# Research

## What is my Research about?

Geometric Analysis is an area of mathematics where geometric objects are studied with analytical methods, often involving partial differential equations. As these equations are generally nonlinear, it is typical for singularities to occur in solutions, thus current efforts aim to transform the theory from one where solutions are required to be smooth to one where singularities play a central role. In fact, the presence of singularities often has deep and insightful geometric reasons. My research mainly focusses on singularities in geometric heat flows and critical points of geometric functionals.

Geometric flow methods have become an important and exciting tool in mathematics, the motivation being to evolve rough initial data towards nice objects, e.g. manifolds with constant curvature, harmonic maps or minimal surfaces. Such flows, and in particular the Ricci Flow, have proved spectacularly successful with Perelman's resolutions of both the Poincaré and Thurston's Geometrisation Conjectures or the Differentiable Sphere Theorem obtained by Brendle-Schoen. In my work, I pioneered the theory of the singularity formation along the Ricci Flow [3, 5, 16], in particular proving Hamilton's Conjecture that fast-forming singularities are always modelled on self-similar solutions. I also obtained optimal concentration-compactness results for the space of these singularity models [4, 7, 19] and studied the dynamical stability and instability properties of stationary points of the Ricci Flow [6]. Furthermore, I introduced the Harmonic Ricci Flow [2] and proved that for surfaces this flow does not develop any singularities in finite time [9]. I also proved a variety of other results for the Harmonic Ricci Flow and other Super Ricci Flows, such as the reduced volume monotonicity [1] or Gaussian heat kernel bounds along the flows [17].

I am also interested in Mean Curvature Flow, which is the most natural extrinsic evolution equation given by the gradient flow of the area functional. In particular, using a min-max approach, I have proved existence of singularity models with arbitrary genus [18], resolving a conjecture by Ilmanen. I have furthermore constructed a modification of Mean Curvature Flow with surgery based on a new two-convex connected sum construction to prove a Smale type theorem for the moduli spaces of two-convex embedded spheres [11]. In a further article [13], I have then extended these results to study embedded two-convex tori, matching the path-components of the moduli space to the knot classes of such tori. In the more recent [20], I have investigated the path-components of mean convex spheres and (Heegaard) tori in three-manifolds. The stationary points of (co-dimension one) Mean Curvature flow are the critical points of the area functional, i.e. Minimal Hypersurfaces. There has been substantial progress in this area over the past years, especially in two dimensions, the most prominent example of this progress probably being the resolution of the Willmore Conjecture by Marques-Neves. One can find many closed, smooth, embedded minimal hypersurfaces in a closed Riemannian manifold N via min-max constructions - these minimal hypersurfaces then typically have bounded Morse index and bounded area and it is thus natural to study the properties of minimal hypersurfaces satisfying these two conditions. In my work [10], I have obtained qualitative lower bounds on the index and area of n-dimensional minimal hypersurfaces in a closed Riemannian (n+1)-manifold in terms of total curvature for 2≤n≤6. This is based on a bubbling argument which also leads to an energy identity for the total curvature along a sequences of minimal hypersurfaces, a result I later exploited in [14], to obtain new smooth multiplicity one compactness theorems for minimal surfaces, generalising classical results of Choi-Schoen from ambient spaces N with positive Ricci curvature, to ambient spaces with positive scalar curvature under an additional bound on either the index or the area. I have then generalised all of these results to free boundary minimal hypersurfaces [15], in particular analysing the formation of "half-bubbles".

I have also worked in Singular Conformal Geometry: The Gauss-Bonnet theorem, one of the most fundamental results in differential geometry, gives a link between the geometry of a surface (given by its total Gauss curvature) and its topology (given by its Euler characteristic). In particular, it shows that there are topological obstructions to the existence of certain metrics, for example no two-dimensional torus carries a metric of positive Gauss curvature. A generalisation of the Gauss-Bonnet theorem to higher-dimensional compact Riemannian manifolds was discovered by Chern and has been known for over fifty years. However, little is known about the corresponding formula for non-compact or singular Riemannian manifolds. I proved a new four-dimensional Chern-Gauss-Bonnet formula involving the Paneitz Q-curvature [8] for metrics with finitely many conformally flat ends and singular points, under very weak (and necessary) curvature restrictions. This is the first such result in a dimensions higher than two which allows the underlying manifold to have isolated branch points or conical singularities. The formula obtained includes a precise characterisation of the error terms, expressing them as isoperimetric deficits near the singular points, infinitesimally measuring the deviation from flat Euclidean space. Later, I generalised the result to dimensions higher than four in the locally conformally flat case [12].

All references above refer to the articles and preprints on my publictions page.

## PhD Students and Postdocs

I have been the main supervisor of two PhD Students at Queen Mary University of London: Gianmichele Di Matteo who finished his PhD in 2021 and then left for a postdoc at KIT (Karlsruhe, Germany), and Louis Yudowitz who finishes his PhD in 2023 and left to a postdoc position at KTH (Stockholm, Sweden). Since 2022, I am also the supervisor of Alessandro Bertellotti at SISSA Trieste who is supposed to finish his PhD in 2025. At the University of Torino (UniTo), where I have recently become a member of the Differential and Complex Geometry Group led, I currently do not have PhD Students. If you are interesed in working with me here, please contact me! [Note: the last call for PhD students has closed in July and there are currently no funded positions.]

Fully financed by my EPSRC Grant, I have also supervised two Postdoctoral Researchers at Queen Mary University of London: Mario Schulz who has left to another postdoc position in Münster and Shengwen Wang who, after a further year as postdoc in Warwick, is now a lecturer at Queen Mary University of London. From September 2024 onwards, I will be the supervisor of Shujing Pan at the University of Torino for one year. She will be financed by our PRIN project "Differential-geometric Aspects of Manifolds via Global Analysis". [Note: We currently have four funded 2-year postdoc positions with application deadline on 23rd September 2024 - please contact me if you are interested in working with me! - and otherwise I am also happy to supervise postdocs with external funding and a suitable mathematical background.]