News

27 Jun 2024: Geometry seminar

Ilya Kossovskiy (Masaryk University in Brno and TU Wien): Sphericity and analyticity of a strictly pseudo-convex hypersurface in low regularity

Abstract

It is well known that the sphericity of a strictly pseudoconvex CR-hypersurface amounts to the vanishing of its Chern-Moser tensor. The latter is computed pointwise in terms of the 6-jet of the hypersurface at a point, and thus requires regularity of the hypersurface of class at least C^6. In our joint work with Zaitsev, we apply our recent theorem on the analytic regularizability of a strictly pseudoconvex hypersurface to find a necessary and sufficient condition for the sphericity of a strictly pseudoconvex hypersurfaces of arbitrary regularity starting with C^2. Further, we obtain a simple condition for the analytic regularizability of hypersurfaces of the respective classes. Surprisingly, despite of the seemingly analytic nature of the problem, our technique is geometric and is based on the Reflection Principle in SCV.

20 Jun 2024: Geometry seminar

Denis Polly (TU Wien): Linear Weingarten surfaces in isotropic spaces

Abstract

According to a result by Burstall, Hertrich-Jeromin and Rossman, linear Weingarten surfaces in Riemannian and Lorentian space forms are the envelopes of isothermic sphere congruences with constant curvature. This description is derived via the use of Lie sphere geometry (LSG) and symmetry breaking to obtain results about metric subgeometries of LSG. While this method recovers all linear Weingarten surfaces in Riemannian and Lorentzian space forms, surfaces in isotropic space, as described by Strubecker, do not appear. Our goal is to close this gap.

To this end, we introduce a way of breaking symmetry that has not recieved much attention. This leads us to the study of isotropic space forms and the linear Weingarten surfaces therein. As an application we describe Weierstrass-type representations for certain linear Weingarten surfaces in these space forms.

13 Jun 2024: Geometry seminar

Fabian Achammer (TU Wien): Formula equations and the affine solution problem

Abstract

Formula equations are certain kinds of equations whose solutions are logical formulas. They serve as a common framework for many different problems in computational logic, ranging from software verification to inductive theorem proving.

We start with a short introduction to mathematical logic and computational problems, then introduce formula equations and give a glimpse of their wide applicability.

In the main part of the talk we explore the solution of a particular problem in the area of formula equations - the affine solution problem - which translates into a problem about affine spaces.

Finally, we discuss some results surrounding a generalization of the affine solution problem, which is still open, namely the convex solution problem which is a computational problem about convex polytopes.

06 Jun 2024: Geometry seminar

Kiumars Sharifmoghaddam (TU Wien): Rigid-foldable Quad Meshes with Control Polylines: Interactive Design and Motion Simulation

Abstract

Generic discrete surfaces composed of quadrilateral plates connected by rotational joints in the combinatorics of a square grid are rigid, but there also exist special ones with 1-parametric flexibility. This dissertation focuses on two particular classes of so-called T-hedra (trapezoidal quad surfaces) and V-hedra (discrete Voss surfaces). T-hedra can be thought of as a generalization of discrete surfaces of revolution in such a way that the axis of rotation is not fixed at one point but rather sweeping a polyline path on the base plane. Moreover, the action does not need to be a pure rotation but can be combined with an axial dilatation. After applying these transformations to the breakpoints of a certain discrete profile curve, a flexible quad-surface with planar trapezoidal faces is obtained. Therefore, the design space of T-hedra also includes as subclasses discretized translational surfaces and moulding surfaces beside the already mentioned rotation surfaces. V-hedra are the discrete counterpart of Voss surfaces which carry conjugate nets of geodesics. In discrete case the opposite interior angles of a vertex star are equal. From a V-hedral vertex one can always generate an anti-V-hedral vertex with the same kinematics, in which the sum of corresponding opposite angles equal to pi and therefore is a known case of valence four flat-foldable and developable origami vertex. The author developed Rhino/Grasshopper plugins, implemented with C-sharp, which make the design space of T-hedra, V-hedra and anti-V-hedra accessible for designers and engineers. The main components enable the user to design these quad surfaces interactively and visualize their deformation in real time based on a recursive parametrization of the quad-mesh vertices under the associated isometric deformation. Furthermore, this research investigates semi-discrete T-hedral surfaces and other topologies, such as tubular structures composed of T-hedra.

23 May 2024: Geometry seminar (14:30 Dekanatsraum 9th floor)

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.