## Jesse Zhang : Geodesic complexity of convex polyhedra

- Mathematical Biology ( 102 Views )Geodesic complexity of the d-dimensional boundary S of a convex polytope of dimension d+1 is intimately related to the combinatorics of nonoverlapping unfolding of S into a Euclidean space R^d following Miller and Pak (2008). This combinatorics is based on facet sequences, which are lists of adjacent facets traversed by geodesics in S. Our main result bounds the geodesic complexity of S from above by the number of distinct maximal facet sequences traversed by shortest paths in S. For d=2, results from the literature on nonoverlapping unfolding imply that this bound is polynomial in the number of facets. In arbitrary dimension d, a reinterpretation of conjectures by Miller and Pak (2008) leads to the conjecture that the geodesic complexity of S is polynomial in the number of facets. The theory and results developed here hold more generally for convex polyhedral complexes. This is joint work with Ezra Miller.

## Stephan Huckemann : Statistical challenges in shape prediction of biomolecules

- Mathematical Biology ( 164 Views )The three-dimensional higher-order structure of biomolecules
determines their functionality. While assessing primary structure is
fairly easily accessible, reconstruction of higher order structure is
costly. It often requires elaborate correction of atomic clashes,
frequently not fully successful. Using RNA data, we describe a purely
statistical method, learning error correction, drawing power from a
two-scale approach. Our microscopic scale describes single suites by
dihedral angles of individual atom bonds; here, addressing the
challenge of torus principal component analysis (PCA) leads to a
fundamentally new approach to PCA building on principal nested spheres
by Jung et al. (2012). Based on an observed relationship with a
mesoscopic scale, landmarks describing several suites, we use Fréchet
means for angular shape and size-and-shape, correcting
within-suite-backbone-to-backbone clashes. We validate this method by
comparison to reconstructions obtained from simulations approximating
biophysical chemistry and illustrate its power by the RNA example of
SARS-CoV-2.

This is joint work with Benjamin Eltzner, Kanti V. Mardia and Henrik
Wiechers.

Literature:

Eltzner, B., Huckemann, S. F., Mardia, K. V. (2018):
Torus principal component analysis with applications to RNA
structure. Ann. Appl. Statist. 12(2), 1332?1359.

Jung, S., Dryden, I. L., Marron, J. S. (2012):
Analysis of principal nested spheres. Biometrika, 99 (3), 551-568

Mardia, K. V., Wiechers, H., Eltzner, B., Huckemann, S. F. (2022).
Principal component analysis and clustering on manifolds. Journal of
Multivariate Analysis, 188, 104862,
https://www.sciencedirect.com/science/article/pii/S0047259X21001408

Wiechers, H., Eltzner, B., Mardia, K. V., Huckemann, S. F. (2021).
Learning torus PCA based classification for multiscale RNA backbone
structure correction with application to SARS-CoV-2. To appear in the
Journal of the Royal Statistical Society, Series C,
bioRxiv https://doi.org/10.1101/2021.08.06.455406

## Chun-Hung Liu : Assouad-Nagata dimension of minor-closed metrics

- Mathematical Biology ( 78 Views )Assouad-Nagata dimension addresses both large-scale and small-scale behaviors of metric spaces and is a refinement of Gromov’s asymptotic dimension. A metric space is a minor-closed metric if it is defined by the distance function on the vertices of an edge-weighted graph that satisfies a fixed graph property preserved under vertex-deletion, edge-deletion, and edge-contraction. In this talk, we determine the Assouad-Nagata dimension of every minor-closed metric. It is a common generalization of known results about the asymptotic dimension of H-minor free unweighted graphs, about the Assouad-Nagata dimension of complete Riemannian surfaces with finite Euler genus, and about their corollaries on weak diameter coloring of minor-closed families of graphs and asymptotic dimension of minor-excluded groups.