Abstract:
In this paper we develop the foundations for microlocal analysis on supermanifolds. Making use of pseudodifferential operators on supermanifolds as introduced by Rempel and Schmitt, we define a suitable notion of super wavefront set for superdistributions which generalizes Dencker's polarization sets for vector-valued distributions to supergeometry. In particular, our super wavefront sets detect polarization information of the singularities of superdistributions. We prove a refined pullback theorem for superdistributions along supermanifold morphisms, which as a special case establishes criteria when two superdistributions may be multiplied. As an application of our framework, we study the singularities of distributional solutions of a supersymmetric field theory.

Abstract:
The wavefront set provides a precise description of the singularities of a distribution. Because of its ability to control the product of distributions, the wavefront set was a key element of recent progress in renormalized quantum field theory in curved spacetime, quantum gravity, the discussion of time machines or quantum energy inequalitites. However, the wavefront set is a somewhat subtle concept whose standard definition is not easy to grasp. This paper is a step by step introduction to the wavefront set, with examples and motivation. Many different definitions and new interpretations of the wavefront set are presented. Some of them involve a Radon transform.

Abstract:
In recent years directional multiscale transformations like the curvelet- or shearlet transformation have gained considerable attention. The reason for this is that these transforms are - unlike more traditional transforms like wavelets - able to efficiently handle data with features along edges. The main result in [G. Kutyniok, D. Labate. Resolution of the Wavefront Set using continuous Shearlets, Trans. AMS 361 (2009), 2719-2754] confirming this property for shearlets is due to Kutyniok and Labate where it is shown that for very special functions $\psi$ with frequency support in a compact conical wegde the decay rate of the shearlet coefficients of a tempered distribution $f$ with respect to the shearlet $\psi$ can resolve the Wavefront Set of $f$. We demonstrate that the same result can be verified under much weaker assumptions on $\psi$, namely to possess sufficiently many anisotropic vanishing moments. We also show how to build frames for $L^2(\mathbb{R}^2)$ from any such function. To prove our statements we develop a new approach based on an adaption of the Radon transform to the shearlet structure.

Abstract:
It is known that the continuous wavelet transform of a function $f$ decays very rapidly near the points where $f$ is smooth, while it decays slowly near the irregular points. This property allows one to precisely identify the singular support of $f$. However, the continuous wavelet transform is unable to provide additional information about the geometry of the singular points. In this paper, we introduce a new transform for functions and distributions on $\R^2$, called the Continuous Shearlet Transform. This is defined by $\mathcal{S}\mathcal{H}_f(a,s,t) = \ip{f}{\psi_{ast}}$, where the analyzing elements $\psi_{ast}$ are dilated and translated copies of a single generating function $\psi$ and, thus, they form an affine system. The resulting continuous shearlets $\psi_{ast}$ are smooth functions at continuous scales $a >0$, locations $t \in \R^2$ and oriented along lines of slope $s \in \R$ in the frequency domain. The Continuous Shearlet Transform transform is able to identify not only the location of the singular points of a distribution $f$, but also the orientation of their distributed singularities. As a result, we can use this transform to exactly characterize the wavefront set of $f$.

Abstract:
We define and prove some properties of the semi-classical wavefront set. We also define and study semi-classical Fourier integral operators, of which we give a complete characterization. Lastly, we prove a generalization of the semi-classical Egorov's Theorem to manifolds of unequal dimensions.

Abstract:
A Maslov cycle is a singular variety in the lagrangian grassmannian L(V) of a symplectic vector space V consisting of all lagrangian subspaces having nonzero intersection with a fixed one. Givental has shown that a Maslov cycle is a Legendre singularity, i.e. the projection of a smooth conic lagrangian submanifold S in the cotangent bundle of L(V). We show here that S is the wavefront set of a Fourier integral distribution which is "evaluation at 0 of the quantizations".

Abstract:
We consider the problem of characterizing the wavefront set of a tempered distribution $u\in\mathcal{S}'(\mathbb{R}^{d})$ in terms of its continuous wavelet transform, where the latter is defined with respect to a suitably chosen dilation group $H\subset{\rm GL}(\mathbb{R}^{d})$. In this paper we develop a comprehensive and unified approach that allows to establish characterizations of the wavefront set in terms of rapid coefficient decay, for a large variety of dilation groups. For this purpose, we introduce two technical conditions on the dual action of the group $H$, called microlocal admissibilty and (weak) cone approximation property. Essentially, microlocal admissibilty sets up a systematical relationship between the scales in a wavelet dilated by $h\in H$ on one side, and the matrix norm of $h$ on the other side. The (weak) cone approximation property describes the ability of the wavelet system to adapt its frequency-side localization to arbitrary frequency cones. Together, microlocal admissibility and the weak cone approximation property allow the characterization of points in the wavefront set using multiple wavelets. Replacing the weak cone approximation by its stronger counterpart gives access to single wavelet characterizations. We illustrate the scope of our results by discussing -- in any dimension $d\ge2$ -- the similitude, diagonal and shearlet dilation groups, for which we verify the pertinent conditions. As a result, similitude and diagonal groups can be employed for multiple wavelet characterizations, whereas for the shearlet groups a single wavelet suffices. In particular, the shearlet characterization (previously only established for $d=2$) holds in arbitrary dimensions.

Abstract:
The superbosonisation identity of Littelmann-Sommers-Zirnbauer is a new tool to study universality of random matrix ensembles via supersymmetry, which is applicable to non-Gaussian invariant distributions. In this note, we identify the right-hand side with a super-generalisation of the Riesz distribution. Using the Laplace transformation and tools from harmonic superanalysis, we give a short and conceptual new proof of the formula.

Abstract:
By using a particular class of directional wavelets (namely, the conical wavelets, which are wavelets strictly supported in a proper convex cone in the -space of frequencies), in this paper, it is shown that a tempered distribution is obtained as a finite sum of boundary values of analytic functions arising from the complexification of the translational parameter of the wavelet transform. Moreover, we show that for a given distribution ∈(？), the continuous wavelet transform of with respect to a conical wavelet is defined in such a way that the directional wavelet transform of yields a function on phase space whose high-frequency singularities are precisely the elements in the analytic wavefront set of .

Abstract:
The space $D'_\Gamma$ of distributions having their wavefront sets in a closed cone $\Gamma$ has become important in physics because of its role in the formulation of quantum field theory in curved space time. In this paper, the topological and bornological properties of $D'_\Gamma$ and its dual $E'_\Lambda$ are investigated. It is found that $D'_\Gamma$ is a nuclear, semi-reflexive and semi-Montel complete normal space of distributions. Its strong dual $E'_\Lambda$ is a nuclear, barrelled and bornological normal space of distributions which, however, is not even sequentially complete. Concrete rules are given to determine whether a distribution belongs to $D'_\Gamma$, whether a sequence converges in $D'_\Gamma$ and whether a set of distributions is bounded in $D'_\Gamma$.