Archives for September 2019

Mélanie Theillière (Université Claude Bernard Lyon 1) will speak in the geometry seminar on Tuesday 19 November 2019 at 11am in the Salle de Profs. Melanie’s title is Convex Integration without Integration and her abstract is below.

The Convex Integration Theory was developped by Gromov in 70s. This theory allows to solve differential constraints seen as subsets of the jet space and called Differential Relations. In the case of a relation of order 1, it allows to build a solution F from a section (x,f(x),L(x)) of the bundle J^1(M,W) -> M whose image lies inside the differential relation using an iteration of suitable integrations called “Convex Integrations”. Recently this theory led to explicit constructions of C^1-isometric embeddings. In this talk, we will propose an alternative formula to the Convex Integrations called Corrugation Process and we will introduce a kind of differential relation that we call Kuiper relation. For these relations, the formula is greatly simplified. As an application of this result, we will give an idea of the construction of a new immersion of RP^2 and we will state a Nash-Kuiper C^1-isometric embedding theorem in the case of totally real applications.

Dmytro Yeroshkin (ULB) will speak in the geometry seminar on Tuesday 1 October 2019 at 11am in the Salle de Profs. Dmytro’s title is Manifolds with Density and Measure and his abstract is below.

Riemannian manifolds with measure, and more general metric spaces with measure have been studied under different guises for decades. In this talk I’ll present a new geometric approach to these spaces that introduces a torsion free affine connection, as opposed to a measure, as the fundamental object of study. This approach allows us to connect the notion of a manifold with measure and the Bakry-Emery notion of manifold with density. This allows us to extend the work on comparison results to the new case of negative synthetic dimension, as well as reveals a number of new structures for future study. This is joint work with Will Wylie and Lee Kennard.

Benjamin Sibley (ULB) will speak in the geometry seminar on Tuesday 15 October 2019 at 11am in the Salle de Profs. Ben’s title is Compactifcations of Hermitian-Yang-Mills moduli space and the Yang-Mills flow on projective manifolds and his abstract is below.

One of the cornerstones of gauge theory and complex geometry in the late 20th century was the so-called “Kobayashi-Hitchin correspondence”, which provides a link between Hermitian-Yang-Mills connections (gauge theory) and stable holomorphic structures (complex geometry) on a vector bundle over projective (or merely Kähler) manifold. On the one hand, this gives an identification of (non-compact) moduli spaces. On the other, one proof of the correspondence goes through a natural parabolic (up to gauge) flow called Yang-Mills flow. Namely, Donaldson proved the convergence of this flow to an Hermitian-Yang-Mills connection in the case that the initial holomorphic structure is stable. Two questions that this leaves open are: 1. Do the moduli spaces admit compactifications, and if so what sort of structure do they have? Are they for example complex spaces? Complex projective? What is the relationship between the compactifications on each side? 2. What is the behaviour of the flow at infinity in the case when the initial holomorphic structure is unstable? I will touch on aspects of my previous work on these problems and explain how they connect up with each other. This work is spread out over several papers, and is partly joint work with Richard Wentworth, and with Daniel Greb, Matei Toma, and Richard Wentworth.

Nicolas Marque (Université Paris 7 Diderot) will speak in the geometry seminar on Tuesday 8 October 2019 at 11am in the Salle de Profs. Nicolas’ title is Bubbling phenomenons for Willmore surfaces and his abstract is below.

The Willmore energy arises naturally as a measure of how curved an immersed surface in $\mathbb{R}^3$ is, with applications in relativity (the Hawking mass). Willmore immersions are critical points of this energy. We will study sequences of Willmore surfaces, which are subject to concentration-compactness phenomenons i.e. : bubbling phenomenons. We will focus on simple minimal bubbles, and detail consequences on the compactness below certain thresholds.