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Welcome to the research group
Differential Geometry and Geometric Structures
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Members & friends of the group in Jul 2021
photograph © by Narges Lali
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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.
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Gallery of some research interests and projects
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)
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 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)
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.
(Hertrich-Jeromin, 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.
(Hertrich-Jeromin, Fig Lara Miro)
Non-rigidity and Symmetry breaking:
We investigate relations between non-rigidity and
symmetry breaking, in particular, whether deformability
of a surface or object in more than one way invariably
leads to symmetry breaking.
(Fuchs, Hertrich-Jeromin, Pember, Fig Mundilova)
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.
(Hertrich-Jeromin)
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News
- 27 Oct 2021: Geometry seminar 15:00, ZS1
- Kostiantyn Drach (IST Austria):
Reversing the classical inequalities under curvature constraints
Abstract
A convex body $K$ is called uniformly convex if all the
principal curvatures at every point along its boundary
are bounded by a given constant lambda either above
(lambda-concave bodies), or below (lambda-convex bodies). We
allow the boundary of $K$ to be non-smooth, in which case the
bounds on the principal curvatures are defined in the barrier
sense, and thus the definition of lambda-convex/concave
bodies makes sense in a variety of discrete settings. The
intersection of finitely many balls of radius 1 is an example
of a 1-convex body, while the convex hull of finitely many
balls of radius 1 is an example of a 1-concave body.
Under uniform convexity assumption, for convex bodies of, say,
given volume, there are non-trivial upper and lower bounds
for various functionals, such as the surface area, in-, and
outer-radius, diameter, width, meanwidth, etc. The bound in
one direction usually constitutes the classical inequality:
for example, the lower bound for the surface area is the
isoperimetric inequality. The bound in another direction
becomes a well-posed and in many cases highly non-trivial
reverse optimization problem. In the talk, we will give an
overview of the results and open questions on the reverse
optimization problems under curvature constraints in various
ambient spaces.
- 13 Oct 2021: Geometry seminar 15:00, ZS 1
- Matty Van-Son (TU Wien): Geometry and Markov numbers
Abstract
We discuss the history of Markov
numbers, which are solutions to the equation $x^2+y^2+z^2=3xyz$.
These solutions can be arranged to form a tree, and we show
that similar trees of $SL(2,Z)$ matrices, quadratic forms,
and sequences of positive integers relate very closely to
Markov numbers. We use the tree structure of sequences, along
with a geometric property of the minimal value of forms at
integer points, to propose an extension to Markov numbers.
This is a joint work with Oleg Karpenkov (University of Liverpool).
- 18 Jun 2021 (Fri!): Geometry seminar
- Graham Andrew Smith (Federal University of Rio de Janeiro):
On the Weyl problem in Minkowski space
Abstract
We show how the work of Trapani & Valli may be
applied to solve the Weyl problem in Minkowski space.
This work is available on arXiv at
https://arxiv.org/abs/2005.01137.
- 02 Jun 2021: Geometry seminar
- Sergey Tabachnikov (Penn State University):
Variations on the Tait-Kneser theorem
Abstract
The Tait-Kneser theorem, first demonstrated by Peter
G. Tait in 1896, states that the osculating circles along
a plane curve with monotone non-vanishing curvature
are pairwise disjoint and nested.
I shall present Tait's proof and discuss variations on
this result.
For example, the osculating circles can be replaced by the
osculating Hooke and Kepler conics along a plane curve;
the proof uses the Lorentzian geometry of the space
of these conics.
I shall also present a version of this theorem for
the graphs of Taylor polynomials of even degrees
of a smooth function.
- 16-18 Sep 2019: Conference
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ISHM2019: Integrable systems and harmonic maps,
Vienna University of Technology
- 01 Feb 2019: Web site
- The
catalogues of our planar/spatial kinematic models are online
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Technische Universität Wien
