Geometry and Analysis

Sub-Riemannian structures of high codimension (greater than one) are rare on 7-manifolds. Until recently, only three such examples were known on any of the homotopy 7-spheres: two on the standard 7-sphere and one on the Gromoll–Meyer exotic sphere. In this talk I will describe new examples of 2-step, codimension-3 sub-Riemannian structures on every homotopy (exotic) 7-sphere.

Let $X$ be a six-dimensional Riemannian manifold with nonnegative sectional curvature that is a Riemannian product of a closed manifold with an Euclidean factor. We prove that every complete, finite index, non-minimal CMC hypersurface immersed in $X$ is compact. This answers affirmatively a question of do Carmo for this class of ambient Riemannian spaces, extending known lower dimensional results.
As a consequence, we complete the classification of two-sided, complete weakly stable CMC hypersurfaces immersed in the space forms of positive curvature in dimension six.
We also show that a complete, finite index CMC hypersurface immersed in the hyperbolic space $\mathbb{H}^6$ with mean curvature $|H|>7$ is compact. This gives a partial answer to a question posed by Chodosh in his survey for the ICM.

Let M be a moduli space of quadratic differentials on the sphere with poles of fixed odd orders. Such a space has a concrete realisation as a space of rational functions. In this talk, I will explain an elementary construction that gives rise to a meromorphic hyper-Kähler metric on the algebraic torus bundle that is the quotient of TM by the natural period lattice. We will meet the theory of isomonodromic deformations, the geometry of Riemann surfaces and, crucially, twistor theory. All of this is motivated by the wall-crossing behaviour of Donaldson-Thomas invariants under the variation of Bridgeland stability conditions but the technical prerequisites to understand this particular construction should be minimal. This talk is partly based on joint work with Maciej Dunajski.

NOTES: Note the unusual room!.
Abstract: Multicomplexes are variants of bicomplexes arising naturally in many geometric, topological and algebraic contexts. I will explain some recent joint work with Joana Cirici and Muriel Livernet  which explores homotopy theories related to the two spectral sequences of a truncated multicomplex.

There are potential applications to the study of homotopy types of almost and generalized complex manifolds.

NOTES: Note the unusual room!.
The Weyl energy on four-manifolds is a geometric functional related to the Chern-Gauss-Bonnet formula. Similarly to Willmore’s functional for surfaces embedded in the three-dimensional Euclidean space, it enjoys conformal invariance properties. We are interested in the interaction of the Weyl’s functionals of two manifolds under the operation of connected sum, showing conditions that decrease the energy.
Such estimates might be useful in understanding compactness properties of minimizing or critical families of metrics.
This is based on a joint project with Matt Gursky and Francesco Malizia.

We aim to introduce and discuss some concepts and results on the stability of minimal submanifolds, mainly when the ambient manifold is a topological sphere or a convex domain of the Euclidean space. In particular, we will talk about a joint work with Alcides de Carvalho and Federico Trinca, in which we showed that a conformally Euclidean manifold with convex boundary does not admit volume-minimizing free boundary minimal submanifolds.

Orbifolds extend the concept of manifolds by allowing singularities that arise in a controlled way from group actions. They naturally appear in many geometric and physical settings, for example, as quotient spaces of symmetries, moduli spaces with isotropy, and as local models of singular spaces in mathematical physics. A fundamental problem in the spectral theory of orbifolds is whether the spectrum of a differential operator uniquely determines the underlying geometric or topological structure. This raises two natural questions:

(1) Can spectral data distinguish orbifolds with singularities from smooth manifolds?

(2) What geometric and topological features of the singular set can be recovered from spectral data?

Using heat invariants of the spectra of the Hodge–Laplace operator, we address these questions and examine how spectral information encodes the singular structure of orbifolds. This talk is based on joint work with Katie Gittins, Carolyn Gordon, Juan Pablo Rossetti, Mary Sandoval, and Liz Stanhope.

In this talk, I will present a result proved in my recent paper arXiv:2502.13580. I will discuss biharmonic maps between (and submanifolds of) conformally compact manifolds, a large class of complete manifolds generalizing hyperbolic space. After an introduction to conformally compact geometry, I will discuss one of the main results of the paper: if S is a properly embedded sub-manifold of a conformally compact manifold (N,h), and moreover S is transverse to the boundary and (N,h) has non-positive curvature, then S must be minimal. This result confirms a conjecture known as the Generalized Chen’s Conjecture, in the conformally compact context.

The Bernstein–Gelfand–Gelfand (BGG) sequences provide a framework for encoding the differential structure of tensor fields with symmetries, generalizing the de Rham complex to higher-order and symmetric tensors. These complexes underlie important examples such as the Calabi complex in differential geometry and the Kröner complex in continuum mechanics, yielding cohomological information and decomposition properties essential for well-posedness in related PDE systems.
We present a systematic procedure to derive new differential complexes from existing ones, including BGG-type sequences, and establish analytic properties. Building on this, we focus on form-valued forms (double forms), which include fields like the metric tensor and curvature tensor. We discuss finite element discretizations of these structures, extending Whitney forms and compatible complexes to canonical discretizations of general symmetric tensors, enabling discrete differential-geometric structures and tensor decompositions in 2D, 3D, and higher dimensions with applications in computational electromagnetics, elasticity, and linearized gravity.

In the last few years, in collaboration with B. Franchi (University of Bologna) and P. Pansu (University of Orsay), we have proved Poincaré and Sobolev inequalities in Heisenberg groups, for  Rumin differential forms. These inequalities can be seen as the analytical version of the well-known topological problem of whether a given closed form is exact. More precisely, one can ask whether a primitive can be upgraded to one that satisfies certain estimates; hence, geometric applications follow.

In the first part of the talk I would like to discuss the validity of these type of integral inequalities  in the Heisenberg groups, that descend from the existence of a fundamental solution of a suitable Hodge-type Laplacian defined by M. Rumin in this setting. The estimates that we obtain are sharp.  In the second part of the talk, I will discuss how we can be extended to general Carnot groups the same approach. The results presented in the second part include also  recent joint work with F. Tripaldi (University of Leeds) and A. Rosa (University of Bologna).