[4] Triangulated surfaces with complex projective structures. With Feng Luo. In preparation.

[3] Dimers and Circle patterns. With R. Kenyon, S. Ramassamy, and M. Russkikh. (2018). arXiv: 1810.05616

-- The dimer model concerns the perfect matching on a bipartite graph with face weights. We show that there is a bijection between bipartite graph with face weights to circle patterns. A local move of the graph that preserves the probability distribution is shown to be equivalent to an application of Miquel’s six circle theorem. (See animation here)

[2] Holomorphic quadratic differentials on graphs and the chromatic polynomial. With Richard Kenyon. (2018). arXiv:1803.00115

-- We show that the chromatic polynomial at negative integers counts a set of "compatible" acyclic orientations, and give an explicit integral formula for it.

[1] Minimal surfaces from infinitesimal deformations of circle packings. (2017). arXiv:1712.08564

-- We construct discrete minimal surfaces of Koebe type from circle packings on any triangle mesh.
It unifies the remaining type of discrete minimal surfaces defined by Bobenko et al via Steiner's formula.

[5] Infinitesimal conformal deformations of triangulated surfaces in space. With U. Pinkall. To appear in *Discrete & Computational geometry* (2018).

-- We investigate the change in the intrinsic and the extrinsic geometry under conformal deformations in space.
It is based on my master thesis.

[4] Trivalent maximal surfaces in Minkowski space. With Masashi Yasumoto. In: *Lorentzian Geometry and Related Topics.* Springer Proceedings in Mathematics & Statistics (2018).

-- We define vertex normals for any trivalent graph in space that are compatitble with the theory of quadratic differentials. We use the vertex normals to study the singularity of maximal surfaces, which are analogues of minimal surfaces in Minkowski space.

[3] Discrete minimal surfaces: critical points of the area functional from integrable systems. *International Mathematics Research Notices IMRN* (2018). arXiv

-- An associated family of discrete minimal surfaces are constructed from quadratic differentials on any planar graph. They unify several notions of discrete minimal surfaces. The mean curvature is derived from Steiner's mixed area formula. (See animation here)

[2] Holomorphic vector fields and quadratic differentials on planar triangular meshes. With U. Pinkall. In: *Advances in Discrete Differential Geometry*. Ed. by A. I. Bobenko. 2016.

-- Discrete holomorphic quadratic differentials are introduced. They are derived as the derivative of cross ratios (Schwarzian derivative) and related to infinitesimal conformal deformations, discrete harmonic functions (cotangent Laplacian) and minimal surfaces.

[1] Isothermic triangulated surfaces. With Ulrich Pinkall. *Math. Annalen* (2016). arXiv

-- We introduce a class of surfaces that admit infinitesimal deformations preserving edge lengths and the mean curvature. We show that they are singularities of the space of conformal immersions.

[1] From isothermic triangulated surfaces to discrete holomorphicity. Research report in Oberwolfach Report No. 13/2015