Related papers: The sectorial projection defined from logarithms

We prove the existence and uniqueness of a projectively equivariant symbol map (in the sense of Lecomte and Ovsienko) for the spaces $D_p$ of differential operators transforming p-forms into functions. These results hold over a smooth…

In this note, we consider a Fourier integral operator defined by \begin{align*} T_{\phi,a}f(x) = \int_{\mathbb{R}^{n}}e^{i\phi(x,\xi)}a(x,\xi)\widehat{f} \xi)d\xi, \end{align*}here $a$ is the amplitude, and $\phi$ is the phase. Let…

Quaternionic analysis offers a function theory focused on the concept of $\psi-$hyperholomorphic functions defined as null solutions of the $\psi-$Fueter operator, where $\psi$ is an arbitrary orthogonal base (called structural set) of…

We develop foundations for oriented category theory, an extension of $(\infty,\infty)$-category theory obtained by systematic usage of the Gray tensor product, in order to study lax phenomena in higher category theory. As categorical…

Category theory provides a means through which many far-ranging fields of mathematics can be related by their similar structure. In a paper by Robinson [2], this interconnectivity afforded by categorical perspectives allowed for the…

We prove an index theorem for families of pseudodifferential operators generalizing those studied by C. Callias, N. Anghel and others. Specifically, we consider operators on a manifold with boundary equipped with an asymptotically conic…

One can think of power series or polynomials in one variable, such as $P(x)=2x^3+x+5$, as functors from the category $\mathsf{Set}$ of sets to itself; these are known as polynomial functors. Denote by $\mathsf{Poly}_{\mathsf{Set}}$ the…

The main result of this paper is a recursive description of all decompositions \[ \Delta^+ = \Phi_1 \sqcup \Phi_2 \sqcup \dots \sqcup \Phi_k \] of the positive roots $\Delta^+$ of an arbitrary root system $\Delta$ into a disjoint union of…

We derive a priori bounds for the $\Phi^4$ equation in the full sub-critical regime using Hairer's theory of regularity structures. The equation is formally given by \begin{equation} \label{e}(\partial_t-\Delta)\phi = -\phi^3 + \infty \phi…

We consider a smooth hyper-surface Z of a closed Riemannian manifold X. Let P be the Poisson operator associating to a smooth function on Z its harmonic extension on X\Z. If A is a pseudo-differential operator on X of degree <3, we prove…

We extend the theorem of Hausel and the author from arXiv:2212.11836 that relates equivariant cohomology rings and algebras of functions on zero schemes. This paper combines three separate results. We prove that for a reductive group G…

In this paper we consider the solvability of pseudodifferential operators when the principal symbol vanishes of at least second order at a non-radial involutive manifold $\Sigma_2$. We shall assume that the subprincipal symbol is of…

In this paper we introduce the notion of an operator category and two different models for homotopy theory of $\infty$-operads over an operator category -- one of which extends Lurie's theory of $\infty$-operads, the other of which is…

For an additive category $\mathbf{P}$ we provide an explict construction of a category $\mathcal{Q}( \mathbf{P} )$ whose objects can be thought of as formally representing $\frac{\mathrm{im}( \gamma )}{\mathrm{im}( \rho ) \cap \mathrm{im}(…

To infer a function value on a specific point $x$, it is essential to assign higher weights to the points closer to $x$, which is called local polynomial / multivariable regression. In many practical cases, a limited sample size may ruin…

Given a map $\phi:X\rightarrow Y$ between $F$-analytic manifolds over a local field $F$ of characteristic $0$, we introduce an invariant $\epsilon_{\star}(\phi)$ which quantifies the integrability of pushforwards of smooth compactly…

Let $\mathfrak{F}_m$ be the set of all cuspidal automorphic representations of $\mathrm{GL}_m(\mathbb{A}_{\mathbb{Q}})$, and let $F(s,\boldsymbol{\pi})$ be a polynomial in the derivatives of $L$-functions associated with representations…

Let \theta:\pi_1(R) \to \PSL(2,\C) be a homomorphism of the fundamental group of an oriented, closed surface R of genus exceeding one. We will establish the following theorem. Theorem. Necessary and sufficient for \theta to be the monodromy…

Let $Y$ be a complex algebraic variety, $G \curvearrowright Y$ an action of an algebraic group on $Y$, $U \subseteq Y({\mathbb C})$ a complex submanifold, $\Gamma < G({\mathbb C})$ a discrete, Zariski dense subgroup of $G({\mathbb C})$…

Let Phi be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Phi is a Dirichlet series in r complex variables s_1,...,s_r, initially converging for Re(s_i) sufficiently large, that has meromorphic continuation to…