conjugate points in projective sprays.

We give an integral representaion of the zeta-reguralized determinant of Laplacians on three dimensional Heisenberg manifolds, and study a behaivior of the values when we deform the uniform discrete subgroups. Heiseberg manifolds are the total space of a fiber bundle with a torus as the base space and a circle as a typ…

Paper removes singularities from compact area minimizers in positive scalar curvature manifolds.

The paper shows deep connections between exotic smoothings of a small R^4 (the spacetime), the leaf space of codimension-1 foliations (related to noncommutative algebras) and quantization. At first we relate a small exotic R^4 to codimension-1 foliations of the 3-sphere unique up to foliated cobordisms and characterize…

Projections to a graph have bounded diameter for certain group structures.

A mathematical model describes deforming manifolds with precise vectors and fields.

Study YB operators and their deformations, finding integrable and nontrivial cases.

Smooth deformation of Moishezon manifolds preserves their Moishezon property.

In this paper, we study deformations of holomorphic Poisson maps which extend Horikawa's series of papers on deformations of holomorphic maps in the context of holomorphic Poisson deformations. In appendices, we present deformations of Poisson morphisms in the language of functors of Artin rings which is the algebraic …

In this paper we study the deformation theory of submanifolds characterized by a system of differential forms and provide a criterion for deformations of such submanifolds to be unobstructed. We apply this deformation theory to special Legendrian submanifolds in Sasaki-Einstein manifolds. In general, special Legendrian…

Study on deformations of Lie groupoid morphisms and their properties.

Study canonical deformations of complex forms and their cohomology properties.

In this paper, we study deformations of compact holomorphic Poisson submanifolds which extend Kodaira's series of papers on semi-regularity (deformations of compact complex submanifolds of codimension 1), deformations of compact complex submanifolds of arbitrary codimensions, and stability of compact complex submanifol…

The $L_\infty$-algebra is an algebraic structure suitable for describing deformation problems. In this paper we construct one $L_\infty$-algebra, which turns out to be a differential graded Lie algebra, to control the deformations of Lie algebroids and a second one to control the deformations of Lie subalgebroids. We a…