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 Friday at 11:00 in the Dissertantenzimmer. If you are interested in giving a talk, please contact the organizer: Ivan Izmestiev.

Seminar "JA, surfaces and beyond"

This is a joint online/hybrid research seminar with colleagues from Japan, focused on surface geometry; the seminar is currently scheduled on Monday at 12:00 CEST/19:00 JST (every 3-4 weeks). If you are interested to participate, please contact (one of) the organizers: Gudrun Szewieczek, Atsufumi Honda, Udo Hertrich-Jeromin or Masatoshi Kokubu.

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.

Seminar talks

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

22 Jul 2026, 12:00 CEST: JA, surfaces and beyond

Naoya Suda (Kobe University) Bäcklund transformations of K-surfaces via Darboux matrices in smooth and discrete settings

Abstract

Surfaces of constant negative Gaussian curvature in $3$-dimensional Euclidean space (hereafter abbreviated as $K$-surfaces) admit an integrable equation known as the sine-Gordon equation, and have been studied from the perspectives of both integrable systems and differential geometry. Furthermore, the existence of Bäcklund transformations, which construct a new $K$-surface from a given one, is an important property of $K$-surfaces, and the use of Darboux matrices is well known as one of the algebraic computation methods for these transformations. Meanwhile, Bobenko and Pinkall (1996) and Hoffmann and Sageman-Furnas (2016) each clarified the discretization of $K$-surfaces and their algebraic computation methods based on different coordinate systems. In this talk, we discuss the computation methods for Bäcklund transformations using Darboux matrices, comparing the smooth and discrete cases. Part of this presentation is based on joint work with Thomas Raujouan (University of Tours) and Wayne Rossman (Kobe University).

23 Jun 2026: Bachelor Seminar im Dissertantenzimmer (UF DG, in Deutsch)

09:15 V A Horak: Möbius-geometrische Symmetrien in einem euklidischen Dreieck
10:15 L Reisinger: Vom Parallelenproblem zur hyperbolischen Geometrie
11:15 B Szabo: Hopffaserung

Abstracts

Möbius-geometrische Symmetrien in einem euklidischen Dreieck: Gesucht werden Symmetrien zwischen den Eckpunkten eines Dreiecks und dessen In- und Ankreismittelpunkten. Tatsächlich lassen sich mithilfe der Möbius-Geometrie acht Abbildungen finden, die diese Punkte und den Punkt Unendlich aufeinander abbilden - obwohl Mittelpunkte Möbius-geometrisch nicht mit deren Kreisen zusammenhängen. Eine dieser Abbildungen, die Antipodenabbildung, wird im Detail vorgestellt und anhand von Visualisierungen veranschaulicht.

Vom Parallelenproblem zur hyperbolischen Geometrie: Über zweitausend Jahre lang wurde Euklids Parallelenpostulat, das nicht in die Reihe der anderen passte, immer wieder hinterfragt. Der Vortrag zeichnet nach, wie Janos Bolyai und Nikolai Lobatschewski im 19. Jahrhundert die Frage umdrehten. Sie behandelten die Negation des Postulats nicht länger als Fehler, sondern als Ausgangspunkt einer eigenständigen Geometrie. Im Zentrum stehen die mathematischen Ergebnisse ihrer Arbeit und die Frage, warum diese neue Geometrie erst spät Teil der wissenschaftlichen Debatte wurde.

Hopffaserung: Die Hopffaserung ordnet Punkten der $2$-Sphäre Kreise der $3$-Sphäre zu, wobei Quaternionen eingesetzt werden. Mithilfe der stereographischen Projektion werden Bilder der Fasern erzeugt. Auch die Bewegung entlang einer Hopffaser wird visualisiert, wobei die doppelte Überlagerung von $\mathrm{SO}(3)$ erkennbar wird.

18 Jun 2026: Geometry seminar

Eleni Pachyli (TU Wien): Lichnerowicz-Obata Theorem for the discrete spherical Laplacian, in dimension two

Abstract

The classical Lichnerowicz-Obata Theorem states that for an $n$-dimensional Riemannian manifold $(M,g)$ with $Ric \geq (n-1)g$, the first eigenvalue of the Laplacian is greater or equal than $n$. We prove a discrete analogue of this theorem in dimension 2, for a spherical cone metric with curvature $\kappa_i := 2\pi-\omega_i >0$ for the discrete spherical Laplacian defined by I. Izmestiev and W. Lam. Our proof draws on ideas from the theory of mixed volumes, in particular from Y. Shenfeld and R. van Handel's proof of the Alexandrov-Fenchel inequality. In the process, we also derive a discrete analogue for the Bochner method.

17 Jun 2026, 12:00 CEST: JA, surfaces and beyond

Keita Takahashi (Institute of Science Tokyo) Completeness conditions for globally hyperbolic spacetimes

Abstract

In this talk, we discuss several completeness conditions for globally hyperbolic spacetimes within the framework of low-regularity settings. These conditions, originally introduced by Busemann and Beem, serve as Lorentzian analogues to those found in the classical Hopf-Rinow theorem. As a related result, we explain an approximation theorem for the space of Cauchy hypersurfaces, a space whose properties have recently been investigated by Lange and Peteranderl.

11 Jun 2026: Geometry seminar

Ivan Izmestiev (TU Wien): Discrete Wirtinger inequality

Abstract

The classical Wirtinger inequality compares the $L_2$-norm of a periodic function with zero average with the $L_2$-norm of its derivative. It was used by Hurwitz to give a new proof of the isoperimetric inequality for smooth curves. A discrete version of the Wirtinger inequality is due to Fan, Taussky, and Todd. In this talk a generalization of the Fan-Taussky-Todd inequality will be stated and proved.

03 Jun 2026, 12:00 CEST: JA, surfaces and beyond

Shintaro Akamine (Nihon University): Constant mean curvature surfaces in the three-dimensional light cone

Abstract

Some classes of surfaces in spaces equipped with degenerate metrics often arise in correspondence with minimal surfaces in Euclidean space and maximal surfaces in Lorentz-Minkowski space. In this talk, we present local and global properties of spacelike constant mean curvature surfaces in the three-dimensional light cone. In particular, we explain Bernstein-type theorems for such surfaces. This talk is based on the joint work with Wonjoo Lee (Jeonbuk National University) and Seong-Deog Yang (Korea University).

28 May 2026: Geometry seminar

Darius Imre (TU Wien): Die Geometrie der Addition und Multiplikation

Abstract

In diesem Vortrag konstruieren wir, ausgehend von einer Menge von Punkten, Geraden und einer Inzidenzrelation, die zusammen eine projektive Ebene bilden, eine Addition und eine Multiplikation. Dazu untersuchen wir einige Eigenschaften von Homologien und Elationen und verwenden diese anschliessend zur Konstruktion eines Schiefkörpers. Dabei werden wir sehen, dass die Existenz der dafür benötigten Abbildungen nur in desarguesschen Ebenen garantiert ist. Abschliessend betrachten wir die erweiterte euklidische Ebene und zeigen, dass der dort entstehende Schiefkörper isomorph zu den reellen Zahlen ist.

21 May 2026: Geometry seminar

Michal Zamboj (Charles University Prague): Von Staudts synthetic constructions of algebraic operations on complex and split-complex numbers

Abstract

Over the years 1847-1856, in "Geometrie der Lage and Beiträge zur Geometrie der Lage", Karl Georg Christian von Staudt formalized projective geometry based on incidence properties. He introduced synthetic constructions - on points and lines with respect to a fixed conic - corresponding to operations on the extended real numbers. We show a generalization of von Staudt's constructions to the complex and split-complex numbers via projection onto the Riemann sphere and subsequent models on quadrics. Consequently, we present a geometric algorithm for the synthetic construction of Gaussian primes on a paraboloid of revolution.

07 May 2026: Geometry seminar

Christopher Latour (TU Wien): Elliptic Curves, Complex Tori and Moduli Spaces

Abstract

A projective nonsingular curve defined by a homogeneous polynomial of degree 3 is called an elliptic curve. The study of elliptic curves plays an integral part in modern mathematics and is at the heart of many famous results, such as Fermats Last Theorem. In this talk we will investigate the structure of elliptic curves over $\mathbb{C}$ and their connection to complex tori and elliptic functions. This connection will enable us to represent the quotient space of all elliptic curves modulo isomorphism as a Riemann surface, where every point of the Riemann surface corresponds to an equivalence class of elliptic curves. Such spaces are called moduli spaces and we will take a look at further moduli spaces of elliptic curves.

06 May 2026, 12:00 CEST: JA, surfaces and beyond

Katrin Leschke (University of Leicester): Links between the integrable systems of a CMC surface

Abstract

A CMC surface in 3-space is constrained Willmore and isothermic. It is well known that these 3 surface classes are each determined by a family of flat connections. In this talk we discuss links between the corresponding families of flat connections: we show that parallel sections of the associated family of flat connections of one family give algebraically the parallel sections of the other families. In particular, we obtain links between transformations of CMC surfaces, isothermic surfaces and constrained Willmore surfaces which are given by parallel sections, such as the associated family, the simple factor dressing and the Darboux transformation.

16 Apr 2026: Geometry seminar

Elizaveta Streltsova (IST Austria): Face numbers of polytopes and levels in arrangements

Abstract

Levels in arrangements are a fundamental notion in discrete and computational geometry and are a natural generalization of convex polytopes. In the talk, I will present the relevant background from convex polytope theory and two new results on the face numbers of levels in arrangements. Collectively, these numbers form the $f$-matrix (which generalizes the $f$-vector of a polytope). We determine the affine space spanned by the $f$-matrices of all arrangements of n hemispheres in $S^d$. This completes a long line of research on linear relations between face numbers and answers a question posed by Andrzejak and Welzl in 2003. Moreover, we proved a special case $n = d + 4$ of the long-standing conjecture of Eckhoff, Linhart, and Welzl on the complexity of the ($\leq k$)-levels, which implies the Harary-Hill Conjecture on the number of crossings of complete graphs for the class of spherical arc drawings. For the proofs, we introduce the $g$-matrix, which encodes the $f$-matrix and generalizes the classical $g$-vector of a polytope.

Joint work with Uli Wagner.

05 Mar 2026: Geometry seminar

Matthias Pichelbauer (TU Wien): Alpha-Shapes

Abstract

This talk presents an introduction to alpha shapes, a concept that generalizes the notion of convex hulls. Based on the ideas developed in the paper "On the Shape of a Set of Points in the Plane" by Herbert Edelsbrunner, David G.Kirkpatrick and Raimund Seidel, alpha shapes provide a flexible way to capture the shape of a finite point set, controlled by a parameter that allows for varying levels of detail.

The talk focuses primarily on the construction of alpha shapes and highlights their close relationship with Delaunay triangulations. In particular, it explains how alpha shapes can be derived as subcomplexes of the Delaunay triangulation, making this connection central to both their theoretical understanding and practical computation.

12 Mar 2026: Geometry seminar

Sarah Quin (TU Wien): Existenz und Eindeutigkeit von Kreispackungen und Kreismustern

Abstract

In meinem Vortrag behandle ich einen Auszug aus dem Paper "Variational principles for circle patterns and Koebe's theorem" von Alexander Bobenko und Boris Springborn. Konkret geht es um die Frage, ob man zu einer vorgegebenen Kombinatorik eine Kreispackung finden kann und inwiefern diese eindeutig ist. Hierbei ist eine Kreispackung eine Familie von Kreisen, die einander beruehren koennen, aber nicht schneiden. Die gesuchten Kreispackungen lassen sich auf orthogonale Kreismuster zurueckfuehren. Die Existenz und Eindeutigkeit dieser Kreismuster laesst sich in weiterer Folge ueber ein Variationsprinzip zeigen.

05 Mar 2026 in Zeichensaal 3: Geometry seminar

Clara Oeverink (TU Wien): Classification of Pencils of Quadrics in the Real Projective Space

Abstract

In the complex projective space (of arbitrary dimension) the Segre symbol has long been known as a sufficient tool for classification of pencils of quadrics. However, it lacks information in the real case. This, the use of the index sequence, as introduced by Tu et al in 2006, can compensate for. Together, they determine the quadric pair canonical form (QPCF) of a pair of matrices representing quadrics in a nondegenerate pencil, as defined by Uhlig in 1976, thereby fixing the pencil itself.

Tu et al used index sequences, together with the related signature sequences, to construct a simple algebraic method of classifying pencils in projective 3-space and accordingly their intersection curves. This allows low-cost determination of basic topological properties of the curves for a more stable parametrisation.

The presentation aims to provide insight into the aforementioned tools via an examination on how Segre symbol, index sequence, signature sequence and QPCF are related. It illustrates how this enables the formal classification of pencils of quadrics in the real projective space. An emphasis is put on index sequences and their equivalence classes, leading to their implementation in a classification algorithm emulating Tu et al.

04 Mar 2026, 11:30 CET: JA, surfaces and beyond

Yoshiki Jikumaru (Toyo University): On the governing equations for membrane O surfaces

Abstract

It is known that a shell membrane in equilibrium where a constant purely normal load qn acts on the membrane, and where the principal curvature lines coincide with the principal stress lines, forms an integrable system called a membrane O surface. In this talk, we formulate the governing equations for membrane O surfaces of the 1st and 2nd kind, which are analogues to Guichard surfaces of the 1st and 2nd kind introduced by Calapso. Furthermore, under this formulation, we show that membrane O surfaces are a subclass of Demoulin's $\Omega$ surfaces, and that the Bäcklund transformation for membrane O surfaces preserves membrane O surfaces of the 1st and 2nd kind, respectively.

Seminar talks

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

11 Feb 2026, 11:30 CET: JA, surfaces and beyond

Joseph Cho (Handong Global University): Geometry at infinity - from a Minkowski geometric viewpoint

Abstract

Many interesting surface classes in Minkowski 3-space including maximal surfaces and constant mean curvature surfaces are known to admit non-degenerate singularities. These singularities differ from the isolated singularities appearing on analogous surfaces in Euclidean 3-space. However, the case of constant mean curvature surfaces shows that some surfaces in Minkowski 3-space also appear with disconnected components, admitting blow-up points. In this talk, we propose Laguerre geometry to gain insight into the fundamental difference between Euclidean 3-space and Minkowski 3-space, and to obtain an intuitive understanding of why non-degenerate singularities and blow-up points appear on surfaces in Minkowski 3-space. This talk is based on joint work with Wonjoo Lee, Gudrun Szewieczek, and Seong-Deog Yang.

17 Dec 2025, 11:30 CET: JA, surfaces and beyond

Mason Pember (Univ of Bath): $\Omega_0$ surfaces

Abstract

An $\Omega_0$ surface is a surface for which one of the curvature sphere congruences is isothermic. To the speaker's knowledge, the only known examples of such surfaces are channel surfaces. In pursuit of finding non-channel examples, we discuss a method for creating examples by applying Lie-Darboux transformations to a particular class of channel surfaces. This leads to an explicit parametrisation of a non-channel $\Omega_0$ surface.

26 Nov 2025, 11:30 CET: JA, surfaces and beyond

Shun Kumagai (Hachinohe Institute of Technology): Self-affinity of planar curves: towards unified description of aesthetic curves

Abstract

Self-affinity is a symmetry of planar curves and is regarded as playing a crucial role in characterizing log-aesthetic curves (LACs), which have been studied as reference curves for designing aesthetic shapes in CAD systems. Inoguchi et al. showed a variational principle and an integrable deformation of LACs in similarity geometry, as well as its application to geometric shape generation. In this talk, we discuss LACs and their self-affinity, and their parallels in Klein geometries, where LACs meet with parabolas.

This talk is based on joint work with Kenji Kajiwara.

12 Nov 2025: Bachelor seminar

Enzo Kitt (TU Wien): Infinitesimal rigidity of surfaces under projective transformations

Abstract

I present a proof that infinitesimal rigidity of smooth surfaces is preserved under projective transformations. Following an argument of Sevennec, the deformation condition for an immersion p and its infinitesimal isometric deformation (IID) q is encoded by a map into a quadric Q. The maximal isotropic subspaces of this quadric correspond to graphs of orthogonal transformations and split into two families. This observation allows the construction of a map into these subspaces whose constancy characterizes infinitesimal rigidity. The result then follows since the entire construction is formulated in projectively invariant terms. Given an IID q, I also establish a formula that explicitly produces an IID of the projective image, which in particular provides a simple way to construct IIDs of the quadrics in three-space.

05 Nov 2025, 11:30 CET: JA, surfaces and beyond

Masaya Hara (NIT, Anan College; Kobe University): Darboux transformations between zero mean curvature surfaces

Abstract

Darboux transformations are transformations that preserve the isothermicity of surfaces and have been studied not only in classical differential geometry but also actively in modern differential geometry.

In this talk, we impose an additional condition on Darboux transformations and study such transformations between zero mean curvature surfaces in the Euclidean, Minkowski, and isotropic three-spaces. In comparing these spaces, we analyze and contrast their geometric behaviors, including singularities and ends.

This talk is based on joint work with J Cho, A Honda, T Raujouan, and W Rossman.

29 Oct 2025: Geometry seminar

Ruzica Mijic-Rasoulzadeh (TU Wien): Sphericity and geometric properties of ratios in Möbius and Laguerre geometry

Abstract

In recent years, several applications ($S$-nets, $S^\ast$-nets, circular nets, conical nets, etc.) have renewed the interest in incidence conditions involving circles and spheres. This talk develops a systematic comparison of incidence theorems in Möbius and Laguerre geometry, in terms of algebraic ratios. As a starting point, we recall a well-known Möbius theorem, stating that four points in the plane lie on a circle if and only if their cross-ratio is real. We dualize this statement for the Laguerre case, and look at four further incidence theorems, increasing the dimension and the number of objects involved, respectively (for example five planes touching a common sphere). In particular, we use the cross-ratio as long as the number of objects remains four, whereas for five objects we need the novel diagonal-ratio. With the appropriate algebraic structures chosen for each configuration, the resulting incidence theorems take strikingly parallel forms, highlighting the close relationship between Möbius and Laguerre geometry.

22 Oct 2025: Geometry seminar

Hans-Peter Schröcker (University of Innsbruck): Recent results on rational PH curves

Abstract

A polynomial or rational parametric curve $r(t)$ is said have a "Pythagorean Hodo-graph" if $\langle r'(t),r'(t)\rangle$ is a square in the ring of polynomials $R[t]$ or in the field of rational functions $R(t)$, respectively. Curves with this property are called PH curves. They offer some advantages over conventional polynomial or rational curves in typical CAGD constructions or in the control of objects moving along PH trajectories. While polynomial PH curves are well-studied and understood, less is known for rational PH curves - probably due to the lack of convenient construction methods. This has changed recently. We present a simple construction for rational PH curves and then continue with three further topics:

Rational curves with a rational arc-length function. Rational curves with a (piecewise) rational arc-length function are necessarily PH but only few rational PH curves enjoy this property (while the arc length function of polynomial PH curves is always at least piecewise polynomial). They have an interpretation as curves of constant slope in 4D and the ensuing constraints can be conveniently incorporated into the construction of general rational PH curves.

Bounded and regular rational PH curves. In contrast to polynomial curves - which are always infinite - rational curves may be bounded. Of particular interest are bounded rational PH curves as they give rise to closed rational framing motions that can be used, for example, as camera trajectories. While previous approaches rely on piecewise constructions with limited smoothness, we design bounded rational PH curves with closed framing motions. The challenge is to ensure curve regularity. We present a simple necessary criterion to ensure this and sketch the proof for its sufficiency.

Minimal surfaces as complex PH curves. Finally, we talk about a recently dis- covered relations between PH curves and minimal surfaces. A result from quaternionic function theory ensures that any polynomial or rational minimal surface in isothermal parametrization can be obtained by a complex extension of our construction method. The parameter lines of these surface parametrizations are necessarily PH curves.

This is joint work, mostly with Zbynek Sir (Charles University in Prague), but also with Amedeo Altavilla (Universita degli Studi di Bari Aldo Moro) and Jan Vrsek (University of West Bohemia).

09 Oct 2025, 12:30 CEST: JA, surfaces and beyond

Hirotaka Kiyohara (Osaka Kyoiku University): Singularities on timelike minimal surfaces in the three-dimensional Heisenberg group

Abstract

Most timelike minimal surfaces in the $3$-dimensional Heisenberg group can be represented via Lorentzian harmonic maps into the de Sitter $2$-sphere. These surfaces naturally admit singularities, and we provide a characterization of several types of them. This talk is based on joint work with Shintaro Akamine.