arXiv research

Examines differential smoothness in a specific skew PBW extension family.

We prove that to every inclusion $A\hookrightarrow L$ of Lie algebroids over the same base manifold $M$ corresponds a Kapranov dg-manifold structure on $A[1]\oplus L/A$, which is canonical up to isomorphism. As a consequence, $Γ(Λ^\bullet A^\vee\otimes L/A)$ carries a canonical $L_\infty[1]$ algebra structure whose una…

Inspired by the recent work of Chen-Stiénon-Xu on Atiyah classes associated to inclusions of Lie algebroids, we give a very simple criterium (in terms of those classes) for relative Poincaré-Birkhoff-Witt type results to hold. The tools we use (e.g. the first infinitesimal neighbourhood Lie algebroid) are straightforwa…

We prove that the HOMFLYPT polynomial of a link, colored by partitions with a fixed number of rows is a $q$-holonomic function. Specializing to the case of knots colored by a partition with a single row, it proves the existence of an $(a,q)$ super-polynomial of knots in 3-space, as was conjectured by string theorists. …

Given any pair $(L,A)$ of Lie algebroids, we construct a differential graded manifold $(L[1]\oplus L/A,Q)$, which we call Fedosov dg manifold. We prove that the cohomological vector field $Q$ constructed on $L[1]\oplus L/A$ by the Fedosov iteration method arises as a byproduct of the Poincaré--Birkhoff--Witt map establ…

We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of regularizing operators is identified with the smooth algebra of the groupoid, in the …

Extends quantization theory to mixed polarizations using transverse differential operators.

Unique vertical isomorphisms between Fedosov dg manifolds are proven for Lie pairs.

To a closed wide Lie subgroupoid $\mathbf{A}$ of a Lie groupoid $\mathbf{L}$, i.e. a Lie groupoid pair, we associate an Atiyah class which we interpret as the obstruction to the existence of $\mathbf{L}$-invariant fibrewise affine connections on the homogeneous space $\mathbf{L}/\mathbf{A}$. For Lie groupoid pairs with…

Proves Poincaré duality for Hopf algebroids with bijective antipode.

New spaces help connect manifold structures on equivariant Poincaré spaces.

Proves Poincaré surgery theorem using homotopy theory.

Extends Poincaré-Lefschetz duality to pairs of ∞-categories.

Uniform Poincaré inequalities established for various metric spaces.

Proof outlined for 4D smooth Poincaré conjecture.

The author connects Poincaré embeddings to Reidemeister traces and diagonal maps.

The paper shows plentiful non-homotopy finite Poincaré duality spaces.

New tool: relative Hopf invariant for Poincaré surgery.

The paper characterizes Ricci solitons on the Poincaré upper half plane.

The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.

We establish a number of foundational results on Poincaré spaces which result in several applications. One application settles an old conjecture of C.T.C. Wall in the affirmative. Another result shows that for any natural number n, there exists a finite CW pair $(X,Y)$ satisfying relative Poincaré duality in dimension …

A special knot in the Poincaré sphere leads to a unique connected sum of lens spaces.

The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.

Expands Euler-Poincare characteristic to supergeometry.

The study improves Poincaré and log-Sobolev inequalities on hyperbolic spaces.

We prove a version of the Arezzo-Pacard-Singer blow-up theorem in the setting of Poincaré type metrics. We apply this to give new examples of extremal Poincaré type metrics. A key feature is an additional obstruction which has no analogue in the compact case. This condition is conjecturally related to ensuring the metr…

Computed formulas for curvature operators and Poincaré polynomials of symmetric spaces.

Proves a theorem for 3D Poincaré duality pairs.

Optimal Poincaré constant estimates on manifolds with ends.

The paper classifies Poincaré complexes as topological manifolds.

We prove a Poincare type inequality for differential forms on compact manifolds by means of a constructive 'globalization' of a local Poincare inequality on convex sets.

The Poincaré series for surfaces with boundary extends to the complex plane.

The paper removes a condition for the Nielsen realisation problem using equivariant Poincaré duality.

Paper proves rigidity of Poincaré-Einstein manifolds with cylindrical conformal infinities.

The paper studies Fox pairings of Poincaré duality groups using group cohomology.

We prove two kinds of fibering theorems for maps X --> P, where X and P are Poincare spaces. The special case of P = S^1 yields a Poincare duality analogue of the fibering theorem of Browder and Levine.

Researchers describe and compare decompositions of Poincaré duality pairs.

The paper studies degenerate and nonlocal elliptic operators on Poincaré-Einstein manifolds.

Study Poincaré inequality in metric spaces via separating sets.

In this paper, using exclusively homotopy theoretical methods, we study degrees of maps between $(n-2)$-connected $(2n-1)$-dimensional Poincar\' e complexes which have torsion free integral homology. Necessary and sufficient algebraic conditions for the existence of map degrees between such Poincar\' e complexes are es…

Proves energy expression on Poincaré-Einstein spaces.

The study proves rigidity for Poincaré-Einstein manifolds with flat Euclidean conformal infinity.

We describe classes of potential structures (covector fields) on Minkowski space that admit subgroups of the Poincaré group. We describe also seven classes of Maxwell spaces that admit subgroups of the Poincaré group.

Reduces field theories using Poisson-Poincaré method.

Study of Poincaré-Reeb graphs for algebraic domains.

The Poincaré map is widely used to study the qualitative behavior of dynamical systems. For instance, it can be used to describe the existence of periodic solutions. The Poincaré map for dynamical systems with impulse effects was introduced in the last decade and mainly employed to study the existence of limit cycles (…

Study non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.

Using an inverse system of metric graphs as in: J. Cheeger and B. Kleiner, "Inverse limit spaces satisfying a Poincaré inequality", we provide a simple example of a metric space $X$ that admits Poincaré inequalities for a continuum of mutually singular measures.