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

27 Jun 2024: Geometry seminar
Ilya Kossovskiy (Masaryk University in Brno and TU Wien): TBA
20 Jun 2024: Geometry seminar
Denis Polly (TU Wien): TBA

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13 Jun 2024: Geometry seminar
Fabian Achammer (TU Wien): TBA

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06 Jun 2024: Geometry seminar
Kiumars Sharifmoghaddam (TU Wien): TBA

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23 May 2024: Geometry seminar
Georg Nawratil (TU Wien): A global approach for the redefinition of higher-order flexibility and rigidity

Abstract

The famous example of the double-Watt mechanism given by Connelly and Servatius [Higher-order rigidity - What is the proper definition? Discrete & Computational Geometry 11:193-200, 1994] raises some problems concerning the classical definitions of higher-order flexibility and rigidity, as they attest the cusp configuration of the mechanism a third-order rigidity, which conflicts with its continuous flexion. Some attempts were done to resolve the dilemma but they could not settle the problem. According to Müller [Higher-order analysis of kinematic singularities of lower pair linkages and serial manipulators. Journal of Mechanisms and Robotics 10:011008, 2018] cusp mechanisms demonstrate the basic shortcoming of any local mobility analysis using higher-order constraints. Therefore we present a global approach inspired by Sabitov's finite algorithm for testing the bendability of a polyhedron given in [Local Theory of Bendings of Surfaces. Geometry III, pp. 179-250, Springer, 1992], which allows us (a) to compute iteratively configurations with a higher-order flexion (e.g. all configurations of a given 3-RPR manipulator with 3rd-order flexion) and (b) to come up with a proper redefinition of higher-order flexibility and rigidity.
16 May 2024: Geometry seminar
Martina Iannella (TU Wien): Classification of non-compact $3$-manifolds

Abstract

A classification problem consists of an equivalence relation on some set of mathematical objects; a solution to such a problem is an assignment of complete invariants. In this talk we consider the problem of classifying non-compact 3-manifolds up to homeomorphism from the perspective of descriptive set theory. We first look at the parametrization of 3-manifolds as objects of a Borel subset of a Polish space. We then discuss the framework of Borel reducibility, a standard tool for comparing the complexity of different classification problems, and present our recent result which determines the exact complexity of the classification of non-compact 3-manifolds up to homeomorphism. This is joint work with Vadim Weinstein.
02 May 2024: Geometry seminar
Niklas Affolter (TU Wien): Discrete maximal Lorentz surfaces and incircular nets

Abstract

Incircular nets (s-embeddings) were introduced by Chelkak as a generalization of Smirnov's approach to study the conformal invariance of the Ising model in the continuous limit. We build upon the work of Chelkak, Laslier and Russkikh to present a class of incircular nets that corresponds to discrete isothermic surfaces in Lorentz space. As a special case, we identify discrete maximal surfaces, which are discrete surfaces with vanishing discrete mean curvature. In this way, we introduce a result on the discrete level that was obtained by CLR in the limit. We also introduce an associated family of discrete maximal surfaces and the corresponding family of incircular nets. Joint work with Dellinger, Müller, Polly, Smeenk and Techter.
25 Apr 2024: Geometry seminar
Alessandro Andretta (University of Turin): The Banach-Tarski paradox

Abstract

One of the most surprising results of modern mathematics is the following result proved by Hausdorff, Banach and Tarski: the unit ball of the euclidean space can be partitioned in a finite number of pieces so that these can be rearranged, using rigid motions so to form two balls identical to the original. The proof is non-constructive, relying on the Axiom of Choice, and the pieces of the decomposition are inconceivably sharp and edgy! Geometry plays a substantial role, as the core of the proof is based on the existence of a free subgroup of the group of rotations. (A similar result cannot be proved for the plane, i.e. it is not possible to duplicate a disk.)

In this talk I will sketch the proof of the Banach-Tarski paradox, and survey many related results that have been proved in the following years.

18 Apr 2024: Geometry seminar
Ivan Izmestiev (TU Wien): Cayley-Bacharach theorem and sums of squares

Abstract

The Cayley-Bacharach theorem (first proved by Chasles) says that if two cubics meet at nine points, then any other cubic passing through eight of these nine points also passes through the ninth. This theorem includes as special cases the Pappus and the Pascal theorems.

The sums of squares problem was posed by Hilbert: can every positive definite homogeneous polynomial of degree $2d$ in n variables be represented as a sum of squares of polynomials of degree $d$? While the answer is positive for $d=1$ and n arbitrary as well as for d arbitrary and $n=2$, Hilbert has proved the negative for $d=3$ and $n=3$. And a crucial point in his proof was the Cayley-Bacharach theorem.

This talks is based on the articles by Eisenbud-Green-Harris and Blekherman.

21 Mar 2024: Geometry seminar
Gudrun Szewieczek (TU Munich): Discrete isothermic nets with a family of spherical parameter lines from holomorphic maps

Abstract

Smooth surfaces with a family of planar or spherical curvature lines are an active area of research, driven by both purely differential geometric aspects and practical applications such as architectural design. In integrable geometry it is a natural question to ask which of these surfaces admit a conformal curvature line parametrization and are therefore isothermic surfaces.

It is an open problem to explicitly describe all those smooth isothermic surfaces. However, over time, prominent examples were found in this rich integrable surface class: above all Wente's torus. More recently, further specific examples have led to the discovery of compact Bonnet pairs and to free boundary solutions for minimal and CMC-surfaces.

This talk covers a discrete version of the problem: we shall generate all discrete isothermic nets with a family of spherical curvature lines from special discrete holomorphic maps via the concept of "lifted-folding". In particular, we point out how this novel approach leads to quasi-periodic solutions and to topological tori with symmetries.

This is joint work with Tim Hoffmann.

14 Mar 2024: Geometry seminar (Sem.R. DB gelb 03)
David Sykes (TU Wien): CR Hypersurface Geometry, an Introduction

Abstract

CR geometry concerns structures on real submanifolds in complex spaces that are preserved under biholomorphisms. This talk will present a light introduction to CR geometry of real hypersurfaces. We will survey some of the area's major classical results, namely solutions to local equivalence problems of E Cartan, Tanaka, and Chern-Moser and their applications. And we will preview some of the area's current-day research trends related to Levi degenerate structures.

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