We use sheaves of spectra to quantize a Hamiltonian $\coprod _n BO(n)$-action on $\varinjlim _{N}T^*\mathbf {R}^N$ that naturally arises from Bott periodicity. We employ the category of correspondences developed by Gaitsgory and Rozenblyum [A study in derived algebraic geometry, vol. I. Correspondences and duality, Mathematical Surveys and Monographs, vol. 221 (American Mathematical Society, 2017)] to give an enrichment of stratified Morse theory by the $J$-homomorphism. This provides a key step in the work of Jin [Microlocal sheaf categories and the $J$-homomorphism, Preprint (2020), arXiv:2004.14270v4] on the proof of a claim of Jin and Treumann [Brane structures in microlocal sheaf theory, J. Topol. 17 (2024), e12325]: the classifying map of the local system of brane structures on an (immersed) exact Lagrangian submanifold $L\subset T^*\mathbf {R}^N$ is given by the composition of the stable Gauss map $L\rightarrow U/O$ and the delooping of the $J$-homomorphism $U/O\rightarrow B\mathrm {Pic}(\mathbf {S})$. We put special emphasis on the functoriality and (symmetric) monoidal structures of the categories involved and, as a byproduct, we produce several concrete constructions of (commutative) algebra/module objects and (right-lax) morphisms between them in the (symmetric) monoidal $(\infty, 2)$-category of correspondences, generalizing the construction out of Segal objects of Gaitsgory and Rozenblyum, which might be of independent interest.

We show a result on propagation of the anisotropic Gabor wave front set for linear operators with a tempered distribution Schwartz kernel. The anisotropic Gabor wave front set is parametrized by a positive parameter relating the space and frequency variables. The anisotropic Gabor wave front set of the Schwartz kernel is assumed to satisfy a graph type criterion. The result is applied to a class of evolution equations that generalizes the Schrödinger equation for the free particle. The Laplacian is replaced by any partial differential operator with constant coefficients, real symbol and order at least two.

The sheaf-function correspondence identifies the group of constructible functions on a real analytic manifold M with the Grothendieck group of constructible sheaves on M. When M is a finite dimensional real vector space, Kashiwara-Schapira have recently introduced the convolution distance between sheaves of $\mathbf {k}$-vector spaces on M. In this paper, we characterize distances on the group of constructible functions on a real finite dimensional vector space that can be controlled by the convolution distance through the sheaf-function correspondence. Our main result asserts that such distances are almost trivial: they vanish as soon as two constructible functions have the same Euler integral. We formulate consequences of our result for Topological Data Analysis: there cannot exist nontrivial additive invariants of persistence modules that are continuous for the interleaving distance.

We study Fourier transforms of regular holonomic ${\mathcal{D}}$-modules. In particular, we show that their solution complexes are monodromic. An application to direct images of some irregular holonomic ${\mathcal{D}}$-modules will be given. Moreover, we give a new proof of the classical theorem of Brylinski and improve it by showing its converse.

In Albano, Bove and Mughetti [J. Funct. Anal. 274(10) (2018), 2725–2753]; Bove and Mughetti [Anal. PDE 10(7) (2017), 1613–1635] it was shown that Treves conjecture for the real analytic hypoellipticity of sums of squares operators does not hold. Models were proposed where the critical points causing a non-analytic regularity might be interpreted as strata. We stress that up to now there is no notion of stratum which could replace the original Treves stratum. In the proposed models such ‘strata’ were non-symplectic analytic submanifolds of the characteristic variety. In this note we modify one of those models in such a way that the critical points are a symplectic submanifold of the characteristic variety while still not being a Treves stratum. We show that the operator is analytic hypoelliptic.

We define the notion of a trace kernel on a manifold $M$. Roughly speaking, it is a sheaf on $M\times M$ for which the formalism of Hochschild homology applies. We associate a microlocal Euler class with such a kernel, a cohomology class with values in the relative dualizing complex of the cotangent bundle ${T}^{\ast } M$ over $M$, and we prove that this class is functorial with respect to the composition of kernels.

This generalizes, unifies and simplifies various results from (relative) index theorems for constructible sheaves, $\mathscr{D}$-modules and elliptic pairs.

Here we define and prove some properties of the semi-classical wavefront set. We also define and study semi-classical Fourier integral operators and prove a generalization of Egorov’s theorem to manifolds of different dimensions.

We prove that the k-truncated microsupport of the specialization of a complex of sheaves F along a submanifold is contained in the normal cone to the conormal bundle along the k-truncated microsupport of F. In the complex case, applying our estimates to , where is a coherent -module, we obtain new estimates for the truncated microsupport of real analytic and hyperfunction solutions. When is regular along Y we also obtain estimates for the truncated microsupport of the holomorphic solutions of the induced system along Y as well as for the nearby-cycle sheaf of when Y is a hypersurface.

We compute the category of perverse sheaves on Hermitian symmetric spaces in types $\text{A}$ and $\text{D}$, constructible with respect to the Schubert stratification. The calculation is microlocal, and uses the action of the Borel group to study the geometry of the conormal variety $\Lambda$.

We estimate the density of resonances close to a critical curve, for strictly convex obstacles with analytic boundary. Contrary to the C∞-case, already treated with Zworski, the estimates are in terms of dynamical quantities. A new feature in the proof is a certain averaging procedure.