Filtered Techniques for Noncommutative Rings (GRK 2240 Retreat: Gong Talk)
Summary: We illustrate how to study noncommutative rings via filtrations and their associated graded rings, using the example of the (first) Weyl algebra over the complex numbers.

Notions from p-adic Geometric Representation Theory (Oberseminar Algebra und Geometrie Düsseldorf)
Summary: We give an overview of the notions and constructions arising in the context of geometric representation theory of p-adic groups. The basis principle is the Beilinson–Bernstein Localisation Theorem, which we briefly discuss. We then shift our focus to the local p-adic theory, which constitutes the study of p-adic Weyl algebras, their completions, and their Arens–Michael envelopes. This is the algebro-analytical machinery needed to introduce Ardakov–Wadsley's construction of coadmissible D-modules on rigid spaces. Finally, we summarise some key ideas from Clausen–Scholze's condensed mathematics and how they arise in the context of topological algebra. This includes a discussion of condensed abelian groups, solid abelian groups, and analytic rings.

Adic Spaces  (Oberseminar Algebraic Geometry Düsseldorf: Perfectoid Spaces)
Summary: The first part of the talk is devoted to defining adic spaces. For this, we recall aspects of the theory of topological rings in general, and Huber rings and Huber pairs specifically. We then discuss when the associated affinoid spectrum of a Huber pair gives rise to an affinoid adic space, meaning, when the naturally defined structure presheaf is a sheaf. To close this part of the talk, we discuss the five types of points on the affinoid unit disk. The second part of the talk introduces étale cohomology of adic spaces. We briefly discuss how the classical notion of étale morphism translates to the adic world. As an application, we present some comparison theorems for adic spaces associated to rigid spaces.

Regular Singular Connections and the Riemann–Hilbert Correspondence (Research Seminar Wuppertal: Algebraicity of solutions to differential equations after Katz, Lam–Litt )
Summary: We consider the construction of monodromy representations for the fundamental group of a complex manifold arising as the analytification of a smooth complex variety. For a smooth complex projective variety, we show that associating to an algebraic differential equation its monodromy representation defines an equivalence of categories. For a smooth complex quasi-projective, one has to impose certain regularity conditions to recover an equivalence of this type; this is titular the Riemann–Hilbert Correspondence. We also briefly sketch how one can generalise the Riemann–Hilbert Correspondence to regular (holonomic) D-modules.

l-adic cohomology and Deligne–Lusztig induction (GRK 2240 Workshop: Deligne–Lusztig theory)
Summary: We briefly motivate and introduce l-adic cohomology (with and without compact support) as a tool to study varieties defined over fields of positive characteristic and discuss some computations. For a finite group of Lie type, we then define the Deligne–Lusztig induction and restriction functors for virtual representations. We show that Deligne–Lusztig induction generalises Harish-Chandra induction. Finally, we discuss well-definedness, adjointness, and transitivity of Deligne–Lusztig induction (and restriction).

Analytic Rings (Research Seminar Wuppertal: Condensed Mathematics)
Summary: As a generalisation of the theory of solid abelian groups, we introduce the framework of analytic rings. We consider examples and define, for every analytic ring A, a (derived) category of solid A-modules. If A is commutative, we also define a natural symmetric monoidal structure on this (derived) category. We briefly discuss the incompatibility of the formalism of solid abelian groups with the real numbers.

D-modules (GRK 2240 Retreat: The Theorem of Beilinson–Bernstein)
Summary: We introduce the sheaf of differential operators, their associated D-modules, and discuss the examples of affine and projective space. Using that the global sections of the tangent sheaf on the projective line are isomorphic to the Lie algebra sl_2, we illustrate how to construct the natural map from the universal enveloping algebra of sl_2 to the global sections of the sheaf of differential operators on P^1. This is leads to the Beilinson–Bernstein theorem, in the special case of SL_2.