The Geometry and Analysis seminar is the main seminar of the Leeds Geometry and Analysis Group. Unless otherwise stated, seminars take place on Wednesdays at 15:00. The seminar is organised by Ben Lambert and Francesca Tripaldi.
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).
NOTES: Unusual time.
I will revisit some of the intriguing facts supporting the viewpoint that vortices quantise as fermionic (rather than bosonic) particles. One such fact is an analogue of Sen’s conjecture, for which a complete proof exists in the simplest setting of vortices in line bundles.
In a Riemannian manifold, the first filling function $F(l)$ measures the least area needed to fill all null-homotopic loops of length at most $l$ using minimal discs. In this talk I will focus on the first filling function for large loops in Riemannian Lie groups. This function is also known as the Dehn function and quantifies the complexity of the word problem from combinatorial group theory. I will review results of Cornulier and Tessera showing it is either exponential or polynomially bounded, and discuss some recent progress on the problem of estimating the degree of polynomial growth when it is polynomially bounded. This is joint work with Ido Grayevsky (Bristol).
I will begin by introducing the framework of algebraic quantum field theory (AQFT), including its formulation on curved spacetimes, and outline how measurements can be described within this setting following the approach of Fewster and Verch.
Within this framework, I will show that the presence of symmetries necessitates the use of quantum reference frames in order to distinguish between measurements related by symmetry transformations. Incorporating quantum reference frames has nontrivial implications for the structure of the observable algebras that arise, shedding light on recent observations by Chandrasekaran, Longo, Penington, and Witten in the context of QFT on de Sitter spacetime. The talk will be primarily based on my joint work with Fewster, Janssen, Loveridge, and Waldron, “Quantum reference frames, measurement schemes and the type of local algebras in quantum field theory.”
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.
The existence of positive solutions for the Schrödinger equation with a prescribed L2 norm has been a subject of great interest in recent years. While the problem is well-understood in the whole Euclidean space, very little is known concerning the existence of solutions in more general domains. In this talk, we will present some recent results on the existence of solutions in exterior domains with a mass supercritical nonlinearity. This is a joint work with Prof. Riccardo Molle.
I will discuss the geometry of CMC surfaces embedded in Euclidean space with finite genus and, among other things, prove a local area estimate for such surfaces. This is joint work with Bill Meeks.
Riemannian structures of limited regularity arise naturally in the realm of geometric PDEs, especially in relation to physical models. Since the work of Sabitov-Shefel and De Turck-Kazdan in the late seventies, it is well known that the optimal regularity of a Riemannian structure is governed by that of the Ricci tensor in harmonic coordinates. In this talk, we will discuss the conformal analogue problem: we consider a closed 3-dimensional Riemannian manifold (M, g) with g in the Sobolev class W^{2, q} with q > 3 and show that the optimal regularity of the conformal structure is governed by that of the conformally invariant Cotton tensor.
The proof requires the resolution of the renowned Yamabe problem for W^{2, q} metrics, which is interesting on its own. In turn, the resolution of the Yamabe problem relies on new results on the existence, regularity and expansion of the Conformal Green's function for W^{2, q} metrics, which we provide in any dimension n \geq 3 and for q>n/2.
This is based on joint work with R. Avalos and A. Royo Abrego.
The goal of the talk is to give an overview of the metric theory of currents by Ambrosio-Kirchheim, together with some recent progress. Metric currents are a generalization to the metric setting of classical currents. Classical currents are the natural generalization of oriented submanifolds, as distributions play the same role for functions. We present a structure result for metric 1-currents as superposition of 1-rectifiable sets in complete and separable metric spaces, which generalizes a previous result by Schioppa. This is based on an approximation result of metric 1-currents with normal 1-currents and a more refined analysis in the Banach space setting. This is joint work with D. Bate, J. Takáč, P. Valentine, and P. Wald (Warwick).
