# Events

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

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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 Monday at 16:15 in the Zeichensaal 1 or 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

13 Jun 2022: Geometry seminar (ZS1)
Jonas Tervooren (TU Wien): Cone-Nets

## Abstract

Cone-nets are conjugate nets on a surface such that along each individual curve of one family of parameter curves there is a cone in tangential contact with the surface. The corresponding conjugate curve network is projectively invariant and is characterized by the existence of particular transformations. We study properties of the corresponding transformation theory and illustrate how several known surface classes appear within our framework. We present cone-nets in a classical smooth setting of differential geometry as well as in the context of a consistent discretization with counterparts to all relevant statements and notions of the smooth setting. Special emphasis deserve smooth and discrete tractrix surfaces as those cone-nets which are characterized as principal nets with constant geodesic curvature along one family of parameter curves.

13 Mai 2022 (Fri!) 11:00: Geometry seminar (Dekanatsraum 9th floor)
Andrew Sageman-Furnas (NC State University): Constructing isometric tori with the same curvatures

## Abstract

Which data determine an immersed surface in Euclidean three-space up to rigid motion? A generic surface is locally determined by only a metric and mean curvature function. However, there are exceptions. These may arise in a family like the isometric family of vanishing mean curvature surfaces transforming a catenoid into a helicoid, or as a so-called Bonnet pair of surfaces. For compact surfaces, Lawson and Tribuzy proved in 1981 that a metric and non-constant mean curvature function determine at most one immersion with genus zero, but at most two compact immersions (compact Bonnet pairs) for higher genus.

In this talk, we discuss our recent construction of the first examples of compact Bonnet pairs. It uses a local classification by Kamberov, Pedit, and Pinkall in terms of isothermic surfaces. Moreover, we describe how a structure-preserving discrete theory for isothermic surfaces and Bonnet pairs led to this discovery. The smooth theory is joint work with Alexander Bobenko and Tim Hoffmann and the discrete theory is joint work with Tim Hoffmann and Max Wardetzky.

02 May 2022: Geometry seminar (online/zoom)
Denis Polly (TU Wien): Channel linear Weingarten surfaces in smooth and discrete differential geometry

## Abstract

In this talk, we consider linear Weingarten (lW) surfaces in space forms. These surfaces are defined via a linear relation between their Gauss and mean curvature. This class of surfaces contains many important subclasses: minimal surfaces in Euclidean space, flat fronts in hyperbolic space or, more generally, surfaces with constant mean or Gauss curvature. Channel lW surfaces are those lW surfaces that are given as the envelopes of a 1-parameter family of spheres. Delaunay's surfaces, constant mean curvature surfaces of revolution, are an example of this.

Channel lW surfaces have been studied with many different methods. We will use a Lie sphere geometric setup that allows a unified treatment of surfaces in different space forms. We aim for a complete classification of all channel lW surfaces in terms of transparent and well-behaved parametrisations.

In the realm of discrete differential geometry, circular nets allow for a notion of face-wise Gauss and mean curvature that is inspired by Steiner's formula for parallel surfaces. A notion of discrete channel surfaces as envelopes of a one-parameter family of Dupin cyclides has recently been developed.

In the discrete portion of the talk, we will investigate discrete channel lW surfaces and present results that are similar to the smooth case (with some notable changes). We will then give an outline for further research that is planned in this area.

04 Apr 2022: Geometry seminar Zoom
Anna Felikson (Durham): Friezes for a pair of pants

## Abstract

Frieze patterns are numerical arrangements that satisfy a local arithmetic rule. Conway and Coxeter showed that frieze patterns are tightly connected to triangulated polygons. Recently, friezes were actively studied in connection to the theory of cluster algebras, and the notion of a frieze obtained a number of generalisations. In particular, one can define a frieze associated with a bordered marked surface endowed with a decorated hyperbolic metric. We will review the construction and will show that some nice properties can be extended to friezes associated to a pair of pants. This work is joint with Ilke Canakci, Ana Garcia Elsener and Pavel Tumarkin, arXiv:2111.13135.
21 Mar 2022: Geometry seminar Zoom
Boris Springborn (TU Berlin): The hyperbolic geometry of Markov's theorem on Diophantine approximation and quadratic forms

## Abstract

Markov's theorem classifies the worst approximable irrational numbers, and the indefinite binary quadratic forms that are "most nonzero" for integer arguments. This talk is about the theorem and a new proof using hyperbolic geometry. The main ingredients are a dictionary to translate between hyperbolic geometry and algebra/number theory, and some very basic tools borrowed from modern geometric Teichmüller theory. Simple closed geodesics and ideal triangulations of the modular torus play an important role, and so does the problem: How far can a straight line crossing a triangle stay away from the vertices?

## Talks in the 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).