Coarse homotopy theory connects Euclidean cones to shape theory of compact spaces.
problem Establishing connections between coarse homotopy theory and shape theory.
method Using pointed shape invariants and inverse mapping telescopes.
result Proving two compact spaces are strong shape equivalent if their Euclidean cones are coarsely homotopy equivalent.
New 3D shape not homotopy equivalent to any hyperbolic shape.
problem Finding 3D shapes not homotopy equivalent to hyperbolic ones.
method Constructed a locally hyperbolic 3-manifold with specific fundamental group properties.
result The constructed manifold is not homotopy equivalent to any hyperbolic manifold.
Fine shape of local compacta represented by ordinary maps.
problem Representing fine shape of local compacta.
method Constructing a space ∣X∣ for each local compactum X such that fine shape classes correspond to homotopy classes of maps to ∣X∣. result Fine shape classes from any locally compact metrizable space Y to X bijectively correspond to homotopy classes of maps from Y to ∣X∣. M. Bestvina has shown that for any given torsion-free CAT(0) group G, all of its boundaries are shape equivalent. He then posed the question of whether they satisfy the stronger condition of being cell-like equivalent. In this article we prove that the answer is "Yes" in the situation where the group in question splits…
A new method for analyzing shapes and forms using additive models on manifolds.
problem Analyzing shapes and forms under geometric transformations.
method Extending generalized additive regression to models for shapes/forms using squared geodesic distance and Riemannian L2-Boosting algorithm. result Automated model selection and intuitive visualization of covariate effects in shape/form space.
Question 2.6 of Bestvina's Questions in Geometric Group Theory asks whether every pair of boundaries of a given CAT(0) group G is cell-like equivalent. The question was posed by Bestvina shortly after the discovery, by Croke and Kleiner, of a CAT(0) group that admits multiple boundaries. Previously, it had been observe…
Paper adapts SVDD for 3D-shapes filtering and outlier detection.
problem Filtering and detecting outliers in 3D-shapes.
method Adapted SVDD to SimpleMKL, developing Slim-MK-SVDD for tighter boundaries.
result Slim-MK-SVDD produces a tighter boundary around data.
It is known that shape injectivity implies homotopical Hausdorff and that the converse does not hold, even if the space is required to be a Peano continuum. This paper gives an alternative definition of homotopical Hausdorff inspired by a new topology on the set of fixed endpoint homotopy classes of paths. This version…
The paper shows how to use fine shape to understand infinite-dimensional spaces.
problem Understanding infinite-dimensional metrizable spaces and their homology theories.
method Obtained results indicating fine shape is tractable and can be used for Polish spaces.
result Every Polish space is fine shape equivalent to the limit of an inverse sequence of simplicial maps.
Homotopy theory of differentiable sheaves connects manifold properties to underlying homotopy types.
problem Understanding the homotopy type of manifolds using differentiable sheaves.
method Developed model structures and homotopical calculi on the ∞-category Diff∞ to compute and compare shapes. result The shape of any manifold coincides with various other notions of underlying homotopy types.
Let M be a manifold or (more generally) a locally compact, metrizable ANR. If K is an attractor for a flow in M, with basin of attraction A(K), it is well known that the inclusion i:K⊆A(K) is always a shape equivalence. In this paper we investigate to what extent this generaliz…
An algorithm preserves topological features in dimensionality reduction.
problem Preserving topological features in dimensionality reduction.
method Simulated annealing for finding a linear projection preserving persistent homology.
result Measures of topological equivalence between filtrations.
Extends geometric group theory techniques to arbitrary proper metric ARs.
problem Generalizing geometric group actions to non-freely acting groups with torsion.
method Extends techniques from geometric group theory to arbitrary proper metric ARs, eliminating freeness requirements.
result New theorems on proper homotopy equivalence and Z-structures for geometric actions.
This paper connects monetary and star-shaped risk measures by showing their equivalence under certain conditions.
problem Understanding the relationship between monetary and star-shaped risk measures.
method Analyzing the acceptability of 0 and the normalization property.
result Monetary risk measures are only a translation away from star-shapedness under mild conditions.
SVarM uses varifold representations for shape classification and regression.
problem Challenges in analyzing geometric data due to non-Euclidean shape spaces.
method Develops a neural network-based framework for varifold representations of shapes.
result Demonstrates strong performance and robustness in shape classification and regression.
In 2000, Croke and Kleiner showed that a CAT(0) group G can admit more than one boundary. This contrasted with the situation for word hyperbolic groups, where it was well-known that each such group admitted a unique boundary---in a very stong sense. Prior to Croke and Kleiner's discovery, it had been observed by Geoghe…
The class of surfaces in 3-space possessing nontrivial deformations which preserve principal directions and principal curvatures (or, equivalently, the shape operator) was investigated by Finikov and Gambier as far back as in 1933. We review some of the known examples and results, demonstrate the integrability of the c…
Introduces Star-Shaped DDPMs for non-Gaussian distributions.
problem Difficulties in defining DDPMs for non-Gaussian distributions.
method Star-shaped diffusion process, duality with specific Markovian diffusions, efficient algorithms.
result SS-DDPMs can model distributions like Beta, von Mises-Fisher, Dirichlet, Wishart.
We present and study a family of metrics on the space of compact subsets of RN (that we call ``shapes''). These metrics are ``geometric'', that is, they are independent of rotation and translation; and these metrics enjoy many interesting properties, as, for example, the existence of minimal geodesics. We view our s…
New 4D shapes without spine found.
problem Existence of spines in simply-connected 4-manifolds.
method Heegaard Floer d invariants to find obstructions.
result Infinitely many 4-manifolds homotopy to S2 with no spine. We analyze a monetary system of random money transfer on the basis of double entry bookkeeping. Without boundary conditions, we do not reach a price equilibrium and violate text-book formulas of economists quantity theory (MV=PQ). To match the resulting quantity of money with the model assumption of a constant price, w…
Proposes a new metric for comparing shapes in different spaces.
problem Comparing shapes in different metric spaces with unequal mass.
method Developed a Partial Gromov-Wasserstein (PGW) metric and algorithms to solve it.
result PGW is a well-defined metric between metric measure spaces.
We present infinitely many nonlocal conservation laws, a pair of compatible local Hamiltonian structures and a recursion operator for the equations describing surfaces in three-dimensional space that admit nontrivial deformations which preserve both principal directions and principal curvatures (or, equivalently, the s…
New complexity measure MEHC refines MDP upper bounds and rewards informativeness.
problem Refining complexity measures for MDPs and understanding reward informativeness.
method Introducing MEHC, a new complexity measure that tightens MDP diameter by accounting for reward structure.
result MEHC replaces diameter in upper bounds on optimal value span and UCRL2-like algorithms' regret.
New disks found with similar outer shapes.
problem Finding similar outer shapes for Lagrangian disks.
method Modified construction of Abe and Tange.
result Arbitrarily large families of disks found.
We describe the local conformal geometry of a Lorentzian spin manifold (M,g) admitting a twistor spinor φ with zero. Moreover, we describe the shape of the zero set of φ. If φ has isolated zeros then the metric g is locally conformally equivalent to a static monopole. In the other case the zero set consists o…
The paper proves a Whitehead theorem for fine shape spaces.
problem Proving a Whitehead theorem for fine shape spaces.
method Using Steenrod-Sitnikov homotopy groups and ind-groups.
result Fine shape morphisms are equivalences if they induce isomorphisms on π_i.
It is well known that every word hyperbolic group has a well-defined visual boundary. An example of C. Croke and B. Kleiner shows that the same cannot be said for CAT(0) groups. All boundaries of a CAT(0) group are, however, shape equivalent, as observed by M. Bestvina and R. Geoghegan. Bestvina has asked if they also …
This study proves new financial market theorems breaking standard risk definitions.
problem Breaking standard risk definitions in financial markets.
method Presenting proofs for new financial market theorems.
result New definitions are richer and broader than standard ones considering shape.
The paper studies the free elastic flow of closed curves and finds their asymptotic shape converges to a circle.
problem Challenges in studying the asymptotic behavior of the free elastic flow for closed curves.
method Analysis of the free elastic flow as an L2-gradient flow for Euler's elastic energy. result An appropriate rescaling of initial curves geometrically close to circles converges to a unique round circle.
The paper connects curves on a light cone to KdV equations.
problem Understanding differential invariants of curves on a light cone.
method Poisson equivalence and centro-affine action of Lorentzian group.
result Solutions of KdV equations as flows of curves on the cone.
SRNF framework extends surface distance to Lipschitz surfaces.
problem Defining a distance metric for unparametrized surfaces.
method Square Root Normal Fields (SRNF) and Wasserstein Fisher Rao (WFR) metric.
result SRNF distance on Lipschitz surfaces is equivalent to WFR metric.
This survey/expository article covers a variety of topics related to the "topology at infinity" of noncompact manifolds and complexes. In manifold topology and geometric group theory, the most important noncompact spaces are often contractible, so distinguishing one from another requires techniques beyond the standard …
We consider compact, aspherical solenoids obtained as the inverse limit of a system of CW~complexes and covering maps. This includes P-adic solenoids, as well as the universal hyperbolic solenoid of Teichmüller theory. Using ideas from shape theory, we classify maps between such solenoids up to homotopy, and we prove…
A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studies geometric shapes with finite number of elements (such as polyhedra)…
Various semigroups of noninvertible supermatrices of the special (antitriangle) shape having nilpotent Berezinian which appear in supersymmetric theories are defined and investigated. A subset of them continuously represents left and right zero semigroups and rectangular bands. The ideal properties of higher order rect…
New geometric framework for non-stationary processes using dilation matrices and shape space.
problem Characterizing non-stationary stochastic processes.
method Introducing dilation matrices and induced shape space manifolds.
result Rotation matrices can uniquely identify non-stationary processes, even when periodicity is present.
Study handles in 3D shapes, applies to material patterns.
problem Understanding 3D shapes through handlebody decompositions.
method Introduced handlebody decompositions, showed stability, applied to materials.
result Stable equivalence of handlebody decompositions in 3-manifolds.
Compactness proven for isospectral Birkhoff billiard tables.
problem Proving compactness of isospectral Birkhoff billiard tables.
method Derived a hierarchical structure for integral invariants and used interpolating Hamiltonian.
result Compactness of equivalence classes of marked length isospectral Birkhoff billiard tables.
Deep neural networks (DNNs) have achieved unprecedented performance on a wide range of complex tasks, rapidly outpacing our understanding of the nature of their solutions. This has caused a recent surge of interest in methods for rendering modern neural systems more interpretable. In this work, we propose to address th…
The aim of this paper is to find an optimal matching between manifold-valued curves, and thereby adequately compare their shapes, seen as equivalent classes with respect to the action of reparameterization. Using a canonical decomposition of a path in a principal bundle, we introduce a simple algorithm that finds an op…
Characterizes convergence of orbits of holomorphic semigroups in the unit disc.
problem Understanding the convergence behavior of orbits of holomorphic semigroups.
method Analyzes the shape of the starlike domain image of the Koenigs function and uses properties of quasi-geodesics.
result Characterizes convergence types (non-tangential and tangential) in terms of the domain's shape and quasi-geodesic properties.
The paper derives inequalities for Riemannian maps and submersions involving quaternionic space forms.
problem Establishing optimal inequalities for Riemannian maps and submersions involving quaternionic space forms.
method Deriving Casorati inequalities for Riemannian maps and submersions involving quaternionic space forms.
result Geometric characterizations of equality cases for Riemannian maps and submersions involving quaternionic space forms.
Two curves in hyperbolic space share a bounded distance, leading to a minimizing surface.
problem Finding a surface minimizing area between two disjoint curves in hyperbolic space.
method Analyzing the asymptotic boundary of hyperbolic 3-space, applying Definition 1.8 for distance bounds, and proving Theorems 1.7 and 1.11.
result Existence of an area-minimizing surface between two disjoint curves with bounded distance.
Geomstats introduces shape module for analyzing shapes of objects.
problem Analyzing shapes of objects represented as landmarks, curves, and surfaces.
method Implementing shape spaces, group actions, fiber bundles, quotient spaces, and Riemannian metrics.
result Users can compare, average, and interpolate shapes inside shape spaces.
A new shape space allows optimization of non-smooth shapes in fluid mechanics.
problem Optimizing non-smooth shapes in fluid mechanics.
method Constructing a product manifold to include piecewise-smooth shapes.
result Numerical results show applicability in minimizing viscous energy dissipation.
Investigates conditions for risk or utility functionals to be sensitive to large losses.
problem Conditions for risk or utility functionals to be sensitive to large losses.
method Analyzes sensitivity to large losses for various risk and utility functionals.
result Value at Risk and Expected Shortfall generally fail to be sensitive to large losses, but expected utility functionals and certain adjusted versions are sensitive.
Introduces Star-Shaped deviation measures for risk analysis.
problem Risk measurement and analysis in finance.
method Characterizes Star-Shaped deviation measures through acceptance sets and convex deviation measures.
result Exposes the relationship between Star-Shaped risk measures and deviation measures.