Establishes geometric properties of elements in the positive semigroup of a general real semisimple Lie group.
problem Generalizing Lusztig's total positivity to the setting of general real semisimple Lie groups.
method Classifying Lie groups admitting a positive structure and establishing key properties of unipotent positive semigroups.
result Establishes key geometric properties of elements in the positive semigroup.
Establishes a concavity property for positive Hessian quotient operators.
problem Analyzing positive Hessian quotient operators on Riemannian manifolds.
method Proves a special concavity property and a Jacobi inequality.
result Proves a Jacobi inequality for symmetric tensors.
The study classifies graphs on surfaces with positive curvature properties.
problem Classifying graphs on surfaces with specific curvature properties.
method Using medial graphs and classification techniques.
result Complete classification of graphs on surfaces with positive Forman curvature and corner curvature.
Study on positivity properties of vector bundle Monge-Ampère equation.
problem Analyzing positivity in vector bundle Monge-Ampère equation.
method Investigates MA-positivity and MA-semi-positive solutions for different ranks of holomorphic bundles over complex surfaces and manifolds.
result Positivity preservation in rank-two holomorphic bundles but not in higher ranks.
A complex disproves a curvature property.
problem Proving non-equivalence of curvature properties.
method Analyzing a specific 2-complex presentation.
result The contracting non-positive immersions property is not equivalent to positive maximal irreducible curvature.
The book is devoted to constructing embedding finite-dimensional maps into trivial bundles and investigating the corresponding general position properties.
Paper shows L∞-positivity and stochastic completeness are equivalent.
problem Analyzing L∞-positivity preserving property and stochastic completeness. method Using monotone approximation results for distributional solutions of −Δ+1≥0. result The L∞-positivity preserving property is equivalent to stochastic completeness. The study examines Eschenburg orbifolds with positive sectional curvature and their geometric/topological properties.
problem Understanding the geometric and topological constraints of positively curved Eschenburg orbifolds.
method Proved restrictions on singular sets and computed orbifold cohomology rings.
result Distinctive behavior in cohomology groups of positively curved Eschenburg orbifolds.
We prove that positive knots and knots with braid index three in the 3-sphere satisfy the Property P conjecture.
Collar lemma proven for certain surface group representations.
problem Proving a collar lemma for specific surface group representations.
method Using partial hyperconvexity properties and Anosov representations.
result 'Positivity properties' hold for partially hyperconvex representations.
Study of new link types and their invariants, extending previous results.
problem Properties of polynomial invariants and signatures of weakly successively almost positive links.
method Analysis of minimal genus and fibering properties, extending known theorems.
result Extension of Scharlemann-Thompson's theorem to weakly successively almost positive links.
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 show that all finite-dimensional resolvable generalized manifolds with the piecewise disjoint arc-disk property are codimension one manifold factors. We then show how the piecewise disjoint arc-disk property and other general position properties that detect codimension one manifold factors are related. We also note …
In this survey article, we present some panorama of groups acting on metric spaces of non-positive curvature. We introduce the main examples and their rigidity properties , we show the links between algebraic or analytic properties of the group and geometric properties of the space. Finally, we conclude with a few conj…
The paper bounds bandwidth and focal radius for manifolds with positive isotropic curvature.
problem Bounding bandwidth and focal radius for manifolds with positive isotropic curvature.
method Using spectral properties of a twisted de Rham-Hodge operator.
result Upper bounds on bandwidth and focal radius are derived for hypersurfaces in PIC manifolds.
This paper describes metrics with varying curvature properties in a specific manifold.
problem Characterizing regions in a manifold with specific curvature properties.
method Using a projected Ricci flow and studying the dynamics of regions in the manifold.
result Sign curvature maintenance and escaping in regions of the manifold.
We perform an analysis of fractal properties of the positive and the negative changes of the German DAX30 index separately using Multifractal Detrended Fluctuation Analysis (MFDFA). By calculating the singularity spectra f(α) we show that returns of both signs reveal multiscaling. Curiously, these spectra display a s…
Paper proves Gromov's conjecture on manifolds with certain group properties.
problem Gromov's conjecture on positive scalar curvature and simplicial volume.
method Proves conjecture under a fundamental group decay property.
result Proves Gromov's conjecture for manifolds with a weakened rapid decay property.
The paper studies modified Einstein tensors and their positivity properties on compact manifolds.
problem Analyzing the positivity of modified Einstein tensors and their implications on compact manifolds.
method Investigates modified Einstein tensors defined as $\Eink :=\Scal \, g -k\Ric$ for 0<k<n and studies their positivity properties. result Defines a smooth invariant $\cEin(M)$ measuring how far a manifold is from admitting an Einstein metric with positive scalar curvature.
Study finds criteria for surfaces with specific curvature properties.
problem Understanding Kählerian or projective structures on surfaces with non-positive curvature.
method Established a criterion for compact Hermitian surfaces with non-positive second Chern-Ricci curvature.
result Found conditions for Kählerian or projective structures on surfaces with non-positive curvature.
Extends positive and almost positive links to successively almost positive ones.
problem Extending properties of positive and almost positive diagrams and links.
method Introducing successively almost positive diagrams and links, and analyzing their properties.
result Improves known results of positive and almost positive links.
In this paper, we study the existence and non-existence result of positive solutions to a singular elliptic equation with negative power on the bounded smooth domain or in the whole Euclidean space. Our model arises in the study of the steady states of thin films and other applied physics. We can get some useful local …
Groups with specific curvature have a regular language of geodesics.
problem Understanding the language of geodesics in non-positively curved triangle groups.
method Proving finitely many cone types and regularity of geodesic languages.
result The language of lexicographically first geodesics is regular and satisfies the fellow traveller property.
Introduces Θ-positivity in Lie groups, generalizing Lusztig's positivity.
problem Generalizing Lusztig's total positivity to a broader class of Lie groups.
method Introduces and studies Θ-positivity in real simple Lie groups. result Four families of Lie groups admit Θ-positive structures. It is well known that positivity properties of the curvature of a vector bundle have implications on the algebro-geometric properties of the bundle, such as numerical positivity, vanishing of higher cohomology leading to existence of global sections etc. It is also well known that bundles arising in Hodge theory tend t…
The paper extends structure theorem to projective klt varieties with specific tangent sheaf properties.
problem Understanding the structure of projective klt varieties with nef tangent sheaves.
method Developing theory of positivity of coherent sheaves and proving structure theorem.
result Projective klt varieties with specific tangent sheaf properties admit rationally connected fibrations onto abelian varieties.
The study explores knots and manifolds, proving properties and non-left-orderable groups.
problem Properties of knots and manifolds after surgeries and fillings.
method Fixed point method and Dehn surgery.
result Fundamental groups of certain manifolds are not left orderable.
We constructively prove the existence of time-discrete consumption processes for stochastic money accounts that fulfill a pre-specified positively homogeneous projection property (PHPP) and let the account always be positive and exactly zero at the end. One possible example is consumption rates forming a martingale und…
In this paper two metric properties on geodesic length spaces are introduced by means of the metric projection, studying their validity on Alexandrov and Busemann NPC spaces. In particular, we prove that both properties characterize the non-positivity of the sectional curvature on Riemannian manifolds. Further results …
In this paper we study numerical properties of quotients of holomorphic log-tensors.
We study a second order ordinary differential equation corresponding to rotationally symmetric p-harmonic maps. We show unique continuation and Liouville's type theorems for positive solutions. We discuss the existence of bounded positive entire solutions. Asymptotic properties of the positive solutions are investiga…
Paper shows regularizing flow for conical Kähler-Ricci equations.
problem Regularizing property of conical Kähler-Ricci flow.
method Regularizing property of the twisted conical Kähler-Ricci flow from a positive closed current with zero Lelong number.
result Extends regularizing property to conical singularity case.
A large class of positive finite presentations of the braid groups is found and studied. It is shown that no presentations but known exceptions in this class have the property that equivalent braid words are also equivalent under positive relations.
The article shows how to create metrics with positive Ricci curvature on twisted suspensions.
problem Creating metrics with positive Ricci curvature on complex manifolds.
method Using twisted suspensions and Riemannian metrics.
result Maximal symmetry rank of positive Ricci curvature manifolds is (n-2) in all dimensions n≥4.
We examine geometric properties of a knot J that are unchanged by taking a (p,q)-cable K of J. Specifically, we relate w(K) to w(J), where w(K) is the width of K in the sense of Gabai. We use this information to demonstrate that thin position is a minimal bridge position of J if and only if the same is true for K, and …
The paper examines positivity properties of singular Hermitian metrics.
problem Investigating positivity for singular Hermitian metrics.
method Exploring Griffiths, ω-trace, and RC positivity.
result These positivity notions imply cohomology vanishing and rational connectedness.
Study on weakly G-slim complexes and non-positive immersions for group presentations.
problem Conditions for non-positive immersions in group presentations.
method Investigation of weakly G-slim complexes and their relationship to left-orderable groups.
result Conditions on generalized Wirtinger presentations guaranteeing non-positive immersions for their associated 2-complexes.
The paper examines functional properties on manifolds with very negative curvature.
problem Functional properties on manifolds with very negative curvature.
method New Hardy-type inequalities and first and second order inequalities.
result Functional properties typically hold in manifolds with polynomially growing negative curvature.
Manifolds admitting positive sectional curvature are conjectured to have rigid homotopical structure and, in particular, comparatively small Euler charateristics. In this article, we obtain upper bounds for the Euler characteristic of a positively curved Riemannian manifold that admits a large isometric torus action. W…
For non-reversible Finsler metrics of positive flag curvature on spheres and projective spaces we present results about the number and the length of closed geodesics and about their stability properties.
Positive braids minimize knot untangling steps.
problem Finding the minimum number of steps to untangle knots.
method Analyzing positive braids and their knot closures, comparing ascending number to unknotting number.
result Ascending number equals unknotting number for knots from positive braids.
We develop the structure theory of full isometry groups of locally compact non-positively curved metric spaces. Amongst the discussed themes are de Rham decompositions, normal subgroup structure and characterising properties of symmetric spaces and Bruhat--Tits buildings. Applications to discrete groups and further dev…
We study homogenous Weyl connections with non-positive sectional curvatures. The Cartesian product S1×M carries canonical families of Weyl connections with such a property, for any Riemmanian manifold M. We prove that if a homogenous Weyl connection on a manifold, modeled on a unimodular Lie group, …
Study on quantum sl3 invariant for positive links.
problem Characterizing and understanding the quantum sl3 invariant of positive links. method Skein theory of sl3-webs, explicit formulae, diagrammatic quantities, obstructions. result Positive links are fibered if and only if the second coefficient of the polynomial is 1.
The paper defines new types of positivity and proves properties of Schur forms for vector bundles.
problem Defining and characterizing new types of positivity for vector bundles.
method Introducing and characterizing two types of strongly decomposable positivity, proving properties of Schur forms.
result Schur forms of strongly decomposable positive vector bundles are positive or weakly positive, answering a question of Griffiths.
We show that the difference between the Seifert genus and the topological 4-genus of a prime positive braid knot is bounded from below by an affine function of the minimal number of strands among positive braid representatives of the knot. We deduce that among prime positive braid knots, the property of having such a g…
Satellite links of fully positive braids are characterized.
problem Characterizing satellite links of fully positive braids.
method Analyzing fully positive braids and their satellites.
result Satellite links of fully positive braids are characterized by specific conditions.
The paper defines and studies the geometric mean for tensors and its associated Riemannian geometry.
problem Defining and studying the geometric mean for tensors.
method Generalized geometric mean for tensors using T-product, verified properties, and investigated Riemannian manifold.
result Geometric mean of T-positive definite tensors is a unique solution of algebraic Riccati tensor equations and a midpoint of geodesics.