Research shows reducibility of low dimensional Poincaré complexes in certain cases.
problem The reducibility of Spivak normal fibrations of low dimensional Poincaré complexes.
method Analysis of Spivak normal fibrations and their reducibility properties.
result In dimensions less than 4, reducibility always exists; in dimension 4, it exists if orientable.
4D Poincaré complexes have simpler fibrations.
problem Complexity of fibrations in 4D Poincaré complexes.
method Showed Spivak normal fibration reducible.
result Reducible Spivak fibrations in 4D Poincaré complexes.
In this thesis we give a review on Ricci flow, an overview on Poincare conjecture, maximum principle, Li-Yau-Perelman estimate, Two functional F and W of Perelman, Reduced volume and reduced length and k-non collapsing estimate
The paper shows plentiful non-homotopy finite Poincaré duality spaces.
problem The existence of non-homotopy finite Poincaré duality spaces.
method Constructing a finitely dominated Poincaré space with a non-trivial 2-divisible element in the reduced Grothendieck group.
result The existence of finitely dominated Poincaré spaces that are not homotopy finite.
Develops higher-order Euler-Poincaré field equations for principal G-bundles.
problem Formulating field equations for higher-order jet bundles of principal G-bundles.
method Reduction theory applied to G-invariant Lagrangian field theories on jet bundles, transferring Hamilton's principle to reduced configuration bundles. result Higher-order Euler-Poincaré field equations are equivalent to conservation of Noether current.
Extends Poincaré-Lefschetz duality to pairs of ∞-categories.
problem Generalizing Poincaré-Lefschetz duality to ∞-categories.
method Introduces Poincaré duality pairs of ∞-categories and uses them to study various diagrams of spaces.
result Unified treatment of Wall's Poincaré ads and iterated Poincaré cobordisms.
In this paper, we will see that the symplectic creed by Weinstein "everything is a Lagrangian submanifold" also holds for Hamilton-Poincaré and Lagrange-Poincaré reduction. In fact, we show that solutions of the Hamilton-Poincaré equations and of the Lagrange-Poincaré equations are in one-to-one correspondence with dis…
Study Hodge Cousin groups for geometric properties.
problem Geometric properties of Hodge Cousin groups.
method Consider Hodge Cousin groups and abelian Cousin groups, proving a Poincaré complete reducibility theorem.
result Proved an analogue of Poincaré complete reducibility theorem for abelian Cousin groups.
Reduces field theories using Poisson-Poincaré method.
problem Reduction of field theories using Poisson-Poincaré method.
method Poisson-Poincaré reduction for field theories.
result Reduction procedure for field theories.
Reduces symplectic Hamiltonian systems to contact systems, realizing Poincaré's dream.
problem Scaling symmetries in Hamiltonian systems.
method Contact reduction of symplectic Hamiltonian systems.
result Generically possible reduction to contact Hamiltonian systems, reducing inputs needed.
Let π:P→Mn be a principal G-bundle, and let L:J1P→Λn(M) be a G-invariant Lagrangian density. We obtain the Euler-Poincare equations for the reduced Lagrangian l defined on C(P), the bundle of connections on P.
Unique Poincaré type cscK metric with singularity at smooth divisor is unique up to holomorphic transformations.
problem Proving uniqueness of Poincaré type cscK metric with singularity at a smooth divisor.
method Holomorphic transformations, asymptotic behavior analysis, fixed point problem.
result Unique Poincaré type cscK metric with singularity at a smooth divisor is unique up to holomorphic transformations.
We prove the following version of Poincare duality for reduced Lq,p-cohomology: For any 1<q,p<∞, the Lq,p-cohomology of a Riemannian manifold is in duality with the interior Lp′,q′−cohomologyfor1/p+1/p'=1,1/q+1/q'=1$.
Study on 4D Einstein manifolds with Kähler conformal geometry.
problem Exploring 4D Poincaré-Einstein manifolds with Kähler metrics.
method Formulated a Dirichlet boundary value problem and established existence and uniqueness theory.
result Existence and uniqueness of new Poincaré-Einstein metrics.
Let D a divisor with simple normal crossings in a Kahler manifold X. The purpose of this short note is to show that the existence of a Poincare type metric with constant scalar curvature in on the complement of D implies for any component of the divisor that the scalar curvature of Poincare type metric outside of D is …
We establish certain Gaussian type upper bound for the heat kernel of the conjugate heat equation associated with 3 dimensional ancient κ solutions to the Ricci flow. As an application, using the W entropy associated with the heat kernel, we give a different and shorter proof of Perelman's classification of backwar…
Study delta invariant of curves on rational surfaces using topological methods.
problem Calculate delta invariant for curves embedded in rational singularities.
method Use topological techniques and Poincaré series.
result Develop formulae for delta invariant in terms of embedded data.
The aim of this paper is to write explicit expression in terms of a given principal connection of the Lagrange-d'Alembert-Poincarè equations in several stages. This is obtained by using a reduced Lagrange-d'Alembert's Principle in several stages, extending methods introduced for the case of two stages by one of the aut…
We give a short proof of the duality theorem for the reduced Lp-cohomology of a complete oriented Riemannian manifold.
Reduces equations for contact mechanical systems on Lie groups by exploiting symmetries.
problem Contact mechanical systems on Lie groups with symmetries.
method Reduction process using Lie group actions and symmetries.
result Euler-Poincaré-Herglotz equations on the reduced phase space.
Given a smooth positive function f defined on the unit circle satisfying a simple condition, we obtain a Poincaré-type inequality for an arbitrary function u whose weighted average with respect to f is zero. The proof uses Fenchel's theorem about the total curvature of closed space curves in an essential way. Nex…
New method calculates volume-renormalized mass from Hamiltonian perspective.
problem Calculating volume-renormalized mass for asymptotically hyperbolic manifolds.
method Using Michel's mass invariants and a reduced Hamiltonian perspective, the volume-renormalized mass is deduced.
result The reduced Hamiltonian recovers the volume-renormalized mass and its variations.
A generalized Lepage form for second-order Lagrangians is described.
problem Finding a Lepage equivalent for second-order Lagrangians.
method Using equivalence relation and preserving the order of Lagrangians.
result Completes attempts to find a Lepage equivalent for second-order Lagrangians.
We deal with Lagrangian systems that are invariant under the action of a symmetry group. The mechanical connection is a principal connection that is associated to Lagrangians which have a kinetic energy function that is defined by a Riemannian metric. In this paper we extend this notion to arbitrary Lagrangians. We the…
Taking configuration space as a Lie group, the trivialized Euler-Lagrange and Hamilton's equations are obtained and presented as Lagrangian submanifolds of the trivialized Tulczyjew's symplectic space. Euler-Poincaré and Lie-Poisson equations are presented as Lagrangian submanifolds of the reduced Tulczyjew's symplecti…
We consider a simple and natural coboundary operator, on the Lie algebra valued differential forms on a manifold, which in the abelian case reduces to usual exterior derivative of such forms. Using the corresponding de Rham cohomology Lie superalgebra H*(M,G) we obtain numerical smooth invariants--as opposed to homotop…
Study Morse-Novikov cohomology on foliated manifolds and prove Hodge theorem.
problem Understanding cohomology groups on foliated manifolds.
method Defined and studied Morse-Novikov cohomology relative to a foliation, proving homotopy invariance and extending to more general forms.
result Proved Hodge theorem and Poincaré duality for reduced leafwise Morse-Novikov cohomology groups on Riemannian foliations.
For a bounded N-dimensional domain with Lipschitz boundary we extend Korn's first inequality to incompatible tensor fields. For compatible tensor fields our estimate reduces to a non-standard variant of the well known Korn's first inequality. On the other hand, for skew-symmetric tensor fields our new estimate turns to…
Proves Poincaré duality for Hopf algebroids with bijective antipode.
problem Proving Poincaré duality for Hopf algebroids.
method Using twisted Poincaré duality and bijective antipode properties.
result Recovering and extending known Poincaré dualities for Hopf algebroids.
Geometric derivation of quantum dynamics from Lie group actions.
problem Deriving quantum dynamics from geometric principles.
method Euler-Poincaré reduction on adjoint-coupled semidirect products.
result Reproduces the Lindblad equation from geometric reduction.
New spaces help connect manifold structures on equivariant Poincaré spaces.
problem Creating manifold structures on equivariant Poincaré spaces.
method Introducing semifree isovariant G-Poincaré spaces and gap conditions. result Space of isovariant structures on semifree G-Poincaré spaces is highly connected. Proves Poincaré surgery theorem using homotopy theory.
problem Fundamental Theorem of Poincaré surgery in simply connected spaces.
method Homotopy theoretic proof.
result Deduced Poincaré transversality exact sequence.
This paper uses second-order Poincaré inequalities to establish quantitative central limit theorems for Gaussian neural networks.
problem Establishing quantitative central limit theorems for Gaussian neural networks.
method Using second-order Poincaré inequalities to reduce the problem to computing the gradient and Hessian of the NN's output.
result Suboptimal rates of convergence for the NN's output due to the use of second-order Poincaré inequalities.
Uniform Poincaré inequalities established for various metric spaces.
problem Establishing uniform Poincaré inequalities on different metric spaces.
method Proper geodesic metric spaces equipped with a Borel measure. Local Poincaré inequality and volume conditions are used to derive uniform Poincaré inequalities.
result Uniform Poincaré inequalities are established for various metric spaces including hyperbolic spaces and covers of compact spaces.
In this paper, we introduce local expressions for discrete Mechanics. To apply our results simultaneously to several interesting cases, we derive these local expressions in the framework of Lie groupoids, following the program proposed by Alan Weinstein in [19]. To do this, we will need some results on the geometry of …
The paper proves new results on Poincaré duality pairs and spaces.
problem Establishing Poincaré duality in various contexts and dimensions.
method Analyzes Poincaré spaces and CW pairs, proving relative Poincaré duality and related results.
result Found a finite CW pair (X,Y) where Y fails to satisfy Poincaré duality in any dimension. Develops a reduction theory for covariant field theories with gauge symmetries.
problem Handling gauge symmetries in covariant field theories.
method Utilizes generalized principal connections and fiberwise action of Lie groups.
result Relates vertical reduced equations to the Noether theorem.
Proof outlined for 4D smooth Poincaré conjecture.
problem 4-dimensional smooth Poincaré conjecture.
method Outline of proof.
result Proof of 4D smooth Poincaré conjecture.
New extremal Poincaré type metrics found via blow-up theorem.
problem Finding new extremal Poincaré type metrics.
method Proved Arezzo-Pacard-Singer blow-up theorem for Poincaré type metrics and applied to new examples.
result Found new examples of extremal Poincaré type metrics with an additional obstruction.
In this paper we derive the symplectic framework for field theories defined by higher-order Lagrangians. The construction is based on the symplectic reduction of suitable spaces of iterated jets. The possibility of reducing a higher-order system of PDEs to a constrained first-order one, the symplectic structures natura…
The author connects Poincaré embeddings to Reidemeister traces and diagonal maps.
problem Existence of Poincaré embeddings for specific spaces.
method Relates total obstruction to Reidemeister trace and uses Poincaré duality.
result Diagonal maps admit Poincaré embeddings under certain conditions.
A second part of detailed elementary introduction into Khovanov homologies. This part is devoted to reduced Jones superpolynomials. The story is still about a hypercube of resolutions of a link diagram. Each resolution is a collection of non-intersecting cycles, and one associates a 2-dimensional vector space with each…
Uniform Poincaré inequalities on manifolds with bounded Ricci curvature.
problem Establishing uniform Poincaré inequalities on manifolds with specific curvature properties.
method Analyzing manifolds with polynomial growth and bounded Ricci curvature.
result Uniform Poincaré inequalities on manifolds with polynomial growth and bounded Ricci curvature.
Uniform rectifiability proven for sets with Poincaré inequalities.
problem Uniform rectifiability of sets with Poincaré inequalities.
method Weak (1,d)-Poincaré inequality and surface measure. result Uniform rectifiability achieved for sets supporting such inequalities.
New tool: relative Hopf invariant for Poincaré surgery.
problem Non-simply connected Poincaré surgery.
method Relative Hopf invariant in equivariant setting.
result Established Poincaré embedding results in relative setting.
The study improves Poincaré inequalities on hyperbolic space.
problem Improving p-Poincaré inequalities on hyperbolic space. method Investigates and proves several improved inequalities, including a Poincaré-Hardy inequality.
result Proves a Poincaré-Hardy inequality improving the best p-Poincaré inequality. Study shows knots in Poincaré sphere can't be 'cosmetically' altered.
problem Cosmetic surgeries on knots in the Poincaré homology sphere.
method Proof that essential knots cannot undergo purely cosmetic surgeries.
result Essential knots in the Poincaré sphere cannot be 'cosmetically' altered.
Investigates conditions for Poincaré map existence and uniqueness in systems with impulse effects.
problem Existence and uniqueness of Poincaré maps for systems with impulse effects.
method Investigates sufficient conditions for the existence and uniqueness of Poincaré maps for dynamical systems with impulse effects evolving on a differentiable manifold.
result Shows sufficient conditions for the existence and uniqueness of Poincaré maps for systems with impulse effects.