Events

Conferences, Research Colloquia & Seminars, Defenses, and other events

Geometry Seminar

This is the research seminar of the group and focuses on recent research in (differential) geometry; during the semester the seminar is usually scheduled to take place on Wednesdays at 15:30 in the Dissertantenzimmer (or, during times of a pandemic, online). If you are interested in giving a talk, please contact the organizers: Ivan Izmestiev and Gudrun Szewieczek

Student Seminars

These seminars are usually part of the assessment and are open to the public, in particular, to interested students; topics typically focus on geometry but cover a wider range of areas, depending on the students' and the advisor's interests. Presentations are often delivered in German.

Talks in the geometry seminar

(hover/tap name or title to view more information)

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.

12 May 2021: Geometry seminar

Hana Kourimska (IST Austria): Uniformization with a new discrete Gaussian curvature

Abstract

The angle defect - $2\pi$ minus the cone angle at a vertex - is the commonly used discretization of the Gaussian curvature for piecewise flat surfaces. However, it does not possess one of the principal features of its smooth counterpart - upon scaling the surface by a factor $r$, the smooth Gaussian curvature is scaled by the factor of ${1\over r^2}$, whereas the angle defect is invariant under global scaling.

In my talk, I will introduce a new discretization of the Gaussian curvature, that preserves the properties of the angle defect and, in addition, reflects the scaling behavior of the smooth Gaussian curvature. I will also answer the accompanying Uniformization question: Does every discrete conformal class of a piecewise flat surface contain a metric with constant discrete Gaussian curvature? And if so, is this metric unique?

The results I will present in this talk constitute a part of my PhD re-search, which was supervised by Prof. Boris Springborn.

05 May 2021: Geometry seminar

Matteo Raffaelli (TU Wien): Nonrigidity of flat ribbons

Abstract

Developable, or flat, surfaces are classical objects in differential geometry, with lots of real-world applications within fields such as architecture or industrial design. In this talk I will discuss the problem of constructing a developable surface that contains a given space curve. The natural question here is the following. Given a curve, how many locally distinct developables can be defined along it? It turns out that, for any suitable choice of ruling angle (function measuring the angle between the ruling line and the curve's tangent vector), there exists a full circle of flat ribbons.

In the second part of the talk we will examine the set of flat ribbons along a fixed curve in terms of energy. In particular, we will see that the classical rectifying developable of a curve maximizes the bending energy among all infinitely narrow flat ribbons having the same ruling angle.

I will conclude by presenting some important open questions.

28 Apr 2021: Geometry seminar

Gudrun Szewieczek (TU Wien): Smooth and discrete cyclic circle congruences

Abstract

A 2-dimensional congruence of circles in 3-space is called cyclic if it admits a 1-parameter family of smooth orthogonal surfaces. By imposing further (geometric) conditions on such circle congruences, those can be employed to construct families of surfaces of various special types, as for example, pseudospherical surfaces, Guichard surfaces and flat fronts in hyperbolic space.

In this talk we shall give an integrable discretization of cyclic circle con-gruences and characterize them by the existence of a flat connection comprised of "reflections" of the underlying ambient geometry. These explicit flat connections will then provide an efficient way to construct the orthogonal discrete surfaces and could be used to reveal geometric properties of them. As an application of the developed theory, we will construct parallel families of discrete flat fronts in hyperbolic space.

Furthermore, we will discuss how the concept of those discrete flat connections can be carried over to the smooth case.

24 Mar 2021: Geometry seminar

Felix Dellinger (TU Wien): A checkerboard pattern approach to discrete differential geometry

Abstract

Given a quad mesh of regular combinatorics one can obtain a checkerboard pattern by performing a midpoint subdivision, i.e., by connecting all midpoints of neighbouring edges. Such a checkerboard pattern has the property hat every second face is a parallelogram, compare Figure. It turns out, that this checkerboard pattern is very well suited numerically as well as theoretically to define discrete differential geometric properties.

In particular, discrete versions of the shape operator, conjugate nets, principal nets, Koenigs nets and isothermic nets can be consistently defined via this checkerboard pattern approach. Some nice results from the smooth theory also hold for their discrete counter parts:

1. Trace and determinant of the shape operator fit a discrete version of the Steiner formula for offset surfaces.
2. The parameter lines of a principal mesh follow the eigenvectors of the shape operator.
3. Conjugate meshes/Koenigs meshes are mapped to conjugate meshes/Koenigs meshes under projective transformations.
4. Isothermic meshes and in general Koenigs meshes are dualizable.
5. Isothermic meshes/principal meshes are mapped to isothermic meshes/principal meshes under Moebius transformations.

10 Mar 2021: Geometry seminar

Efilena Baseta (TU Wien): TBA

Talks in the geometry seminar

• Wed 20/01/2021, Raman Sanyal (Goethe-Universität Frankfurt)

  Inscribable polytopes, routed trajectories, and reflection arrangements

• Wed 9/12/2020, Hellmuth Stachel (TU Wien)

  The motion of billiards in ellipses (slides)

• Wed 2/12/2020, Joseph Cho (TU Wien)

  Darboux transformations and discrete mKdV equations

• Wed 25/11/2020, Daniil Rudenko (University of Chicago)

  Trigonometry of tetrahedra and rational elliptic surfaces

• Wed 18/11/2020, Michael Jimenez (TU Wien)

  Navigating a Family of Particular Linear Weingarten Surfaces

• Wed 11/11/2020, Christian Müller (TU Wien)

  On a Möbius invariant subdivision algorithm

• Wed 4/11/2020, Kurt Leimer (TU Wien)

  Reduced-Order Simulation of Flexible Meta-Materials

• Wed 28/10/2020, Mohammad Ghomi (Georgia Inst of Technology)

  Total curvature and the isoperimetric inequality (slides)

• Wed 21/10/2020, Stefan Pillwein (TU Wien)

  On Elastic Geodesic Grids and Their Planar to Spatial Deployment

• Wed 14/10/2020, Silviana Amethyst (Univ of Wisconsin Eau Claire)

  Using Numerical Algebraic Geometry to Solve Polynomial Systems (slides)