
Welcome to the research group
Differential Geometry and Geometric Structures

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
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, HertrichJeromin, Pember;
Fig ©Nimmervoll)
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
(HertrichJeromin, Szewieczek)
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)
Discrete Weierstrasstype 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 Omegadual transformation applied to
to a prescribed Gauss map.
(Pember, Polly, Yasumoto)
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).
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)
This is a surface of (hyperbolic) rotation in hyperbolic
space that has constant Gauss curvature,
a
recent classification project.
(HertrichJeromin, Pember, Polly)
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.
(HertrichJeromin, Fig Lara Miro)
Affine Differential Geometry:
In affine differential geometry a main point of research is
the investigation of special surfaces in three dimensional
affine space.
(Manhart)
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.
(HertrichJeromin)


News
12 Dec 2022
16 Jan 2023: Geometry seminar
 Christian Müller (TU Wien):
The Geometry of Discrete AGAGWebs in Isotropic 3Space
Abstract
We investigate webs from the perspective of the geometry of webs
on surfaces in three dimensional space. Our study of AGAGwebs
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 AGAGwebs in isotropic space and
propose a method to construct them. Furthermore, we prove that
some subnets of an AGAGweb are timelike minimal surfaces in
Minkowski space and can be embedded into a oneparameter 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 curveshortening 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, HarPeled, and Nivasch,
that, as the grid is refined, this process converges to
the Affine CurveShortening 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
 HansPeter Schröcker (Universität Innsbruck):
Devil in paradise II  recent results in motion factorization
 17:30  18:30
 Giorgio Figliolini (Universität Cassino):
Kinematics of mechanisms with higherpairs:
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
GromovKleeWagon volumetric conjecture for ball polyhedra.
