PhD Student
Imperial College London
Department of Mathematics
Huxley Building
Room 6M09
E-mail address
I am a PhD student in mathematics at Imperial College London. My research focuses on the intersection of rough path analysis, differential geometry and Malliavin calculus. Previously I have also done some research in discrete differential geometry.
Before I came to London I have obtained a B.Sc. and M.Sc. in mathematics from Technische Universität Berlin. In the course of my undergraduate studies I have also spent an exchange year at École normale supérieure in Paris.
Abstract. We introduce a new definition for solutions \(Y\) to rough differential equations (RDEs) of the form \(\mathrm d Y_{t} =V\left (Y_{t}\right )\mathrm d\mathbf X_{t} ,Y_{0} =y_{0}\). By using the Grossman-Larson Hopf algebra on labelled rooted trees, we prove equivalence with the classical definition of a solution advanced by Davie when the state space \(E\) for \(Y\) is a finite-dimensional vector space. The notion of solution we propose, however, works when \(E\) is any smooth manifold \(\mathcal{M}\) and is therefore equally well-suited for use as an intrinsic defintion of an \(\mathcal{M}\)-valued RDE solution. This enables us to prove existence, uniqueness and coordinate-invariant theorems for RDEs on \(\mathcal{M}\) bypassing the need to define a rough path on \(\mathcal{M}\). Using this framework, we generalise a result of Cass-Hairer-Litterer-Tindel proving the smoothness of the density of \(\mathcal{M}\)-valued RDEs driven by non-degenerate Gaussian rough paths under Hörmander’s bracket condition. In doing so, we reinterpret some of the foundational results of the Malliavin calculus to make them appropriate to study \(\mathcal{M}\)-valued Wiener functionals.
Abstract. We introduce a smooth quadratic conformal functional and its weighted version \[W_2=\sum_e \beta^2(e)\quad W_{2,w}=\sum_e (n_i+n_j)\beta^2(e),\] where \(\beta(e)\) is the extrinsic intersection angle of the circumcircles of the triangles of the mesh sharing the edge \(e=(ij)\) and \(n_i\) is the valence of vertex \(i\). Besides minimizing the squared local conformal discrete Willmore energy \(W\) this functional also minimizes local differences of the angles \(\beta\). We investigate the minimizers of this functionals for simplicial spheres and simplicial surfaces of nontrivial topology. Several remarkable facts are observed. In particular for most of randomly generated simplicial polyhedra the minimizers of \(W_2\) and \(W_{2,w}\) are inscribed polyhedra. We demonstrate also some applications in geometry processing, for example, a conformal deformation of surfaces to the round sphere. A partial theoretical explanation through quadratic optimization theory of some observed phenomena is presented.