# Events

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

    January 2022
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## 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 15:00 in the Zeichensaal 1 (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.

## This semester's schedule

16 Mar 2022: Geometry seminar
15 Dec 2021
postponed to 26 Jan 2022: Geometry seminar
Alexander Glazman (Uni Wien): Phase transitions in two dimensions

## Abstract

Phase transitions are natural phenomena when a small change of external parameter, like temperature, leads to a drastic change of the properties of the material: ice melting to water, ferromagnets becoming paramagnets above the Curie temperature, etc. To study these phenomena, one introduces lattice models - particles are placed in a lattice and only adjacent particles interact. In two dimensions, this leads to beautiful conjectures of universality and conformal invariance, with random fractal SLE curves appearing in the limit. The talk aims to describe some of the classical results and the state of the art in the area now. If time permits, we will also discuss the main ideas in the proofs.
24 Nov 2021
postponed to 12 Jan 2022 (online): Geometry seminar
Corentin Fierobe-Kozyreva (IST Austria): From billiards to projective billiards

## Abstract

A billiard is a bounded domain in which we study the trajectory of a ray of light obtained after successive reflections on the boundary of the domain. Reflections are given by the classical law of optics: angle of incidence = angle of reflection. These dynamical systems are intensively studied, and many interesting questions arise. Ivrii's conjecture is one of them: it states that given a billiard in a Euclidean space, the set of its periodic orbits has zero measure. During this talk, we will present another type of billiards introduced by S. Tabachnikov called projective billiards, which generalizes different types of billiards, and show how they can be useful to understand questions related to usual billiards like Ivrii's conjecture.

08 Nov 2021 (Mon!): Geometry seminar 16:00, ZS1
Ilya Kossovskiy (TU Wien): Mapping problem for real submanifolds in complex space

## Abstract

In this talk, I give a broad overview of the CR (Cauchy-Riemann) geometry, which is the geometry of an embedded real submanifold in complex space subject to transformations (CR maps) preserving the arising CR structure on a submanifold. CR geometry goes back to the 1907 work of H.Poincare, and was later developed in celebrated papers by E.Cartan, Tanaka, Chern-Moser, and a large number of subsequent publications. CR geometry is particularly interesting in that it can be viewed from 3 totally different prospectives: respectively Complex Analysis, Differential Geometry, and Linear PDEs. In the talk, I will outline all the three approaches, mention certain classical and more recent notable advances in the field, and formulate a few open problems.
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).

## Talks in the geometry seminar

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