(All times are Lisbon summer time.)
Abstract: I will discuss some open problems of microlocal sheaf theory and D-module theory related to the question of extending well-known result about holomorphic solutions of coherent D-modules to temperate holomorphic solutions.
Abstract: One-parameter persistence is a fundamental tool in topological data analysis that enables the construction of barcodes - combinatorial descriptors that encode the topological properties of point clouds in data. These barcodes are obtained by associating a filtered space to the data and taking its singular cohomology, leading to objects for which there are notions of distances that can be efficiently computed.
However, in many cases, it is necessary to consider multiple filtrations simultaneously, which gives rise to the theory of multi-persistence. In this framework, the objects obtained no longer have simple combinatorial descriptions and are identified as sheaves on a vector space. One of the challenges in multi-persistence theory is constructing invariants and metrics for these objects that are both discriminative and computable.
In this talk, we will explore how these challenges can be tackled using sheaf theory.
Suggestions: FCUL restaurant in building C6. Otherwise, we suggest Restaurante Borges at 10m walking distance (adresse: Rua João Soares, 6).
Abstract: The theory of ind-sheaves, developed by M. Kashiwara and P. Schapira, was one of the key ingredients for a Riemann-Hilbert correspondence for holonomic D-modules with possibly irregular singularities. As such, they also provide a tool for a topological study of Stokes phenomena. In this talk, we will recall and explain the notion of Stokes data in this context and how they classify meromorphic connections on a Riemann surface. We then present some results about Stokes matrices obtained by using sheaf and (enhanced) ind-sheaf theory, and in particular some explicit computations of Fourier transforms, to which the framework of the irregular Riemann-Hilbert correspondence is particularly adapted.
Suggestions: FCUL restaurant in building C6. Otherwise, we suggest Restaurante Borges at 10m walking distance (adresse: Rua João Soares, 6).
Abstract: Euler calculus techniques—integration of constructible functions with respect to the Euler characteristic—have led to important advances in topological data analysis. For instance, Schapira's Radon transform has provided a positive answer to the following question: is a compact subanalytic subset of the Euclidean space determined by the Euler characteristic of its intersections with all affine half-spaces?
In this talk, I will introduce integral transforms combining Lebesgue integration and Euler calculus for constructible functions. On the theoretical side, these objects enjoy invariance and regularity properties, while they appear in practice as efficiently computable vectorisations of weighted simplicial complexes in the form of multivariate continuous functions. Focusing on one example, the Euler-Fourier transform, I will show various examples illustrating that it is strictly more discriminating than its classical analogue. Finally, I will expose that this transform is injective on a subgroup of constructible functions.
Abstract: I would like to explain an equivariant enhancement of the category introduced by Tamarkin. Our category is useful to study symplectic geometry of nonexact Lagrangian and WKB analysis. This talk is partly based on a joint work with Y. Ike.