University of Notre Dame 

Title:  Manifolds with Ricci bounded below in the spectral sense.

Curvature is one of the central concepts in geometry. Over the last century, a vast body of research has developed around manifolds with lower bounds on curvature, particularly on the Ricci curvature tensor.

In this talk, I will present results on smooth complete manifolds where the first eigenvalue of the operator -γΔ + Ric, for γ > 0, is bounded below. Here, Ric denotes the lowest eigenvalue of the Ricci tensor. This condition is weaker than a pointwise lower bound on Ricci curvature.

I will explore spectral analogues of the Bishop-Gromov volume comparison, the Bonnet-Myers theorem, and the Cheeger-Gromoll splitting theorem; and discuss the relevance of this study in the recent solution of the stable Bernstein problem in R^6. If time permits, I will conclude with possible future directions and open problems.

The material comes from collaborations with M. Pozzetta and K. Xu. 

University of Warwick

Title: Lipschitz differentiation without Poincaré inequalities.

Cheeger’s seminal 1999 paper initiated the study of metric measure spaces that admit a generalised differentiable structure. In such spaces, Lipschitz functions—real-valued and, in some cases, Banach-valued—are differentiable almost everywhere. Since then, much work has gone into determining the precise geometric and analytic conditions under which such structures exist.

The work of Bate-Li and Eriksson-Bique constitutes a major step in this direction. They prove that differentiation into every Banach space (with RNP) is possible only if the space supports suitable Poincaré inequalities. However, such a condition is not necessary for differentiation into some specific targets, such as Hilbert space, as shown by a construction of Schioppa.

In this talk, I will give a brief overview of the theory and present new results from joint work with David Bate. We develop a general procedure to construct new examples with properties akin to Schioppa's and investigate the Banach targets for which differentiation is possible. Our results provide a first step towards a geometric understanding of differentiation in the absence of Poincaré inequalities.

University of Trento

Title: Low-dimensional quantitative rectifiability in Heisenberg groups.

Starting from the foundational work of David and Semmes at the end of the last century, the study of quantitative (or uniform) rectifiability has seen significant development.

In this talk, we explore sufficient and necessary conditions for quantitative rectifiability of low-dimensional sets in Heisenberg groups, equipped with a left-invariant non-Euclidean metric. These assumptions are formulated in terms of geometric lemmas for suitable flatness coefficients—the stratified β-numbers—which quantify the approximation of a set by horizontal planes at every point and scale.

Based on joint works with Yibo Chen, Katrin Fässler, and Andrea Pinamonti.