Calculating the spectrum of the Laplace-Beltrami operator is a fundamental problem in geometric analysis, explicitly solvable mainly for normal homogeneous spaces. This talk addresses the broader class of compact naturally reductive homogeneous spaces. We present a Freudenthal-type formula for the Laplacian's spectrum in this setting. This framework provides a powerful tool to analyze how the spectrum behaves under metric deformations, particularly for canonical variations of normal homogeneous metrics. Applications will demonstrate how these methods, particularly when combined with refined branching techniques, lead towards an understanding of the spectrum for $3$-$(\alpha,\delta)$-Sasaki manifolds. This is joint work with Ilka Agricola.

Biharmonic and conformal biharmonic maps between Riemannian manifolds are two fourth order generalizations of the well-studied harmonic map equation. Since both equations arise as critical points of a non-coercive energy functional and due to the large number of derivatives it is a challenging task to obtain existence theorems for the latter. In the first part of the talk we will provide an introduction to both biharmonic and conformal biharmonic maps and highlight some of the key results that have been established up to now. Finally, we will focus on maps to the Euclidean sphere and will present a number of recent existence results.

A (2,3,5) distribution is a maximally non-integrable tangent 2-plane field on a 5-manifold. These distributions are also known as generic rank two distributions in dimension five. They can equivalently be characterized as regular normal parabolic geometries of type $(G_2,P)$ where $G_2$ denotes the split real form of the exceptional Lie group and P is a particular parabolic subgroup. The Rumin complex associated with a (2,3,5) distribution is a Rockland complex, the analogue of an elliptic complex in the Heisenberg calculus. In this talk we focus on the eta invariant of the Rumin differential in middle degrees, twisted by unitary flat vector bundles. Conjecturally, this spectral invariant coincides with the eta invariant of the (Riemannian) odd signature operator. We present recent computations using harmonic analysis which show that this conjecture holds true for (2,3,5) nilmanifolds.

In this talk, we examine the Yamabe problem for rough Riemannian metrics with limited Sobolev regularity, specifically establishing a solution on closed 3-manifolds for metrics in the Sobolev class $W^{2,q}$ with $q>3$. This analysis is motivated by the growing interest in low-regularity aspects of scalar curvature, including recent developments in low-regularity positive mass-type theorems. Moreover, conformal deformations of scalar curvature in low-regularity settings have become central in the study of rough initial data for the Einstein equations, driven by both physical considerations and analytical conjectures regarding the evolution of realistic space-times. In this rough setting — particularly for Yamabe positive metrics — the Yamabe problem requires developing new elliptic theory for the conformal Laplacian, including a fine blow-up analysis of its Green function. The aim of this talk is to motivate, contextualize, and present these results. Most of the analytical work applies in dimensions $n\geq3$ for metrics in $W^{2,q}$ with $q>n/2$, and is expected to be of independent interest. Time permitting, we will also discuss applications to a broader low-regularity program for conformally covariant geometric equations.

The Standard Model (SM) is one of the greatest successes of modern theoretical physics. Despite this, mathematical references studying the full SM (rather than just its sectors in isolation) on curved spacetimes are somewhat scarce, even though it seems important to understand the theory at a classical level in a more geometric setting. The talk will start with a brief review of the mathematical structure of the SM Lagrangian, the corresponding Euler-Lagrange equations, and some of their basic properties, particularly related to conformality. Then, a global existence result for the SM equations on four-dimensional spacetimes of expanding type will be presented. The talk is focused on a geometrically intrinsic approach to the theory, and the main ingredient for the proof is a gauge-invariant energy estimate.

The Dirac-Yang-Mills functional constitutes a part of the action functional of the standard model of particle physics and the associated Euler-Lagrange equations are known as the Dirac-Yang-Mills equations. On its own, it describes the interaction of a force-field with a (massless) matter field modelled as a section of a certain spinor bundle. Mathematically, it can be described as a variational problem involving fibre bundles over a base Riemannian spin manifold. In the talk we will focus on the Dirac-Yang-Mills equations considered on a closed Riemannian spin manifold. As the functional is unbounded in both directions, showing existence of critical points is a challenging problem. However, we will see how one can apply the tools of analytic perturbation theory to give a characterisation of when the Euler-Lagrange equations decouple into the Yang-Mills and Dirac equations. These equations are much easier to treat and have been separately studied in the Riemannian setting. We will conclude with an application of the Atiyah-Singer index theorem to yield existence results for uncoupled solutions on closed 4-manifolds admitting positive scalar curvature metrics.