New proof for Ending Lamination Theorem using hyperbolic 3-simplices.
problem Proving the Ending Lamination Theorem for hyperbolic 3-manifolds of infinite volume.
method Relies on Maximum Volume Law for hyperbolic 3-simplices.
result New proof of Ending Lamination Theorem.
Develop criteria to distinguish Gromov-Thurston manifolds using algebraic Dehn fillings.
problem Distinguishing homotopy types of Gromov-Thurston manifolds
method Using virtual Dehn fillings of relatively hyperbolic groups
result Criteria to distinguish homotopy types
We consider Gromov-Thurston examples of negatively curved n-manifolds which do not admit metrics of constant sectional curvature. We show that for each n some of the Gromov-Thurston manifolds admit strictly convex real-projective structures.
Gromov-Thurston covers have Betti numbers as expected.
problem Understanding Betti numbers of branched covers of hyperbolic manifolds.
method Analyzing Gromov-Thurston branched covers and their Betti numbers.
result Betti numbers match expectations for non-divisible degree covers.
New Einstein metrics found on complex manifolds.
problem Locally symmetric metrics on complex manifolds.
method Construction of manifolds with specific curvature properties.
result Infinitely many manifolds with negatively curved Einstein metrics but no locally symmetric metrics.
The study constructs AdS manifolds from Gromov-Thurston manifolds.
problem Creating hyperbolic and anti-de Sitter structures from Gromov-Thurston manifolds.
method Explicit correspondence between quasifuchsian AdS manifolds and compact quotients of Ø(2d,2)/U(d,1).
result Existence of quasifuchsian AdS manifolds and hyperbolic ends with specified boundary.
In this paper we prove that for all n=4k−2, k≥2 there exists closed n-dimensional Riemannian manifolds M with negative sectional curvature that do not have the homotopy type of a locally symmetric space, such that π1(T<0(M)) is non-trivial. T<0(M) denotes the Teichmüller space…
Let G be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If H_2(G;Q) is nonzero, then G contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov-Thurston norm on H_2(G;R) is a finite-sided rational polyhedron.
We prove that there exists a metric of positive curvature in a three-sphere which admits a given torus knot as a closed geodesic.We also sketch a construction of a metric in a four sphere, very likely of positive curvature, which admits a totally geodesic projective plane with Euler number four. Surpisingly, the techni…
Study geometric manifolds in arbitrary dimensions, focusing on maps and diffeomorphisms.
problem Existence and properties of maps and diffeomorphisms in geometric manifolds.
method Analysis of geometric structures and homotopy invariants in arbitrary dimensions.
result Existence of Anosov diffeomorphisms and monotonicity of homotopy invariants.
We give conceptual proofs of some well known results concerning compact non-positively curved locally symmetric spaces. We discuss vanishing and non-vanishing of Pontrjagin numbers and Euler characteristics for these locally symmetric spaces. We also establish vanishing results for Stiefel-Whitney numbers of (finite co…
The fundamental group of a Riemannian manifold with δ-pinched negative curvature, δ>1/4, cannot be the fundamental group of a quasicompact Kähler manifold. The proof also implies that a non-uniform lattice in F4(−20) cannot be the fundamental group of a quasicompact Kähler manifold. We also construct examples …
We introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the relative hyperbolicity of G in many natural ways. Second, we construct two useful bic…
This paper proves a theorem about Dehn surgery using a new theorem about PSL(2, C) character varieties. Confirming a conjecture of Boyer and Zhang, this paper shows that a small hyperbolic knot in a homotopy sphere having a non-trivial cyclic slope r has an incompressible surface with non-integer boundary slope strictl…
Research examines how Islamic banking principles spread among managers and scholars.
problem Diffusion of Islamic banking principles among managers and scholars.
method Literature review focusing on knowledge diffusion and Islamic banking governance principles.
result Emergence of common Islamic banking governance principles from diverse knowledge streams.
The paper establishes maximum principles and stochastic completeness for pseudo-Hermitian manifolds.
problem Maximum principles and stochastic completeness for pseudo-Hermitian manifolds.
method Established generalized maximum principles and proved stochastic completeness equivalence.
result Stochastic completeness for the heat semigroup is equivalent to generalized maximum principles.
The h-principle helps solve complex geometric problems.
problem Solving complex geometric problems using the h-principle.
method Developed from the Oka-Grauert principle and Gromov's theory, the h-principle is applied to Oka manifolds and maps.
result Recent developments and applications of the h-principle in complex analysis and geometry.
The study establishes uncertainty principles on harmonic manifolds of rank one.
problem Developing uncertainty principles for harmonic manifolds of rank one.
method Derivation of various uncertainty principles including Heisenberg, Morgen, Schrödinger, and Hömanders principles.
result Generalization of Hausdorff-Young inequality to harmonic manifolds of rank one.
The paper rigorously investigates the Frequency Principle in deep neural networks.
problem Understanding the training dynamics of deep neural networks.
method Theoretical investigation of Frequency Principle at three stages of training.
result Theorem providing quantitative understanding of Frequency Principle for general DNNs.
Conditions for polynomial to be isometry of lattice, answering Hasse principles.
problem Conditions for integral polynomials to be characteristic polynomials of isometries of lattices.
method Necessary and sufficient conditions derived from lattice isometries and Hasse principles.
result Proved a Hasse principle for signatures of knots.
Shows flexible sheaves as fibrant objects for Gromov's h-principle.
problem Applying the h-principle to partial differential relations.
method Interprets flexible sheaves as fibrant objects in a model structure.
result Flexible sheaves can be understood as fibrant objects.
A new method to break down insurance costs into risk and uncertainty.
problem Understanding and quantifying insurance costs in uncertain environments.
method An axiomatic approach to decompose premium principles into risk and deviation measures.
result Maximal risk and minimal deviation measures can be uniquely identified in decompositions.
New objective function preserves Bellman's principle for policy gradient.
problem Lack of objective function capturing Bellman's principle optimally.
method Proposed a new objective function and its gradient.
result Preserves Bellman's principle of optimality in policy gradient.
Investigates stability properties of Haezendonck-Goovaerts premium principles in Orlicz spaces.
problem Stability properties of Haezendonck-Goovaerts premium principles in various Orlicz spaces.
method Analysis of stability properties including Fatou and Lebesgue properties, and continuity with respect to Φ-weak convergence. result Haezendonck-Goovaerts principles satisfy the Fatou property and Lebesgue property under certain conditions.
Study on maximum principles for nonlinear equations on Riemannian manifolds.
problem Investigating strong maximum principles for fully nonlinear equations on Riemannian manifolds.
method Analyzing scaling conditions and applying to various nonlinear operators.
result Established new strong comparison principles for second order uniformly elliptic problems.
Derives Fredholm criteria for isotypical components from a Simonenko principle.
problem Finding Fredholm conditions for isotypical components of invariant pseudodifferential operators.
method General Simonenko's local principle and equivariant local principle for restriction to isotypical components.
result Full proof of equivariant local principle and extension of results.
Establishes a boundary maximum principle for varifolds with fixed contact angle.
problem Boundary behavior of varifolds with contact angle constraints.
method Maximum principle for stationary pairs of varifolds with fixed contact angle condition.
result Boundary maximum principle proven for stationary varifolds.
Study proves Maximum Principles for unbounded Riemannian domains.
problem Proving Maximum Principles for unbounded Riemannian domains.
method Examines both ambient manifold and differential operator assumptions.
result Valid Maximum Principles established for unbounded domains.
A pricing principle is introduced for non-attainable claims in incomplete markets.
problem Pricing non-attainable contingent claims in incomplete markets.
method Distorted Radon-Nikodym derivative and Tsallis relative entropy over a family of equivalent martingale measures.
result The pricing principle is closely related to backward stochastic differential equations and is arbitrage-free and time-consistent.
Proves a principle for one-phase Bernoulli problem minimizers.
problem One-phase Bernoulli problem minimizers.
method Strong maximum principle, Alt-Caffarelli functional, Hardt-Simon-type foliation.
result Constructs a foliation for global minimizers.
We show that a well known uncertainty principle for functions on the circle can be derived from an uncertainty principle for the Euclidean motion group.
New method proves h-principles for stable forms on manifolds.
problem Proving h-principles for stable forms on manifolds. method Convex integration applied to stable forms.
result Proved h-principles for 4 classes of stable forms. We give a version of the comparison principle from pluripotential theory where the Monge-Ampère measure is replaced by the Bergman kernel and use it to derive a maximum principle
In this paper, we develop a new mathematical technique which allows us to express the joint distribution of a Markov process and its running maximum (or minimum) through the marginal distribution of the process itself. This technique is an extension of the classical reflection principle for Brownian motion, and it is o…
The Weyl principle holds in some Finsler settings despite general failure.
problem Applying the Weyl principle to Finsler manifolds.
method Investigation of the Weyl principle in Finsler geometry.
result A weak form of the Weyl principle persists in certain Finsler settings.
Note establishes a local maximum principle for Ricci flow under curvature conditions.
problem Preserving nonnegativity of curvature along Ricci flow with unbounded curvature.
method Combining scaling invariant curvature condition with Dirichlet heat kernel estimates.
result Unified and more direct proof of localized maximum principle.
Derives time-averaged active inference from control principles.
problem Finite-horizon or discounted-surprise problems in active inference.
method Derives infinite-horizon, average-surprise active inference from optimal control principles.
result Unified objective functional for sensorimotor control.
Study proves rigidity for solitons with uncertainty principle.
problem Rigidity of shrinking Ricci solitons with uncertainty principle.
method Proves rigidity theorems for shrinking gradient Ricci solitons.
result Proves rigidity theorems with sharp constant in R^n.
New principle for harmonic maps helps study higher-dimensional submanifolds.
problem Understanding unboundedness of totally geodesic projections in higher codimension.
method Introducing a flexible notion of convexity and applying it to harmonic and conformal maps.
result New maximum principle for harmonic maps applicable to various geometric settings.
In this paper we give three applications of a method to prove h-principles on closed manifolds. Under weaker conditions this method proves a homological h-principle, under stronger conditions it proves a homotopical one. The three applications are as follows: a homotopical version of Vassiliev's h-principle, the contra…
In this paper we prove two extensions of Hamilton's maximal principle for systems pf parabolic equations which sould be useful for the study of the Ricci flow and some other geometric evolution equations. One extension is a time-dependent maximum principle and the other is a time-dependent maximum principle subject to …
We present a new statistical learning paradigm for Boltzmann machines based on a new inference principle we have proposed: the latent maximum entropy principle (LME). LME is different both from Jaynes maximum entropy principle and from standard maximum likelihood estimation.We demonstrate the LME principle BY deriving …
New equivalences found linking parabolicity, comparison principle, and capacity on Riemannian manifolds.
problem Understanding parabolicity and related concepts on Riemannian manifolds.
method Establishing new equivalences between parabolicity, comparison principle, and capacity.
result Equivalence between p-parabolicity and the comparison principle for the p-Laplace equation. 3-manifolds study Hasse norm principle, akin to number fields.
problem Topological analog of Hasse norm principle for 3-manifolds.
method Analogy with number fields and finite cyclic coverings.
result Showed topological Hasse norm principle for 3-manifolds.
Establishes a principle for metric convergence in Kähler geometry.
problem Convergence of evolving Riemannian metrics in Kähler geometry.
method General 'boundedness implies convergence' principle applied to collapsing Calabi-Yau metrics and normalized Kähler-Ricci flows.
result Obtains convergence results for collapsing Calabi-Yau metrics and normalized Kähler-Ricci flows.
New Bianchi-convex sets generalize Ricci flow maximum principle.
problem Generalizing maximum principle for Ricci flow.
method Introducing Bianchi-convex sets.
result Hamilton's maximum principle extended to Bianchi-convex sets.
Paper proves h-principles for symplectic structures and foliations.
problem Existence of conformal symplectic structures and foliations.
method Application of h-principles and foliated Morse theory.
result Linear deformation of foliations to contact structures.
Maps to manifolds transverse to certain distributions satisfy an h-principle.
problem Maps to manifolds transverse to certain distributions.
method Proving the complete h-principle for transverse maps. result Maps to manifolds transverse to certain distributions satisfy the h-principle.