## Martin P. Weidner

PhD Student

Imperial College London
Department of Mathematics
Huxley Building
Room 6M09

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.

#### Publications

• Hörmander’s theorem for rough differential equations on manifolds
Joint work with Thomas Cass
Preprint, 2016
arXiv | Abstract

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.

• On a new conformal functional for simplicial surfaces
Joint work with Alexander I. Bobenko
Curves and Surfaces. 8th International Conference 2014, Paris, vol. 9213 of Lect. Notes Comp. Sci.,
pp. 47–59. Springer, 2015
arXiv | Published version | Abstract

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.

#### Talks

• Imperial-ETH Workshop in Mathematical Finance
London, March 2017
• Imperial-ETH Workshop on Mathematical Finance
Zürich, September 2016
• ERC Berlin-Oxford Young Researchers Meeting on Applied Stochastic Analysis
Berlin, August 2016
• Doktorandentreffen Stochastik
Bielefeld, August 2016
• European Congress of Mathematics
Berlin, July 2016
• World Congress of Probability and Statistics
Toronto, July 2016
• Young Researchers' Meeting in Probability, Numerics and Finance
Le Mans, June 2016
• Doktorandentreffen Stochastik
Berlin, August 2015

Last update: March 2017