arXiv research

Homotopy theory applied to singular foliations leads to new results.

Introduces relative stability conditions on triangulated categories.

A Lie 2-group $G$ is a category internal to the category of Lie groups. Consequently it is a monoidal category and a Lie groupoid. The Lie groupoid structure on $G$ gives rise to the Lie 2-algebra $\mathbb{X}(G)$ of multiplicative vector fields, see (Berwick-Evans -- Lerman). The monoidal structure on $G$ gives rise to…

For a Lie group $G$ and a vector bundle $E$ we study those actions of the Lie group $TG$ on $E$ for which the action map $TG\times E \to E$ is a morphism of vector bundles, and call those \emph{affine actions}. We prove that the category $\mathrm{Vect}_{TG}^{\mathrm{aff}}\left(X\right)$ of such actions over a fixed $G$…

We study natural bases for two constructions of the irreducible representation of the symmetric group corresponding to $[n,n,n]$: the {\em reduced web} basis associated to Kuperberg's combinatorial description of the spider category; and the {\em left cell basis} for the left cell construction of Kazhdan and Lusztig. I…

The paper connects sectional category and parametrized Borsuk-Ulam property for fibrations.

New 3D TQFTs derived from non-semisimple categories.

Fine shape theory extends strong shape to noncompact metrizable spaces.

New representations of Lie algebras via monoidal category actions.

Cube category simplifies set modeling.

The quotient $L/A[-1]$ of a pair $A\hookrightarrow L$ of Lie algebroids is a Lie algebra object in the derived category $D^b(\mathscr{A})$ of the category $\mathscr{A}$ of left $\mathcal{U}(A)$-modules, the Atiyah class $α_{L/A}$ being its Lie bracket. In this note, we describe the universal enveloping algebra of the L…

Study Drinfeld centralizers and Rouquier complexes in homotopy categories.

Functor connects 4D 2-handlebodies to ribbon categories, detecting non-deformation diffeomorphisms.

Study Nijenhuis operators and their linearization problem using left-symmetric algebras.

This paper introduces tangent display maps to simplify tangent category theory.

We prove that the forgetful functor from groupoids to pregroupoids has a left adjoint, with the front adjunction injective. Thus we get an enveloping groupoid for any pregroupoid. We prove that the category of torsors is equivalent to that of pregroupoids. Hence we also get enveloping groupoids for torsors, and for pri…

Language models allocate information storage, not collapsing into uniform representations.

In 2001, Khovanov and Seidel constructed a faithful action of the (m+1)-strand braid group on the derived category of left modules over a quiver algebra, A_m. We interpret the Hochschild homology of the Khovanov-Seidel braid invariant as a direct summand of the sutured Khovanov homology of the annular braid closure.

Solves a challenging case of Nijenhuis operator linearization in 2D.

We present a construction of a 2-Hilbert space of sections of a bundle gerbe, a suitable candidate for a prequantum 2-Hilbert space in higher geometric quantisation. We introduce a direct sum on the morphism categories in the 2-category of bundle gerbes and show that these categories are cartesian monoidal and abelian.…

The aim of this note is to introduce the notion of a $\operatorname{D}$-Lie algebra and to prove some elementary properties of $\operatorname{D}$-Lie algebras, the category of $\operatorname{D}$-Lie algebras, the category of modules on a $\operatorname{D}$-Lie algebra and extensions of $\operatorname{D}$-Lie algebras. …

New geometric variant of factorization homology for conformally flat manifolds.

Develops a category-theoretic approach to interpret conformal prediction.

We present an invariant of connected and oriented closed 3-manifolds based on a coribbon Weak Hopf Algebra H with a suitable left-integral. Our invariant can be understood as the generalization to Weak Hopf Algebras of the Hennings-Kauffman-Radford evaluation of an unoriented framed link using a dual quantum-trace. Thi…

Smooth structures on diffeological spaces and sheaves, resolving conjectures.

Relates two types of skein algebras using explicit correspondences.

In this paper we elaborate a general homotopy-theoretic framework in which to study problems of descent and completion and of their duals, codescent and cocompletion. Our approach to homotopic (co)descent and to derived (co)completion can be viewed as $\infty$-category-theoretic, as our framework is constructed in the …

A bottom tangle is a tangle in a cube consisting only of arc components, each of which has the two endpoints on the bottom line of the cube, placed next to each other. We introduce a subcategory B of the category of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite set of generators of…

We show that, associated with any complex root of unity $ω$, there exists a particularly simple 4d-TQFT model $M_ω$ defined on the cobordism category of Delta complexes. For an oriented closed 4-manifold $X$ of Euler characteristic $χ(X)$, it is conjectured that the quantity $N^{3χ(X)/2}M_ω(X)$, where $N$ is the order …

A celebrated theorem of Kapranov states that the Atiyah class of the tangent bundle of a complex manifold $X$ makes $T_X[-1]$ into a Lie algebra object in $D^+(X)$, the bounded below derived category of coherent sheaves on $X$. Furthermore Kapranov proved that, for a Kähler manifold $X$, the Dolbeault resolution $Ω^{\b…

Hyperelastic bodies in Riemannian manifolds can levitate due to curvature-induced forces.

Classifies contact seaweeds based on their algebraic properties.

We complete our recent classification of compact inner symmetric spaces with weakly complex tangent bundle by filling up a case which was left open, and extend this classification to the larger category of compact homogeneous spaces with positive Euler characteristic. We show that a simply connected compact equal rank …

Proves HNN extensions of nilpotent groups are left-orderable, constructs non-left-orderable examples.

We present the characterization of metric spaces that are micro-, macro- or bi-uniformly equivalent to the extended Cantor set $\{\sum_{i=-n}^\infty\frac{2x_i}{3^i}:n\in\IN ,\;(x_i)_{i\in\IZ}\in\{0,1\}^\IZ\}\subset\IR$, which is bi-uniformly equivalent to the Cantor bi-cube $2^{<\IZ}=\{(x_i)_{i\in\IZ}\in \{0,1\}^\IZ:\e…

Studies amenable category's monotonicity and its relation to topological complexity.

To each oriented surface S, we associate a differential graded category Ko(S). The homotopy category Ho(Ko(S)) is a triangulated category which satisfies properties akin to those of the contact categories studied by K. Honda. These categories are also related to the algebraic contact categories of Y. Tian and to the bo…

Linear ODEs are solved by geodesics in hyperbolic geometry.

Innovative 2-categories create 4-manifold invariants.

Formulates a new connection between topological and geometric categories.

Classifies 85 tie knots into mathematical categories.

We continue the program of structural differential geometry that begins with the notion of a tangent category, an axiomatization of structural aspects of the tangent functor on the category of smooth manifolds. In classical geometry, having an affine structure on a manifold is equivalent to having a flat torsion-free c…

Survey on decorated marked surfaces for Calabi-Yau categories.

The paper revisits the $σ_k$-Yamabe problem and proves the existence of a conformal metric with constant $σ_2$-scalar curvature.

The paper classifies symplectic invariants of specific singularities in integrable Hamiltonian systems.

In this paper, we introduce a notion of a left-symmetric algebroid, which is a generalization of a left-symmetric algebra from a vector space to a vector bundle. The left multiplication gives rise to a representation of the corresponding sub-adjacent Lie algebroid. We construct left-symmetric algebroids from $\mathcal …

The paper defines new TQFTs from non-semisimple categories and proves spherical categories are chromatic.

In this paper, we introduce the notion of a left-symmetric bialgebroid as a geometric generalization of a left-symmetric bialgebra and construct a left-symmetric bialgebroid from a pseudo-Hessian manifold. We also introduce the notion of a Manin triple for left-symmetric algebroids, which is equivalent to a left-symmet…