We describe an $E_k$-cell structure on the free $E_{k+1}$-algebra a point, and more generally how May-Milgram filtration of $\Omega^m \Sigma^m S^{k}$ lifts to $E_{k+m}$-algebra point by iterated pushouts $E_k$-algebras.

We discuss linear algebra of infinite-dimensional vector spaces in terms algebraic (Hamel) bases. As an application we prove the surjectivity a large class partial differential operators with smooth ($\mathcal C^\infty$-coefficients) coefficients, called article \emph{regular}, acting on dual $\mathcal D^*(\Omega)$ space test-functions D(\Omega)$. The guarantees solvability corresponding equati...

Using the Zhu algebra for a certain category of $\mathbb{C}$-graded vertex algebras $V$, we prove that if $V$ is finitely $\Omega$-generated and satisfies suitable grading conditions, then rational, i.e. has semi-simple representation theory, with one dimensional level zero algebra. Here $\Omega$ denotes vectors in are annihilated by lowering real part grading. We apply our result to family ran...

Let $M$ be a smooth manifold. When $\Gamma$ is group acting on the manifold by diffeomorphisms one can define $\Gamma$-co-invariant cohomology of to differential complex $\Omega_c(M)_\Gamma=\mathrm{span}\{\omega-\gamma^*\omega,\;\omega\in\Omega_c(M),\;\gamma\in\Gamma\}.$ For Lie algebra $\mathcal{G}$ $M$, defines $\mathcal{G}$-divergence forms $\mathcal{C}_{\mathcal{G}}(M)=\mathrm{span}\{L_X\om...

Given a domain $\Omega$ in $\mathbb{C}^n$ and collection of test functions $\Psi$ on $\Omega$, we consider the complex-valued $\Psi$-Schur-Agler class associated to pair $(\Omega,\,\Psi)$. In this article, characterize interpolating sequences for Banach algebra which is closed unit ball. When disc $\mathbb{D}$ complex plane $\mathbb{C}$ function includes only identity $\mathbb{D}$, aforemention...

For a manifold $M$ with an integral closed 3-form $\omega$, we construct $PU(H)$-bundle and Lie groupoid over its total space, together curving in the sense of gerbes. If form is non-degenerate, furthermore give natural 2-algebra quasi-isomorphism from observables $(M,\omega)$ to weak symmetries above geometric structure, generalising prequantisation map Kostant Souriau.