arXiv research

We study the boundary asymptotics of ACH metrics which are formally Einstein. In terms of the partially integrable almost CR structure induced on the boundary at infinity, existence and uniqueness of such formal asymptotic expansions are studied. It is shown that there always exist formal solutions to the Einstein equa…

Engel manifolds show transverse tori can be made to have various formal invariants.

Two cross caps in Euclidean $3$-space are said to be formally isometric if their Taylor expansions of the first fundamental forms coincide by taking a suitable local coordinate system. For a given $C^\infty$ cross cap $f$, we give a method to find all cross caps which are formally isometric to $f$. As an application, w…

Newlander-Nirenberg theorem extended to complex b-manifolds.

We generalize Witten's conjectured formula relating Donaldson and Seiberg-Witten invariants to manifolds of non-simple type, via equivariant localization techniques. This approach does not use the theory of non-abelian monopoles, but works directly on the Donaldson-Witten and Seiberg-Witten moduli spaces. We give a for…

Explores local structure of morphisms and formal submanifolds in formal manifolds theory.

Derives localization formulas in Batalin-Vilkovisky formalism.

Proposes local coordinate frames for improving model performance in complex dynamical systems.

Renormalized volume invariant for knots in 3-sphere computed.

New research limits what GNNs can compute and generalizes their performance.

Proves formal self-adjointness of certain differential operators.

Generalizes holographic method to higher codimension submanifolds.

Motivated by the Turaev-Viro invariant of 3-manifolds, we construct a formal topological invariant of closed, oriented 3-manifolds involving spherical tetrahedra as an application of the asymptotic formula of 6j symbols for the Quantum Enveloping Algebra of sl(2). This invariant can be considered as a spherical version…

The aim of the present paper is to provide an intrinsic investigation of projective changes in Finlser geometry, following the pullback formalism. Various known local results are generalized and other new intrinsic results are obtained. Nontrivial characterizations of projective changes are given. The fundamental proje…

Geometric interpretation of 3-manifold invariants using immersed curves.

Study on Yang-Mills equations on conformally compact manifolds, finding obstructions and asymptotics.

Study on cohomology of singular foliations with localization results.

Power series invariant of hyperbolic 3-manifolds matches knot invariants.

Starting from ideas of Furuta, we develop a general formalism for the construction of cohomotopy invariants associated with a certain class of $S^1$-equivariant non-linear maps between Hilbert bundles. Applied to the Seiberg-Witten map, this formalism yields a new class of cohomotopy Seiberg-Witten invariants which hav…

Let $Γ$ be a finite group acting linearly on a vector space $V$. We compute the Lie algebra cohomology of the Lie algebra of $Γ$-invariant formal vector fields on $V$. We use this computation to define characteristic classes for foliations on orbifolds.

We survey recent results in hermitian integral geometry, i.e. integral geometry on complex vector spaces and complex space forms. We study valuations and curvature measures on complex space forms and describe how the global and local kinematic formulas on such spaces were recently obtained. While the local and global k…

Paper constructs solutions for a class of overdetermined systems.

Study non-formal pseudo-differential operators over formal ones.

Hopf's Umlaufsatz relates the total curvature of a closed immersed plane curve to its rotation number. While the curvature of a curve changes under local deformations, its integral over a closed curve is invariant under regular homotopies. A natural question is whether one can find some non-trivial densities on a curve…

Develops thermodynamic formalism for quasimorphisms on negatively curved spaces.

Using the locally compact abelian group $\BT \times \BZ$, we assign a meromorphic function to each ideal triangulation of a 3-manifold with torus boundary components. The function is invariant under all 2--3 Pachner moves, and thus is a topological invariant of the underlying manifold. If the ideal triangulation has a …

This paper explores how local behavior of meromorphic connections on the projective line determines the global connection.

The loop invariants of Dimofte-Garoufalidis is a formal power series with arithmetically interesting coefficients that conjecturally appears in the asymptotics of the Kashaev invariant of a knot to all orders in $1/N$. We develop methods implemented in SnapPy that compute the first 6 coefficients of the formal power se…

We establish that Hitchin's connection exist for any rigid holomorphic family of Kahler structures on any compact pre-quantizable symplectic manifold which satisfies certain simple topological constraints. Using Toeplitz operators we prove that Hitchin's connection induces a unique formal connection on smooth functions…

The paper formalizes robot environments using topological concepts.

In the first part of this paper we study geometric formality for generalized flag manifolds, including full flag manifolds of exceptional Lie groups. In the second part we deal with the problem of the classification of invariant almost complex structures on generalized flag manifolds using topological methods.

In this paper we suggest a new general formalism for studying the invariants of polyhedra and manifolds comming from the theory of von Neumann algebras. First, we examine generality in which one may apply the construction of the extended abelian category, which was suggested in the previous publications of the author, …

We prove the conjecture by Feigin, Fuchs and Gelfand describing the Lie algebra cohomology of formal vector fields on an $n$-dimensional space with coefficients in symmetric powers of the coadjoint representation. We also compute the cohomology of the Lie algebra of formal vector fields that preserve a given flag at th…

We prove that no nilpotent Lie algebra admits an invariant generalized Kaehler structure. This is done by showing that a certain differential graded algebra associated to a generalized complex manifold is formal in the generalized Kaehler case, while it is never formal for a generalized complex structure on a nilpotent…

We provide examples of homogeneous spaces which are neither symmetric spaces nor real cohomology spheres, yet have the property that every invariant metric is geometrically formal. We also extend the known obstructions to geometric formality to some new classes of homogeneous spaces and of biquotients, and to certain s…

Develops a framework for stuck knots with rigid constraints and invariants.

In this thesis, we study the deformation problem of coisotropic submanifolds in Jacobi manifolds. In particular we attach two algebraic invariants to any coisotropic submanifold $S$ in a Jacobi manifold, namely the $L_\infty[1]$-algebra and the BFV-complex of $S$. Our construction generalizes and unifies analogous cons…

Novel construction of Bauer--Furuta invariant using sheaves of spectra.

Many learning algorithms have invariances: when their training data is transformed in certain ways, the function they learn transforms in a predictable manner. Here we formalize this notion using concepts from the mathematical field of category theory. The invariances that a supervised learning algorithm possesses are …

This text proposes geometrical descriptions of all variational problems invariant by conformal transformations in two variables. First a characterisation in terms of C-Finsler manifolds, a suitable generalization of Finsler manifolds, is given. Second Hamiltonian formalisms are explored, with an emphasis on Caratheodor…

Let X be a pseudomanifold. In this text, we use a simplicial blow-up to define a cochain complex whose cohomology with coefficients in a field, is isomorphic to the intersection cohomology of X, introduced by M. Goresky and R. MacPherson. We do it simplicially in the setting of a filtered version of face sets, also cal…

In this short note we prove an equivariant version of the formality of multidiffirential operators for a proper Lie group action. More precisely, we show that the equivariant Hochschild-Kostant-Rosenberg quasi-isomorphism between the cohomology of the equivariant multidifferential operators and the complex of equivaria…

We derive a complete set of invariants for a formal Bishop surface near a point of complex tangent with a vanishing Bishop invariant under the action of formal transformations. We prove that the modular space of Bishop surfaces with a vanishing Bishop invariant and with a fixed Moser invariant $s<\infty$ is of infinite…

Researchers create new operators from Riemannian invariants.

We show that $\mathfrak{aff}(n)$, the Lie algebra of affine transformations of ${\mathbb R}^n,$ is formally and analytically nondegenerate in the sense of A. Weinstein. This means that every analytic (resp., formal) Poisson structure vanishing at a point with a linear part corresponding to $\mathfrak{aff}(n)$ is locall…

We construct two knot invariants. The first knot invariant is a sum constructed using linking numbers. The second is an invariant of flat knots and is a formal sum of flat knots obtained by smoothing pairs of crossings. This invariant can be used in conjunction with other flat invariants, forming a family of invariants…

A primary goal in this paper is to study the question that asks when a real analytic submanifold $M$ in ${\mathbb{C}}^{n+1}$ bounds a real analytic (up to $M$) Levi-flat hypersurface $\hat{M}$ near $p\in M$ such that $\hat{M}$ is foliated by a family of complex hypersurfaces moving along the normal direction of $M$ at …

We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further,…