Gauge/gravity duality is more than AdS/CFT.  In this talk I will discuss how the holographic dictionary generalises to non-conformal settings, focusing on maximally supersymmetric Yang-Mills theories in diverse dimensions and their Dp-brane supergravity duals. Scaling covariance replaces conformal invariance as the unifying principle on both sides of the duality. On the gravity side, I will show how to systematically organise effective actions and Witten diagram rules for arbitrary correlators of scalar and spin-1 Kaluza-Klein modes. On the field theory side, scale covariance fixes the kinematic structure of 2- and 3-point functions at strong coupling, with the latter admitting closed-form expressions in terms of Appell functions. I will illustrate these results with explicit examples, focussing on 3d MSYM.

Theta operators are weight-shifting differential operators on  automorphic forms. They play an important role in studying congruences between Hecke eigenforms and their p-adic variation. For instance, the classical theta operator, which acts on q-expansions of modular forms as q·(d/dq), is used crucially in Edixhoven’s proof of the weight part of Serre’s conjecture, Katz’s construction of p-adic L-functions over CM fields, and Coleman’s classicality theorem.

Recent years have witnessed extensive works on understanding theta operators over general Shimura varieties, from both geometric and representation-theoretic perspectives. In this talk, I will hint at some aspects of this fascinating area of research. If time permits, I will discuss my ongoing work on overconvergent theta operators over Siegel Shimura varieties.

Topological Tverberg theory seeks r-fold analogues of classical nonembeddability results. Given a simplicial complex K, the central question is whether every continuous map from K into R^d necessarily identifies r points lying on r pairwise disjoint faces of K. The corresponding collection of faces is called a Tverberg r-partition. Perhaps surprisingly, the existence of such partitions depends on the arithmetic properties of r.

The admissible-prescribable conjecture proposes a refinement of this theory by predicting exactly which face dimensions must occur in Tverberg r-partitions. The conjecture has been verified in a number of cases, using tools such as shelling constructions and discrete Morse theory to determine the homotopy type of the relevant configuration spaces.

In this talk, we present counterexamples that settle the remaining open cases and disprove the conjecture in full generality. Our approach combines a diagrammatic description of configuration spaces with techniques from the theory of homotopy colimits of covers, allowing us to equivariantly reduce these spaces. We then show how methods from differential and PL topology, including the r-fold Whitney trick and surgery of intersections techniques, can be employed to construct the desired counterexamples.

This talk is based on forthcoming joint work with Pavle Blagojević and Florian Frick.

The celebrated additive kinematic formula expresses the mean volume of the Minkowski sum of two compact convex subsets of the Euclidean space placed at random. What about non convex subsets? What about other Lie groups than the Euclidean space? In a joint work with Andreas Bernig, we prove additive kinematic formulas for compact subanalytic sets of the Euclidean space and of the 3-sphere. The key is to generalize the Minkowski sum of convex bodies by a notion of convolution of subanalytic sets introduced by Schapira in the late 80s using Euler characteristic computations. The above will of course be an excuse to discuss integral geometric formulas and constructible functions.

Fersztand, Jacquard, Nanda, and Tilmann ('24) introduced the Skyscraper Invariant, a filtration of the classical rank-invariant, for multiparameter persistence modules. It is defined by considering the Harder-Narasimhan (HN) filtration of the module along a special set of stability conditions.

This talk will begin with a post-hoc motivation for considering stability conditions on persistence modules. To compute an approximation of the Skyscraper Invariant we present a technique which, exploiting the geometry of low-dimensional bifiltrations, lets us perform a brute-force computation. We compare it against Cheng's algorithm [Cheng24] which can compute HN filtrations of arbitrary acyclic quiver representations in polynomial time in the total dimension.

To avoid unnecessary recomputation in our algorithm, we ask for which stability conditions the HN filtrations are equivalent. This partition of the space of stabililty conditions is called the wall-and-chamber structure. We show that for a finitely presented d-parameter module it is given by the lower envelopes of a set of multilinear polynomials of degree d-1. For d=2 it is then easy to compute this, enabling a faster algorithm to compute the Skyscraper Invariant up to arbitrary accuracy. As a proof of concept for data analysis, we use it to compute a filtered version of the Multiparameter Landscape for large modules from real world data.