*Emerging Field 2 – Digital Shapes *organized a research seminar that took place every two to three weeks. In the seminar, current research topics within the thematic scope of the emerging field were presented.

**Organizers**

June 1st, 2021

**Random simple-homotopy theory**, Frank Lutz (TU Berlin)

A standard task in topology is to simplify a given finite presentation of a topological space. Bistellar flips allow to search for vertex-minimal triangulations of surfaces or higher-dimensional manifolds, and elementary collapses are often used to reduce a simplicial complex in size and potentially in dimension. Simple-homotopy theory, as introduced by Whitehead in 1939, generalizes both concepts.

We take on a random approach to simple-homotopy theory and present a heuristic algorithm to combinatorially deform non-collapsible, but contractible complexes (such as triangulations of the dunce hat, Bing’s house or non-collapsible balls that contain short knots) to a point.

The procedure also allows to find substructures in complexes, e.g., surfaces in higher-dimensional manifolds or subcomplexes with torsion in lens spaces.

(Joint work with Bruno Benedetti, Crystal Lai, and Davide Lofano.)

May 25th, 2021

**The Combinatorics of Foams**, John Sullivan (TU Berlin)

Foams, consisting of bubbles separated by interfaces that minimize their surface area, are model for many cellular materials. Plateau’s laws show that foams are combinatorially dual to triangulations. Focusing on measures related to average degree, we survey various results on the combinatorics of foams, including a few rigorous mathematical theorems and many physical heuristics.

May 11th, 2021

**Combinatorial Reconstruction and Analysis of Metallic Foams from Tomography Data**, Frank Lutz (TU Berlin)

Simple foams respect Plateau’s rules combinatorially in the sense that every cell edge is contained in exactly three cells and every vertex is contained in exactly four cells.

Though (non-degenerated) simple foams can be reconstructed from their adjacency graphs, in practice, the registration process of adjacency graphs from tomography greyscale image data of metallic foams comes along with errors.

We discuss heuristics for the correction of the registration errors as a preprocessing step for the combinatorial analysis and roundness computation of the resulting foam structures.

(Joint work with Ihab Sabik, Paul H. Kamm, Mike Noak, Junichi Nakagawa, Francisco Garcia-Moreno.)

May 4th, 2021

**Applications of Random Field Theory and Uncertainty Quantification**, Fabian Telschow (HU Berlin)

Differentiable random fields are a versatile tool in applications to model random effects in neuro-imaging, astronomy, or biomechanical data. In this talk, I will present some ongoing and future research in the quantification of uncertainties of excursion sets of random fields such as CoPE sets or simultaneous confidence bands. Moreover, I will give a short introduction to the Gaussian kinematic formula, which connects the geometry induced on a space by random fields with the expected Euler characteristic of their excursion sets.

February 17th, 2021

**Organizational Meeting**

Discussion and planning of further activities of the Emerging Field 2.

February 3rd, 2021

**Processing of Manifold-Valued Images**, Gabriele Steidl (TU Berlin)

This is an overview talk over recent research of our group connected with geometry concepts. In particular, we address the restoration, segmentation and metamorphosis of manifold-valued image and the approximation of measures on manifolds by measures supported on curves.

December 9th, 2020

**On decomposition of embedded prismatoids in R³ without additional points**, Hang Si (WIAS)

This talk considers three-dimensional prismatoids which can be embedded in R³. A subclass of this family are twisted prisms, which includes the family of non-triangulable Schönhardt polyhedra. We call a prismatoid decomposable if it can be cut into two smaller prismatoids (which have smaller volumes) without using additional points. Otherwise it is indecomposable. The indecomposable property implies the non-triangulable property of a prismatoid but not vice versa.

In this talk we prove two basic facts about the decomposability of embedded prismatoid in R³ with convex bases. Let P be such a prismatoid, call an edge interior edge of P if its both endpoints are vertices of P and its interior lies inside P. Our first result is a condition to characterise indecomposable twisted prisms. It states that a twisted prism is indecomposable without additional points if and only if it allows no interior edge. Our second result shows that any embedded prismatoid in R³ with convex base polygons can be decomposed into the union of two sets (one of them may be empty): a set of tetrahedra and a set of indecomposable twisted prisms, such that all elements in these two sets have disjoint interiors.

November 25th, 2020

**Bi-invariant Two-sample Tests in Lie Groups for Shape Analysis**, Christoph von Tycowicz (ZIB)

We propose generalizations of the T²-statistics of Hotelling and the Bhattacharayya distance for data taking values in Lie groups. A key feature of the derived measures is that they are compatible with the group structure even for manifolds that do not admit any bi-invariant metric.

This property, e.g., assures analysis that does not depend on the reference shape, thus, preventing bias due to arbitrary choices thereof. Furthermore, the generalizations agree with the common definitions for the special case of flat vector spaces guaranteeing consistency. Employing a permutation test setup, we further obtain nonparametric, two-sample testing procedures that themselves are bi-invariant and consistent.

We validate our method in group tests revealing significant differences in hippocampal shape between individuals with mild cognitive impairment and normal controls.

November 11th, 2020

** Conforming Weighted Delaunay Triangulations, **Marc Alexa (TU Berlin)

October 27nd, 2020

**EF2 Kick-off WS 20/21**

Our second EF2 Day was a kick-off event for the coming winter semester. We exchanged our latest results and discussed the forthcoming MATH+ Day.

July 1st, 2020

**Geodesic analysis in Kendall’s shape space with epidemiological applications**, Esfandiar Navayazdani (ZIB)

We analytically determine Jacobi fields and parallel transports and compute geodesic regression in Kendall’s shape space. Using the derived expressions, we can fully leverage the geometry via Riemannian optimization and thereby reduce the computational expense by several orders of magnitude over common, nonlinear constrained approaches. The methodology is demonstrated by performing a longitudinal statistical analysis of epidemiological shape data. As an example application, we have chosen 3D shapes of knee bones, reconstructed from image data of the Osteoarthritis Initiative.

June 3rd, 2020

**Intrinsic Delaunay triangulations and discrete minimal surfaces**, Carl Lutz (TU Berlin)

Intrinsic Delaunay triangulations are special geodesic triangulations of piecewise linear surfaces which are completely determined by the metric of the surface and generically unique. Though not necessarily regular, these triangulations allow conducting a discrete form of harmonic analysis, in particular, the definition of an intrinsic Laplace-Beltrami operator, leading to a theory of discrete minimal surfaces.

This talk gives an introduction to intrinsic Delaunay triangulations with special focus on their global and local geometric properties. Further, the computation of these triangulations by means of a flip algorithm is discussed. Afterwards, the intrinsic Laplace-Beltrami operator is defined and its properties and connections to discrete minimal surfaces are examined. The talk concludes with the presentation of recent numerical experiments on the computation of these discrete minimal surfaces.

May 13th, 2020

**Skew prismatoids**, Marc Alexa (TU Berlin) and Hang Si (WIAS)

April 15th, 2020

**Smooth polyhedral surfaces**, Felix Günther (TU Berlin)

In modern architecture, facades and glass roofs often model smooth shapes but are realized as polyhedral surfaces. Bad approximations may be observed as wiggly meshes, even though the polyhedral mesh is close to a smooth reference surface. So what does it mean for a polyhedral surface to be smooth?

In this talk, we introduce a theory of smooth polyhedral surfaces. A key role is played by the Gaussian normal image. We present a projectively invariant class of polyhedral surfaces that share several properties with their smooth counterparts. Furthermore, we discuss discrete curvatures in an affinely invariant setting and relate them to new notions of smoothness and discrete mean curvature flows.

April 1st, 2020

**Spline models for shape trajectory analysis**, Martin Hanik (ZIB)

Our brain is a master in analyzing even difficult geometric structures and being able to distinguish different shapes is of great importance in many aspects of our daily lives. Thus, it is no wonder that shape analysis is needed in many scientific fields such as archaeology, biology and computational anatomy.

Mathematically, a shape is an element of a high-dimensional manifold. After endowing it with a suitable Riemannian metric, many basic statistical tools can be defined. Quite often not only single shapes but their change with some parameter (for example time) is of interest. For this purpose, regression tools have been generalized to the manifold setting.

After giving a brief introduction to shape spaces in the form of Kendall’s shape space, I will talk about results on geodesic regression and give an overview of recent advancements for generalizing regression with splines consisting of Bézier curves.

March 18th, 2020

**Conforming regular triangulations** and **Update weighted Delaunay triangulations by flips**, Marc Alexa (TU Berlin) and Hang Si (WIAS)

March 14th, 2020

**EF2 Day**

On our EF2 Day, we exchange on the current status of our research efforts and discuss the joint activities within the emerging field. Besides a presentation of each project’s activities, we have a discussion on how we can foster the aims of our emerging field and communicate its progression to the entire MATH+ community and beyond.