### Talks in 2012

#### Vortrag im Seminar Diskrete Mathematik und Optimierung

Title: Random Walks on critical Percolation-Trees
Speaker: Florian Sobieczky (University of Denver)
Date: 18.12. 2012, 14:15
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

Bounds for the expected return probability of the delayed random walk on
finite clusters of an invariant percolation on transitive unimodular
graphs are derived. They are particularly suited for the case of critical
Bernoulli percolation and the associated heavy-tailed cluster size
distributions. The upper bound relies on the fact that cartesian products
of finite graphs with cycles of a certain minimal size are Hamiltonian.
For critical Bernoulli bond percolation on the homogeneous tree this bound
is sharp. The asymptotic type of the expected return probability for large
times t in this case is of order of the 3/4'th power of 1/t.

#### Joint Seminar

Title: Lumped Markov chains and entropy rate preservation
Speaker: Bernhard Geiger and Christoph Temmel (TU Graz)
Date: 10 Dez, 16:00
Room: IDEG134
Abstract:

A lumping of a Markov chain is a coordinate-wise projection of the chain. We characterize the entropy rate loss of a lumping of a stationary Markov chain on a finite state space in two ways. First, by the asymptotic ratio of the number of trajectories with positive weight between the original and the lumped chain. Second, by the reconstructability of original trajectories from their images under the lumping. Every non-trivial lumping of a Markov chain with positive transition matrix incurs an entropy rate loss. We give sufficient conditions on the non-positive transition matrix and the lumping to preserve the entropy rate. In the sparse setting we state sufficient conditions on the lumping to both preserve the entropy rate and result in a k-th order homogeneous Markov chain.

#### Kolloquium: Mathematische Methoden in den Natur- und Ingenieurwissenschaften

Title: Gauss, Jacobi, Seidel, Richardson, Krylov: The Invention of Iterative Methods
Speaker: Univ.-Prof. Dr. Martin J. GANDER (Universität Genf)
Date: Freitag, 7.12.2012, 16:00 Uhr
Room: TU Graz, Steyrergasse 30, 3. Stock, Seminarraum C307
Abstract:

Title: Optimal Transport, Model-Indepedence, and Trajectorial Inequalities.
Speaker: Mathias Beiglböck (Universität Wien)
Date: Freitag, 7.12.2012, 14:15 Uhr
Room: Seminarraum A206, Steyrergasse 30, 2.Stock, Geodäsie
Abstract:

Abstract:
We will explain a recently discovered connection between Optimal
Transport and the areas of model independence / martingale
inequalities in probability. This link has a number of fruitful
consequences. For instance, the duality theorem from optimal transport
leads to new super-replication results. Optimality criteria from the
theory of mass transport can be translated to the martingale setup and
allow to characterize minimizing/maximizing models in finance.
Moreover, the dual viewpoint provides new
insights to the classical inequalities of Doob and Burkholder-Davis-Gundy.

#### Strukturtheorie-Seminar

Title: The Bee in the Balloon: Recurrence on Growing Subgraphs
Speaker: Florian Sobieczky (University of Denver)
Date: 6.12.2012, 15:00
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

What is the rate at which the radius of a ball has to be continuously increased such that a diffusing particle, reflected back inside when hitting the boundary, becomes transient? It is shown that the question has the same answer for brownian motion in three-dimensional euclidean space and for simple random walk on the euclidean lattice: $\sim t^{1/d}$.

#### DK Seminar

Title: Discretized rotation has infinitely many periodic orbits
Speaker: Attila Pethoe (Debrecen)
Date: 21.12.2012, 10:30
Room: HS Thermoprozesstechnik, Montanuniversität Leoben
Abstract:

After the lecture of Attila Petho in Leoben, there will be a lunch in a nearby
restaurant.

#### Minikurs/Intensivseminar

Title: Fortsetzungen von Quasirandtripeln
Speaker: Till Micheler (Institut für Mathematik, TU Berlin)
Date: 4.12. 13:00-14:00 im A206, 5.12. 8:15-9:45 im A111
Room: und 6.12. 13:00-16:00 im A306
Abstract:

Quasirandtripel sind ein modernes Werkzeug in der
Erweiterungstheorie von symmetrischen Operatoren in Hilberträumen
mit vielen konkreten Anwendungsmöglich- keiten, z.B. im Bereich der
elliptischen partiellen Differentialgleichungen.

In diesem Intensivseminar diskutieren wir
Fortsetzungen und Fortsetzbarkeit von Quasirandtripeln, das heisst
wir betrachten hinreichende und notwendige Bedingungen, unter welchen
sich die Randabbildungen eines Quasirandtripels
auf den maximalen Definitionsbereich des zugrundeliegenden Operators
fortsetzen lassen. Dadurch werden in natürlicher Weise
auch das gamma-Feld und die Weylfunktion des Quasirandtripels
fortgesetzt, und es gelten erweiterte Krein'sche Formeln.

#### Strukturtheorie-Seminar

Title: Rate of convergence and Edgeworth-type expansion in the entropic central limit theorem
Date: 29.11.2012, 15:00
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

An Edgeworth-type expansion is established for the entropy distance to the class of normal distributions of sums of i.i.d. random variables or vectors, satisfying minimal moment conditions.

#### Kolloquium

Title: Festkolloquium anlässlich des 80. Geburtstages von em. Univ. Prof. Dr. Ulrich DIETER
Speaker: ()
Date: Freitag, 30. November 2012, 14.00 Uhr
Room: HS D, Kopernikusgasse 24/III
Abstract:

#### Vortrag im Seminar Diskrete Mathematik und Optimierung

Title: The H-elimination random graph process
Speaker: Tamas Makai (Institut für Optimierung und Diskrete Mathematik, TU Graz)
Date: 27.11. 2012, 14:15
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

Consider the random graph process which starts out from the complete graph on $n$
vertices and in every step of the process an edge, selected uniformly at random from
the set of edges which are contained in a copy of a fixed graph $H$, is removed. The
process stops after no more copies of $H$ are present. This process is called the
$H$-elimination random graph process. In 1990 Bollob\'as and Erd\H{o}s asked for the
typical number of edges present in the graph created by this process when $n$ is
large. We answer this question in case $H$ belongs to a special class of graphs,
namely the strictly 2-balanced graphs.

#### Mathematisches Kolloquium

Title: Integral Geometry and Isoperimetric Inequalities
Speaker: Franz Schuster (TU Wien)
Date: 23.11.2012, 14:30 (Kaffee Institut f. Geometrie), 15:00 (Vortrag Hörsaal A)
Room: HS A, Kopernikusgasse 24, 1. Stock
Abstract:

This lecture presents some of the fascinating developments which happened in Integral Geometry in the last years, establishing connections to Differential Geometry and Functional Analysis. The concept of {\em valuation} is central to this development, which means functions $\phi$ on the set of convex bodies (or of more general subsets of ${\mathbb R}^n$) which have values in a certain abelian semigroup and which enjoy the property $\phi(K) + \phi(L) = \phi(K \cup L) + \phi(K \cap L)$ whenever all of $K$, $L$, $K \cup L$ are convex. Valuations, as generalizations of measures, have long played an important role in geometric analysis, starting with Dehn's solution of Hilbert's third problem and the theory of dissections of polytopes. In the last 5 to 10 years strong connections between affine geometry and valuations emerged: A series of fundamental operators on convex bodies, like the projection body and intersection body, could be classified in terms of the valuation property and compatibility with affine mappings. These results have been applied to the theory of affine isoperimetric inequalities and in particular were used to significantly sharpen several classical inequalities of Euclidean geometry.

#### Strukturtheorie-Seminar (Korrektur)

Title: HIERARCHICAL LAPLACIANS: TWO EXAMPLES
Speaker: Prof Alexander Bendikov (Univ. Wroclaw, Polen)
Date: Donnerstag, 22.11.2012, 15 Uhr
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

We consider two examples of Markov generators defined on a discrete group which is an infinite sum of cyclic groups and give the precise asymptotic behavior at
infinity of the corresponding return probability functions. These results,
in turn, give sharp bounds on the corresponding heat kernels.

#### Vortrag im Seminar Diskrete Mathematik und Optimierung

Title: Higher Inclusion Matrices
Speaker: Yury Person (Institut für Mathematik, FU Berlin)
Date: 20.11.2012, 14:15
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

Abstract:

Let $n\ge r \ge s \ge 0$. The higher inclusion matrix $M_s^r({ \left[n\right]\choose r})$ is a $\{0,1\}$-matrix whose
rows are indexed by all $r$-element subsets of $\left[ n\right]:=\{1,2,\ldots, n\}$ and and columns
are indexed by all $s$-subsets of $\left[n\right]$ and the entry corresponding to an $r$-set $R$ and an $s$-set $S$ is $1$ if $S \subseteq R$ and $0$ otherwise.

Gottlieb's theorem from 1966 states that $M_s^r({\left[n\right]\choose r})$ has the rank
$\min\{{n\choose r}, {n \choose s}\}$ over $\mathbb{Q}$.
Keevash asked how many rows one has to delete from $M_s^r({\left[n\right] \choose r})$ to reduce its rank by $1$. We answer his
question for large $n$ and study some generalizations of this problem. Joint work with Codru{\c t} Grosu and Tibor Szab\'o.

#### Vortrag

Title: Subdivision in shift-invarianten Räumen
Speaker: Kurt Jetter (Univ. Hohenheim, Stuttgart)
Date: Montag 19.11.2012, 11:00 Uhr
Room: Seminarraum 2, Kopernikusgasse 24, 4. Stock
Abstract:

Shift-invariante Räume und ihre Approximationseigenschaften in Bezug auf
wichtige Klassen von Funktionenräumen spielen in der Angewandten Mathematik
und in der Signalverarbeitung eine zentrale Rolle. Erfüllen die Generatoren
solcher Räume zusätzlich eine Zwei- oder Mehr-Skalen-Gleichung, so sind
diese Räume und ihre skalierten Versionen die Grundlage für
Multiskalen-Analysen, wie sie z.B. bei Wave\-let-Zerlegungen oder bei
allgemeineren Frame-Darstellungen sehr effizient eingesetzt werden.

Der Vortrag führt zunächst in die Struktur shift-invarianter Räume und
deren Approximationseigenschaften ein. Anschließend wird der Spezialfall
des Vorliegens einer Zwei-Skalen-Gleichung behandelt und der Zusammenhang

Anschließend werden zwei Themenbereiche herausgegriffen, in denen der
Vortragende --- zusammen mit Koautoren --- in letzter Zeit spezielle Fragen zur
Subdivision aufgegriffen und abschließend bearbeitet hat:

-- Polynomiale Reproduktion und Box-Spline-Generatoren für gewisse
bivariate Varietäten.
-- Nichtnegative Subdivision und ihr Bezug zu endlichen nichtstationären
Markov-Prozessen.

\begin{enumerate}\itemsep-\parsep\footnotesize

\item K. Jetter und G. Plonka, A survey on $L_2$-approximation orders from
shift-invariant spaces, in: {\it Multivariate Approximation and
Applicatons}, N. Dyn et al., eds., pp.
73--111, Cambridge University Press, 2001.

\item M. Charina, C. Conti, K. Jetter und G. Zimmermann, Scalar multivariate
subdivision schemes and box splines, {\it Computer Aided Geometric Design}
{\bf 28} (2011), 285--306.

\item K. Jetter und X. Li, SIA matrices and non-negative subdivision, {\it
Results Math.} (2012).
\end{enumerate}

#### Zahlentheoretisches Kolloquium

Title:
Speaker: ()
Date: Freitag, 16.11.2012
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz
Abstract:

{\bf 14:15: Daniel Dombek} (Czech Technical Univ. Prag)
{ On the generalizations of the unit sum number
problem (joint work with Lajos Hajdu, Attila Pethõ)}[3mm]
Abstract: In this contribution we consider representations of algebraic
integers of a number field as linear combinations of units with
coefficients from a fixed small set, and as sums of elements with
bounded norms. Presented theorems can be viewed as results
concerning a generalization of the so-called unit sum number
problem. Extending previous related results we also give an upper
bound for the length of arithmetic progressions of $t$-term sums of
algebraic integers with bounded norms.

{\bf 14:45: Tomás Vávra} (Czech Technical Univ. Prag)
{ Arithmetics in number systems with negative
Abstract: We study $(-\beta)$-numeration systems as introduced by S. Ito and
T. Sadahiro in 2009. We are interested in the set $\mathbb Z_{-\beta}$ of $(-\beta)$-integers, i.e. the numbers whose
$(-\beta)$-expansion does not contain fractional digits. The
results about the number of fractional digits arising when summing
$(-\beta)$-integers can be extended from the class of quadratic
Pisot units $\beta$ to the non-unit case. We also comment on
exceptional arithmetic and geometric properties of a class of bases
$-\beta$ where $\beta^r=m\beta^{r-1}+\cdots + m\beta+m$, $r\geq 2$,
$m\geq 1$.

{\bf 15:15: Manfred Madritsch} (Université de Lorraine, Nancy)
{ Van der Corput sets}[3mm]
Abstract: A set $H$ of positive integers is a van der Corput set (or a vdC set)
if for any sequence $(x_n)_{n\in\mathbb{Z}}$ of real numbers, and for each
$h\in H$, the sequence $(x_{n+h}-x_n)_{n\in\mathbb{Z}}$ is uniformly
distributed mod 1, in which case the sequence $(x_n)_{n\in\mathbb{Z}}$ is
uniformly distributed mod 1. The aim of the present talk is on the one
hand to link this notion to others like recurrent sets or sets forcing
continuity. On the other hand constructions of vdC sets are
given. Finally extensions of this questions to sets in $\mathbb{Z}^d$ are considered.

#### Zahlentheoretisches Kolloquium

Title:
Speaker: ()
Date: Donnerstag, 15.11.2012
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz
Abstract:

{\bf 16:15: Andrej Dujella} (Univ. of Zagreb)
{ Elliptic curves with large torsion and positive rank
over number fields of small degree}

{\bf 16:45: Tomislav Pejkovic} (Univ. of Zagreb)
{ An inequality for values of Koksma's function of two

{\bf Kaffeepause}

{\bf 17:30: Vinko Petricevic} (Univ. of Zagreb)
{ Householder's approximants and continued fraction expansion

#### Strukturtheorie-Seminar

Title: Convergence in Norm for Non-Commutative Central Limit Theorems
Speaker: Octavio Arizmendi (Universität des Saarlandes)
Date: 15.11.2012, 15:00
Room: Seminarraum C208, Steyrergasse 30/II
Abstract:

In this the talk I will explain how to use cumulants to give a simple proof of an instance of the so-called superconvergence of normalized sums of free random variables. Namely, that the operator norm of normalized sums of bounded free random variables with mean 0 and variance 1, converge to 2. Moreover, our approach generalizes in a straightforward way to monotone and boolean independence and q-convolution. This is a joint work with Carlos Vargas.

#### Strukturtheorie-Seminar

Title: Self-similar groups: constructions, applications and open problems
Date: Dienstag, 13.11.2012, 10:45
Room: SR C307, Steyrergasse 30, 3. Stock
Abstract:

Self-similar groups act by automorphisms on rooted trees or, equivalently, are generated by (finite) automata. Such groups have been used to find groups with exceptional and exotic properties, for example the Grigorchuk group, the first group with intermediate growth, is self-similar. In this talk I want to describe some constructions related to such groups that have applications in combinatorics, dynamics and probability. I also plan to present a list of new developments and open problems.

#### Vortrag im Seminar Diskrete Mathematik und Optimierung

Title: On the tree-packing conjecture of Gyarfas and Lehel
Speaker: Anusch Taraz (Zentrum Mathematik, TU München)
Date: 13.11. 2012, 14:15
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

In 1976, Gyarfas and Lehel made the somewhat stunning conjecture that any family of trees $T_1,T_2,\ldots T_n$, with
$1,2,\ldots, n$ vertices respectively, can be packed in an edge-disjoint manner into the complete graph on n vertices.
This conjecture is still open.

In this talk I will sketch a proof for a slightly weakened version of this conjecture, where we only consider
trees of bounded maximum degree and allow the complete graph to have an additional $o(n)$ vertices.

The proof uses tree-indexed random walks controlled by the nibble method and is joint work with J.Böttcher, J.Hladky and D.Piguet.

#### Seminar of the Doctoral School

Title: Doctoral Day
Speaker: ()
Date: 9.11.2011, 10:13-12:30
Room: Seminarraum 2, Inst. f. Geometrie
Abstract:

10:30 T. T. Luong: On a class of pseudo-analytic functions: representations, generalizations and applications

11:00 M. Juhos: Contributions to a mathematically stringent stochastics course in High School (AHS-Oberstufe)

11:30 coffee and refreshments

12:00 G. Peralta: Smooth Global Solution to a 2 × 2 Hyperbolic System on a Bounded Interval with Damping