arXiv research

Examines differential smoothness in a specific skew PBW extension family.

Constructs Fedosov dg manifolds from Lie pairs.

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…

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…

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 …

Proves colored HOMFLYPT polynomial is $q$-holonomic.

Extends quantization theory to mixed polarizations using transverse differential operators.

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

Investigates conditions for Poincaré map existence and uniqueness in systems with impulse effects.

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…

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

Model for assembly map of bordism-invariant functors.

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.

New variational and multisymplectic formulations for soliton equations derived using the inverse map.

We study a generalization of the familiar Poincaré map, first implicitely introduced by N.N. Nekhoroshev in his study of persistence of invariant tori in hamiltonian systems, and discuss some of its properties and applications. In particular, we apply it to study persistence and bifurcation of invariant tori.

Classifies mapping tori of specific groups, generalizing known results.

The paper proves a mapping from a space of holonomy varieties to Teichmüller spaces, with a non-empty discrete intersection.

For M and N closed oriented connected smooth manifolds of the same dimension, we consider the mapping space Map(M,N;f) of continuous maps homotopic to f:M--> N.We show that the evaluation map from the space of maps to the manifold N induces a nontrivial homomorphism on the fundamental group only if the self coincidence…

Differential refinement of homology satisfies Poincaré duality.

We construct a Poincaré section for the horocycle flow on the modular surface $SL(2, \R)/SL(2, \Z)$, and study the associated first return map, which coincides with a transformation (the {\it BCZ map}) defined by Boca-Cobeli-Zaharescu. We classify ergodic invariant measures for this map and prove equidistribution of pe…

On a compact Riemannian manifold with boundary, the absolute and relative cohomology groups appear as certain subspaces of harmonic forms. DeTurck and Gluck showed that these concrete realizations of the cohomology groups decompose into orthogonal subspaces corresponding to cohomology coming from the interior and bound…

We consider two types of minimal Poincaré $4$-complexes. One is defined with respect to the degree $1$-map order. This idea was already present in our previous papers, and more systematically studied later by Hillman. The second type of minimal Poincaré $4$-complexes were introduced by Hambleton, Kreck and Teichner. It…

The normal map of curves is analyzed as a vector field on a cylinder.

The paper constructs quasiconformal mappings in the Heisenberg group.

We study the linearization of the Dirichlet-to-Neumann map for Poincaré-Einstein metrics in even dimensions on an arbitrary compact manifold with boundary. By fixing a suitable gauge, we make the linearized Einstein equation elliptic. In this gauge the linearization of the Dirichlet-to-Neumann map appears as the scatte…

We prove a categorified Poincaré lemma using $\mathsf{A}_\infty$-structures.

Geometric invariants of fold maps and their signatures.

A conformal description of Poincare-Einstein manifolds is developed: these structures are seen to be a special case of a natural weakening of the Einstein condition termed an almost Einstein structure. This is used for two purposes: to shed light on the relationship between the scattering construction of Graham-Zworski…

For a Poincare duality space X and a map X -> B, consider the homotopy fiber product X x^B X. If X is orientable with respect to a multiplicative cohomology theory E, then, after suitably regrading, it is shown that the E-homology of X x^B X has the structure of a graded associative algebra. When X -> B is the diagonal…

In this note, we fill in a gap in the literature by proving that the Teichmueller modular groups (mapping class groups) are not Poincare duality groups and the complexes of curves of surfaces have infinite homotopy type (i.e. are not homotopy equivalent to a finite CW-complex).

Study projective geometry in a C*-algebra using cross ratios and exponential maps.

Study embeddings of manifolds via acyclic maps and surgery.

We introduce a hyperbolic Gauss map into the Poincare disk for any surface in H^2xR with regular vertical projection, and prove that if the surface has constant mean curvature H=1/2, this hyperbolic Gauss map is harmonic. Conversely, we show that every nowhere holomorphic harmonic map from an open simply connected Riem…

Introduces Epstein-Poincaré surfaces for G-oper, generalizing classical construction.

Poincaré VAEs improve hierarchical data representation.

We show that there are no tight nonholomorphic maps from irreducible domains into exceptional codomains, the only exception being the already known tight nonholomorphic maps from the Poincare disc. This follows up on previous work by the first author where this was shown for classical codomains.

Holonomy maps on colored complexes link manifold topology to group theory.

New proof of chain duality for simplicial complexes.

The purpose of this mostly expository paper is to discuss a connection between Nielsen fixed point theory and symplectic Floer homology theory for symplectomorphisms of surface and a calculation of Seidel's symplectic Floer homology for different mapping classes. We also describe symplectic zeta functions and asympltot…

Classifies knots in the Poincaré sphere, using fixed points and folding automata.

Study of Fubini-Study forms on surfaces with punctures.

Establishing criteria for top cell inertness in complexes.

We establish a long exact sequence for Legendrian submanifolds L in P x R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L off of itself. In this sequence, the singular homology H_* maps to linearized contact cohomology CH^* which maps to linearized contact …

Let G be either a finite cyclic group of prime order or S^1. We find new relations between cohomology of a manifold (or a Poincare duality space) M with a G-action on it and cohomology of the fixed point set, M^G. Our main tool is the notion of Poincare duality on the Leray spectral sequence of the map M_G -> BG. We ap…

Proves simplicity of Lyapunov exponents for specific Anosov flows.

Graph cohomology solves symplectic problems in surface mapping groups.