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 focuses 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 in the last decades. In my work, I pioneered the theory of the singularity formation along the Ricci Flow [3, 5, 16, 23], in particular proving Hamilton's Conjecture that fast-forming singularities are always modelled on self-similar solutions. I also studied the spaces of singularity models, in particular obtaining optimal concentration-compactness results [4, 7, 19] and various results about their ends [21, 22]. I also studied the dynamical stability and instability properties of stationary points of the Ricci Flow [6], introduced the Harmonic Ricci Flow [2] and proved that in dimension two this flow does not develop any singularities in finite time [9]. Finally, I 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, 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 in Euclidean space [11]. In a further article [13], I have then extended these results to study embedded two-convex tori in Euclidean space and, more recent, mean-convex spheres and (Heegaard) tori in ambient three-manifolds [20]. The stationary points of Mean Curvature flow are the critical points of the area functional, i.e. minimal submanifolds. There has been substantial progress in this area over the past years, especially in co-dimension one using min-max constructions. Minimal Hypersurfaces arising this way typically have bounded Morse index and bounded area. In my work [10], I have obtained qualitative lower bounds on the index and area of minimal hypersurfaces in a closed Riemannian manifold in terms of total curvature in dimensions 2≤n≤6 using a bubbling argument. I have then exploited the energy identity that follows from this to obtain new smooth multiplicity one compactness theorems for minimal surfaces [14], generalising classical results of Choi-Schoen. 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 over sixty years ago. In [8], I proved a new four-dimensional Chern-Gauss-Bonnet formula involving the Paneitz Q-curvature for metrics with finitely many conformally flat ends and singular points. 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. 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 three PhD Students: 

• Gianmichele Di Matteo finished his PhD in 2021 at QMUL, left to a postdoc at KIT (Karlsruhe, Germany), 

• Louis Yudowitz finished his PhD in 2023 at QMUL, left to a postdoc position at KTH (Stockholm, Sweden),

• Alessandro Bertellotti finished his PhD in 2025 at SISSA, left to a postdoc position of FU (Berlin, Germany). 

I currently do not have PhD Students. If you are interested in working with me here, please contact me! [Note: the next call for PhD students is expected to open in May/June 2027.]

I have also been the main supervisor of three Postdoctoral Researchers: 

• Mario Schulz (QMUL 2019-21), left to a postdoc position in Münster (Germany), now permanent in Trento (Italy),

• Shengwen Wang (QMUL 2019-21), left to a postdoc position in Warwick (UK), now permanent at QMUL (UK),

• Shujing Pan (UniTo 2024-25), left to a postdoc position in Frankfurt. (Germany).

We currently do not have open postdoc positions, but I am always happy to supervise postdocs with external funding if they have a suitable mathematical background.