## Research

My research interests are rooted in pure mathematics, and in particular in the study of Sub-Riemannian geometry, but with a flavour of computational applications.

My first approach to the subject happened during my Master's degree at the University of Bergen where, under the supervision of Prof. Erlend Grong, I worked on previous results by Citti and Sarti [1] and subsequent works by Boscain et al. [2-4] to develop new procedures and algorithms for PDE-based image restoration.

As a PhD fellow at the University of Bergen my focus has expanded to include new areas of interest: optimization problems on Riemannian manifolds and Geometric Deep Learning.

In a general Riemannian manifold computing distances between two points is not an easy task. One could do that by following the geodesics and optimizing the initial velocity by gradient descent, but this is often a computationally intensive operation. What we are interested in studying is the possibility of defining a commuter metric on such manifolds by preprocessing the manifold and computing distances between a set of points of interest. [5]

In Geometric Deep Learning one deals with data that has a non-Eucleadian structure, either by characterizing the structure of the data or by analyzing functions defined on a non-Euclidean domain. [6]

[1] G Citti and A Sarti. “A cortical based model of perceptual completion in the Roto-translation space”. en. In: J. Math. Imaging Vis. 24.3 (May 2006), pp. 307– 326.

[2] Ugo Boscain et al. “Anthropomorphic Image Reconstruction via Hypoelliptic Diffusion”. In: SIAM j. control optim. 50.3 (Jan. 2012), pp. 1309–1336.

[3] U Boscain et al. “Hypoelliptic diffusion and human vision: A semidiscrete new twist”. en. In: SIAM J. Imaging Sci. 7.2 (Jan. 2014), pp. 669–695.

[4] Ugo V Boscain et al. “Highly corrupted image inpainting through hypoelliptic diffusion”. en. In: J. Math. Imaging Vis. 60.8 (Oct. 2018), pp. 1231–1245.

[5] Grong, E., & Sommer, S. (2021). Most probable paths for anisotropic Brownian motions on manifolds. arXiv, 2110.15634

[6] M. M. Bronstein, J. Bruna, Y. LeCun, A. Szlam, and P. Vandergheynst. Geometric deep learning: going beyond euclidean data. IEEE Signal Processing Magazine, 34(4):18–42, 2017.