
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
(hover/tap name or title to view more information)
 16 Mar 2022: Geometry seminar
 Athanase Papadopoulos (IRMA Strasbourg): TBA
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 FierobeKozyreva (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
AbstractIn this talk, I give a broad overview of the CR (CauchyRiemann) 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, ChernMoser, 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
(lambdaconcave bodies), or below (lambdaconvex bodies). We
allow the boundary of $K$ to be nonsmooth, in which case the
bounds on the principal curvatures are defined in the barrier
sense, and thus the definition of lambdaconvex/concave
bodies makes sense in a variety of discrete settings. The
intersection of finitely many balls of radius 1 is an example
of a 1convex body, while the convex hull of finitely many
balls of radius 1 is an example of a 1concave body.
Under uniform convexity assumption, for convex bodies of, say,
given volume, there are nontrivial upper and lower bounds
for various functionals, such as the surface area, in, and
outerradius, 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 wellposed and in many cases highly nontrivial
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 VanSon (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).

Summer term 2021 
Talks in the geometry seminar
(hover/tap name or title to view more information)
 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 TaitKneser theorem
Abstract
The TaitKneser theorem, first demonstrated by Peter
G. Tait in 1896, states that the osculating circles along
a plane curve with monotone nonvanishing 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 research, 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 realworld 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 2dimensional congruence of circles in 3space
is called cyclic if it admits a 1parameter 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 congruences 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:
 Trace and determinant of the shape operator
fit a discrete version of the Steiner formula for
offset surfaces.
 The parameter lines of a principal mesh follow the
eigenvectors of the shape operator.
 Conjugate meshes/Koenigs meshes are mapped to
conjugate meshes/Koenigs meshes under projective
transformations.
 Isothermic meshes and in general Koenigs meshes are
dualizable.
 Isothermic meshes/principal meshes are mapped to
isothermic meshes/principal meshes under Moebius
transformations.
 10 Mar 2021: Geometry seminar
 Efilena Baseta (TU Wien): TBA


Events in former years
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