Extends RDS filtering to position-orientation space for better image processing.
problem Enhancing and inpainting images with crossing structures.
method Created a version of RDS filtering using gauge frames, studying generalised diffusion.
result RDS filtering on position-orientation space improves denoising and inpainting of crossing structures.
We generalise work of Young-Eun Choi to the setting of ideal triangulations with vertex links of arbitrary genus, showing that the set of all (possibly incomplete) hyperbolic cone-manifold structures realised by positively oriented hyperbolic ideal tetrahedra on a given topological ideal triangulation and with prescrib…
A universal collection of 4 invariants improves neural network accuracy for molecular dynamics.
problem Improving accuracy of neural networks in molecular dynamics.
method Developed a universal collection of 4 smooth scalar invariants on M(3) x M(3) and evaluated their effectiveness in a PONITA neural network architecture.
result Using a universal collection of invariants significantly improves neural network accuracy.
A new metric mav offers a practical alternative to costly Riemannian distance.
problem Efficiently compute Riemannian distance on SE(3) invariant metrics.
method Propose mav distance, defined as Riemannian length of a curve.
result Mav distance offers a trainable invariant for geometric deep learning.
Positive Thompson links are proven for oriented subgroup elements.
problem Proving properties of Thompson links with positive elements.
method Analyzing elements of the oriented subgroup of the Thompson group.
result Positive oriented Thompson links are established.
We provide a local classification of self-dual Einstein Riemannian four manifolds admitting a positively oriented Hermitian structure and characterize those which carry a hyperhermitian, non-hyperkählerian structure compatible with the negative orientation. We finally show that self-dual Einstein 4-manifolds obtained a…
Study the space of simple polygons and their moduli.
problem Understanding the space of simple polygons and their moduli.
method Local and global descriptions of S(n) and M(n) using Morse theory and asymptotic analysis.
result Completely described S(4) and M(4) using a topological Morse function.
Formula for Alexander polynomial of links with twists.
problem Computing Alexander polynomial of links with twists.
method Using vector space representation of Uq(gl(1∣1)). result Alexander polynomials stabilize after adding enough twists.
Anosov subgroup equidistributes geodesics and holonomies on homogeneous spaces.
problem Equidistribution of geodesics and holonomies in Anosov homogeneous spaces.
method Analyzes maximal flat cylinders and their holonomies for Anosov subgroups.
result Joint equidistribution of maximal flat cylinders and holonomies as circumference tends to infinity.
Modeling curvature-sensitive cells in visual cortex using manifold geometry.
problem Understanding how curvature influences cell function in the visual cortex.
method Developed a 4D manifold with canonical Engel structure to represent orientation, position, curvature, and scale.
result Characterized curvature-sensitive receptive profiles using left-invariant generators of the Engel structure.
The paper extends the functional geometry of the visual cortex to more complex architectures using contactization and symplectization.
problem Understanding the functional architecture of the visual cortex.
method Contactization and symplectization processes to extend the dimension of the space.
result Extension of the functional geometry of the visual cortex to more complex architectures.
We present proofs of basic results, including those developed by Harold Bell, for the plane fixed point problem: does every map of a non-separating plane continuum have a fixed point? Some of these results had been announced much earlier by Bell but without accessible proofs. We define the concept of the variation of a…
The fixed-point index of a homeomorphism of Jordan curves measures the number of fixed-points, with multiplicity, of the extension of the homeomorphism to the full Jordan domains in question. The now-classical Circle Index Lemma says that the fixed-point index of a positive-orientation-preserving homeomorphism of round…
Cake wavelets minimize orientation score uncertainty.
problem Minimizing uncertainty in orientation scores.
method Axiomatically derived wavelets for orientation score lifting.
result Uncertainty gap of cake wavelets is less than 1.1.
Let M be a non-compact hyperbolic 3-manifold that has a triangulation by positively oriented ideal tetraedra. We explain how to produce local coordinates for the variety defined by the gluing equations for PGL(3,C)-representations. In particular we prove local rigidity of the "geometric" representation in PGL(3,C), rec…
Study on links formed by pseudocircle arrangements, focusing on three unavoidable cases.
problem Counting non-equivalent positive oriented links with pseudocircle arrangements as shadows.
method Analyzing three unavoidable arrangements of pseudocircles to estimate the number of non-equivalent links.
result Sharp estimates on the number of non-equivalent positive oriented links for the three unavoidable arrangements.
The paper introduces a new concept of frame vorticity and uses it to find optimal sections in specific geometric settings.
problem Finding optimal sections in geometric settings using frame vorticity.
method Defining frame vorticity, relating it to split pseudo-Riemannian metrics, and using split special Lagrangian calibrations.
result Explicit homologically volume maximizing sections and optimal sections for specific manifolds.
Recently, Ian Agol introduced a class of "veering" ideal triangulations for mapping tori of pseudo-Anosov homeomorphisms of surfaces punctured along the singular points. These triangulations have very special combinatorial properties, and Agol asked if these are "geometric", i.e. realised in the complete hyperbolic met…
The paper constructs triangulations for double twist knots using geometric methods.
problem Constructing explicit triangulations of double twist knots.
method Using triangulating Dehn fillings, layered solid tori, and their double covers.
result Proves both triangulations are geometric, using conjecturally minimal triangulation to present A-polynomial equations.
Thurston introduced a technique for finding and deforming three-dimensional hyperbolic structures by gluing together ideal tetrahedra. We generalize this technique to study families of geometric structures that transition from hyperbolic to anti de Sitter (AdS) geometry. Our approach involves solving Thurston's gluing …
A biperiodic alternating link has an alternating quotient link in the thickened torus. In this paper, we focus on semi-regular links, a class of biperiodic alternating links whose hyperbolic structure can be immediately determined from a corresponding Euclidean tiling. Consequently, we determine the exact volumes of se…
The paper extends Gluck and Warner's result on fibrations of spheres by great subspheres.
problem Understanding when two Hopf fibrations of S2n−1 agree on a fiber. method Characterizing the conditions for two Hopf fibrations of S2n−1 to agree on a fiber. result A complete characterization of the conditions for two Hopf fibrations of S2n−1 to agree on a fiber. Study on Langevin dynamics on planar motion group, highlighting geometric mechanism.
problem Understanding Langevin dynamics on the Lie group SE(2).
method Intrinsic formulation on SE(2), using invariant vector fields and natural projection.
result Effective macroscopic diffusion on R^2 emerges through averaging.
In this paper we construct a functor from the category of two-dimensional Riemannian manifolds to the category of three-dimensional manifolds with generalized metric tensors. For each two-dimensional oriented Riemannian manifold (M,g) we construct a metric tensor g^ (in general, with singularities) on the total…
A grand challenge in machine learning is the development of computational algorithms that match or outperform humans in perceptual inference tasks that are complicated by nuisance variation. For instance, visual object recognition involves the unknown object position, orientation, and scale in object recognition while …
Constructs polyhedral chains with prescribed tangent plane distributions.
problem Constructing polyhedral chains with specific tangent plane distributions.
method Explicit construction of polyhedral chains that approximate prescribed measures on Grassmannian.
result Polyconvexity is equivalent to quasiconvexity of associated Q-integrands under certain conditions.
Geometric characterization of sub-Riemannian geodesics on frame bundles.
problem Characterize sub-Riemannian geodesics on frame bundles of 3-manifolds.
method Lie theoretical description, geometric characterization, complex length spectrum computation.
result Sub-Riemannian metrics on frame bundles of isospectral manifolds are length isospectral.
Semantic segmentation is a critical method in the field of autonomous driving. When performing semantic image segmentation, a wider field of view (FoV) helps to obtain more information about the surrounding environment, making automatic driving safer and more reliable, which could be offered by fisheye cameras. However…
Let M3 be an oriented 3-manifold. We investigate when one of the fibers or a combination of fiber components, Fbest, of a \emph{harmonic} map f:M3→S1 with Morse-type singularities delivers the Thurston norm χ−([Fbest]) of its homology class [Fbest]∈H2(M3;Z). In particular, for a map …
We introduce a new loss function for evaluating forecasts and estimate models using it.
problem Lack of a decision-theoretic foundation for evaluating forecasts using the Nash-Sutcliffe efficiency.
method We introduce and analyze the Nash-Sutcliffe loss function and its application in estimating models.
result Nash-Sutcliffe loss provides a decision-theoretic foundation for evaluating and estimating models.
Study of tangent spaces in diffeological spaces under Lie group actions.
problem Understanding tangent spaces in generalized spaces.
method Generalized tangent space construction and isomorphism proof.
result Internal tangent space isomorphic to stratified tangent space.
(1,1) non-L-space knots are foliar in 3D space.
problem Proving (1,1) non-L-space knots are foliar.
method Analyzing (1,1) non-L-space knots in S3 and lens spaces. result (1,1) non-L-space knots are persistently foliar.
Universal spaces for finite topological spaces simplify shape descriptions.
problem Describing shape properties of compact metric spaces.
method Inverse limits of finite spaces and Alexandroff extensions.
result Universal spaces simplify shape descriptions of compact metric spaces.
The paper extends Stone duality to topological convexity spaces.
problem Understanding the relationship between topological convexity spaces and sup-lattices.
method Extending Stone duality to topological convexity spaces using preconvexity spaces.
result An adjunction between topological convexity spaces and sup-lattices.
No Einstein hypersurfaces found in Damek-Ricci spaces.
problem Existence of Einstein hypersurfaces in symmetric spaces.
method Analyzing properties of Damek-Ricci spaces and proving no Einstein hypersurface exists.
result No Einstein hypersurfaces in Damek-Ricci spaces.
The distance function ϱ(p,q) (or d(p,q)) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over Rn, whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are no F…
New kernels defined for various spaces, including measures.
problem Defining kernels on non-standard spaces like measures.
method Integrally strictly positive definite and characteristic kernels on Hilbert, Banach, and metric spaces.
result Explicit classes of kernels on Lp spaces and sets of measures. Metric spaces uniquely split into Hilbert and non-line-split parts.
problem Understanding the structure of metric spaces.
method Proved unique decomposition into Hilbert and non-line-split parts.
result Metric spaces have a unique decomposition into a Hilbert space and a non-line-split part.
In a paper (math.DG/0403528) we obtained explicit examples of Moishezon twistor spaces of some compact self-dual four-manifolds admitting a non-trivial Killing field, and also determined their moduli space. In this note we investigate minitwistor spaces associated to these twistor spaces. We determine their structure, …
Study geometry of tetrahedra in complex hyperbolic space and Hilbert spaces.
problem Understanding geometric relationships between complex hyperbolic spaces and Hilbert spaces.
method Use a complex analog of the cosine of a vertex angle as a novel technical tool.
result Describe possible triangular faces of tetrahedra in hyperbolic space and three-dimensional subspaces in Hilbert spaces with Pick kernels.
New curvature positivity helps classify spherical spaces and complex projective spaces.
problem Classifying spherical space forms and complex projective spaces.
method Introducing a new positivity notion for curvature.
result Characterizations for spherical space forms and complex projective spaces.
Study of complete space-like self-expanders in Minkovski space.
problem Characterize complete space-like self-expanders in Minkovski space.
method Use of maximum principle of Omori-Yau type to prove rigidity theorems.
result Classification of 2-dimensional complete space-like self-expanders with constant squared norm of the second fundamental form.
The abstract discusses the linear and smooth structures of mapping spaces.
problem The structure of mapping spaces in differential geometry.
method Proving diffeomorphisms and fibre bundle properties.
result Path spaces and base point preserving mapping spaces are Fréchet spaces.
Study on convergence of transformed metric spaces as dimensions grow.
problem Conditions for convergence of transformed metric spaces.
method Clarifying conditions for convergence of transformed spaces from original sequence and vice versa.
result Spheres and projective spaces converge to Gaussian space and its quotient as dimensions increase.
Characterizes when almost smooth spaces become RCD spaces.
problem Understanding conditions for almost smooth spaces to be RCD spaces.
method Characterizations via local volume doubling and Poincaré inequality.
result Characterizes Einstein 4-orbifolds.
Stability of Wasserstein spaces under various convergence types.
problem Stability and finiteness of Wasserstein spaces over singular and non-singular spaces.
method Gromov--Hausdorff convergence and equivariant Gromov--Hausdorff convergence.
result Analogue of Perelman's stability theorem on Wasserstein spaces.
New infinite families of flat spaces found from symmetric spaces.
problem Finding new flat homogeneous spaces.
method Starting from compact symmetric spaces, constructing infinite families of compact homogeneous spaces with invariant Bismut connections.
result Infinite families of compact homogeneous spaces with vanishing Ricci tensor.
New quasi space forms solve Thurston's geometrical space form problem.
problem Solving Thurston's geometrical space form problem.
method Introducing quasi space forms as non-real space forms with specific geometric properties.
result Quasi space forms offer a metrical, local geometrical solution to Thurston's problem.