Welcome to the research group
Differential Geometry and Geometric Structures

FB3, photo by Narges Lali

Members & friends of the group in Jul 2021
photograph © by Narges Lali

Differential geometry has been a thriving area of research for more than 200 years, employing methods from analysis to investigate geometric problems. Typical questions involve the shape of smooth curves and surfaces and the geometry of manifolds and Lie groups. The field is at the core of theoretical physics and plays an important role in applications to engineering and design.

Finite and infinite geometric structures are ubiquitous in mathematics. Their investigation is often intimately related to other areas, such as algebra, combinatorics or computer science.

These two aspects of geometric research stimulate and inform each other, for example, in the area of "discrete differential geometry", which is particularly well suited for computer aided shape design.

Gallery of some research interests and projects

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Symmetry breaking in geometry: We discuss a geometric mechanism that may, in analogy to similar notions in physics, be considered as "symmetry breaking" in geometry. (Fuchs, Hertrich-Jeromin, Pember; Fig ©Nimmervoll)

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Cyclic coordinate systems: an integrable discretization in terms of a discrete flat connection is discussed. Examples include systems with discrete flat fronts or with Dupin cyclides as coordinate surfaces (Hertrich-Jeromin, Szewieczek)

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We study surfaces with a family of spherical curvature lines by evolving an initial spherical curve through Lie sphere transformations, e.g., the Wente torus (Cho, Pember, Szewieczek)

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Discrete Weierstrass-type representations are known for a wide variety of discrete surfaces classes. In this project, we describe them in a unified manner, in terms of the Omega-dual transformation applied to to a prescribed Gauss map. (Pember, Polly, Yasumoto)

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Billiards: The research addresses invariants of trajectories of a mass point in an ellipse with ideal physical reflections in the boundary. Henrici's flexible hyperboloid paves the way to transitions between isometric billiards in ellipses and ellipsoids (Stachel).

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Spreads and Parallelisms: The topic of our research are connections among spreads and parallelisms of projective spaces with areas like the geometry of field extensions, topological geometry, kinematic spaces, translation planes or flocks of quadrics. (Havlicek)

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This is a surface of (hyperbolic) rotation in hyperbolic space that has constant Gauss curvature, a recent classification project. (Hertrich-Jeromin, Pember, Polly)

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Singularity Closeness of Stewart-Gough Platforms: This project is devoted to evaluating the closeness of Stewart-Gough platforms to singularities. (Nawratil)

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Geometric shape generation: We aim to understand geometric methods to generate and design (geometric) shapes, e.g., shape generation by means of representation formulae, by transformations, kinematic generation methods, etc. (Hertrich-Jeromin, Fig Lara Miro)

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Affine Differential Geometry: In affine differential geometry a main point of research is the investigation of special surfaces in three dimensional affine space. (Manhart)

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Transformations & Singularities: We aim to understand how transformations of particular surfaces behave (or fail to behave) at singularities, and to study how those transformations create (or annihilate) singularities. The figure shows the isothermic dual of an ellipsoid, which is an affine image of a minimal Scherk tower. (Hertrich-Jeromin)

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Stewart Gough Platforms with Self-Motions: The main aim of the project is the systematic determination, investigation and classification of Steward-Gough platforms with self-motions. (Nawratil)

News

12 Dec 2022
16 Jan 2022: Geometry seminar
Christian Müller (TU Wien): The Geometry of Discrete AGAG-Webs in Isotropic 3-Space

Abstract

We investigate webs from the perspective of the geometry of webs on surfaces in three dimensional space. Our study of AGAG-webs is motivated by architectural applications of gridshell structures where four families of manufactured curves on a curved surface are realizations of asymptotic lines and geodesic lines. We describe all discrete AGAG-webs in isotropic space and propose a method to construct them. Furthermore, we prove that some sub-nets of an AGAG-web are timelike minimal surfaces in Minkowski space and can be embedded into a one-parameter family of discrete isotropic Voss nets. This is a joint work with Helmut Pottman.
28 Nov 2022: Geometry seminar
Günter Rote (FU Berlin): Grid peeling and the affine curve-shortening flow

Abstract

Grid Peeling is the process of taking the integer grid points inside a convex region and repeatedly removing the convex hull vertices. It has been observed by Eppstein, Har-Peled, and Nivasch, that, as the grid is refined, this process converges to the Affine Curve-Shortening Flow (ACSF), which is defined as a deformation of a smooth curve.

As part of the M.Ed. thesis of Moritz Rüber, we have investigated the grid peeling process for special parabolas, and we could observe some striking phenomena. This has lead to a conjecture for the value of the constant that relates the two processes.

18 Nov 2022, Zeichensaal 3: Festkolloquium
zum 80. Geburtstag von Hellmuth Stachel

Programm

13:15 - 14:00
Eröffnung und Laudatio von Otto Röschel
14:00 - 15:00
Johannes Wallner (TU Graz): Flexible nets and discrete differential geometry
15:00 - 16:00
Georg Glaeser (Universität für angewandte Kunst Wien): Forty years between descriptive and computational geometry: the universe of spatial imagination
16:00 - 16:30
Kaffeepause
16:30 - 17:30
Hans-Peter Schröcker (Universität Innsbruck): Devil in paradise II - recent results in motion factorization
17:30 - 18:30
Giorgio Figliolini (Universität Cassino): Kinematics of mechanisms with higher-pairs: fundamentals and applications
19:00
Abendessen im Restaurant Waldviertlerhof, Schönbrunnerstr. 20, 1050 Wien

14 Nov 2022: Geometry seminar
Karoly Bezdek (University of Calgary): Ball polyhedra -- old and new

Abstract

We survey a number of metric properties of intersections of finitely many congruent balls called ball polyhedra in Euclidean spaces. In particular, our talk is centered around the status of the shortest billiard conjecture, the global rigidity conjecture, Hadwiger???s covering conjecture, and the Gromov-Klee-Wagon volumetric conjecture for ball polyhedra.


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