## Martin P. Weidner

PhD Student

Imperial College London
Department of Mathematics
Huxley Building
Room 6M09

#### Publications

• Tree algebras over topological vector spaces in rough path theory
Joint work with Thomas Cass
Preprint, 2017
arXiv | Abstract

Abstract. We work with non-planar rooted trees which have a label set given by an arbitrary vector space $$V$$. By equipping $$V$$ with a complete locally convex topology, we show how a natural topology is induced on the tree algebra over $$V$$. In this context, we introduce the Grossman-Larson and Connes-Kreimer topological Hopf algebras over $$V$$, and prove that they form a dual pair in a certain sense. As an application we define the class of branched rough paths over a general Banach space, and propose a new definition of a solution to a rough differential equation (RDE) driven by one of these branched rough paths. We show equivalence of our definition with a Davie-Friz-Victoir-type definition, a version of which is widely used for RDEs with geometric drivers, and we comment on applications to RDEs with manifold-valued solutions.

• 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

• ERC Berlin-Oxford Young Researchers Meeting on Applied Stochastic Analysis
Berlin, May 2017
• Imperial-ETH Workshop on 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: October 2017