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.

Yann Rollin (Nantes University) will speak in the geometry seminar on Tuesday 18 February 2020 at 11am in the Salle de Profs. Yann’s title is Construction of polyhedral Lagrangian surfaces and moment maps and his abstract is below.

Lagrangian surfaces are important objects in symplectic topology. However polyhedral aspects of symplectic geometry are far from being understood.
I will present several methods for constructing polyhedral Lagrangian surfaces in R^4 and more generally isotropic polyhedral surfaces in higher dimensions, that stem from moment map interpretations of the problem.

Ruadhai Dervan (University of Cambridge) will speak in the geometry seminar on Tuesday 3 December 2019 at 11am in the Salle de Profs. (Ruadhai’s talk was initially planned for the 29th of October but has been moved). Ruadhai’s title is Optimal symplectic connections on holomorphic submersions and his abstract is below:

One can think of a Hermite-Einstein metric on a holomorphic vector bundle over a Kähler manifold as determining a canonical choice of Fubini-Study metric on each fibre of the projectivisation of the bundle. I will describe an equation which makes sense for much more general fibrations and (conjecturally) determines a canonical choice of fibrewise constant scalar curvature Kähler metric on each fibre. The main result will use these new metrics to give a novel construction of constant scalar curvature Kähler metrics on the total space of such fibrations. Time permitting, I will describe a largely conjectural picture which explains the geometric significance of these equations. This is joint work with Lars Sektnan.