### Talks in 2013

#### Mathematisches Kolloquium

Title: Quantum graphs and networks
Speaker: Pavel Exner (Czech Academy of Sciences)
Date: 17.12.2013, 15:30 Kaffee im Foyer, 16:00 Vortrag
Room: Hörsaal BE01
Abstract:

Dynamics of a quantum particle confined to a graph is interesting both
mathematically and physically. The corresponding self-adjoint Hamiltonians
typically contain free parameters coming from coupling of the wave
functions at the graph vertices. It is a longstanding problem whether one
can motivate the parameter choice by approximating the graph Hamiltonian
by operators on a family of networks, i.e. systems of tubular manifolds
the transverse size of which tends to zero. It appears that the answer
depends on the conditions imposed on tube boundaries. In this talk we
present a complete solution for Neumann networks: we demonstrate that
adding properly scaled potentials and changing locally the topology, one
can approximate any admissible vertex coupling. The result comes from a
common work with Taksu Cheon, Olaf Post, and Ond\v{r}ej Turek.

#### Gastvortrag

Title: Verspielte Mathematik
Speaker: Alexander Mehlmann (TU Wien)
Date: 13.12.2013, 14 Uhr
Room: Seminarraum 2, Kopernikusgasse 24
Abstract:

\sloppy

Trotz der zweifellos vorhandenen Bezugspunkte zur den schönen Künsten scheint die moderne Mathematik manchmal davor zurückzuscheuen, sich auf das Spiel mit literarischen Mustern einzulassen. Dieses spröde Verhalten der ernsthaftesten aller Musen entspricht jedoch keineswegs ihrer ursprünglichen Zielsetzung.

Für die Mathematiker Antonio Manetti (1423--1497) und Galileo Galilei (1564--1642) war die Beschäftigung mit Dantes Divina Commedia'' ein selbstverständlicher Schritt im Dienste der Dichtkunst und nicht zuletzt auch ein entscheidender zur Stärkung der eigenen Reputation. Beiden verdanken wir erstaunliche geometrische Einsichten in der Hölle Maß und Dimensionen; ein wahrhaft meisterlicher Ba\-lance\-akt zwischen den Erfordernissen diesseitiger Geodäsie und den Dogmen jenseitsgewandter Theologie.

Die rhetorische Tradition, Mathematik in Verse zu fassen, lässt sich unter anderem am Epigramm des Erzgrüblers Archimedes zum Problem der Rinder des Sonnengottes entdecken und nicht zuletzt durch Tartaglias poetische Formel zur Lösung der kubischen Gleichung bekräftigen.

Im Vortrag zur Verspielten Mathematik'' werden unter anderem Modelle der Spieltheorie vorgestellt, die eine durchaus adäquate, wenn auch augenzwinkernde, Beschreibung literarischer Motive zulassen. Kann sie einer derartig wohltemperierten, mathematischen Partitur folgen, so erweist sich die sogenannte Königin der Wissenschaften'' durchaus als ein geeignetes Instrument, um der Literatur interessante Noten abzugewinnen. Durch die Linse der Mathematik betrachtet, strebt der Mythos vom Wahnsinn des Odysseus einer überraschend anderen Lösung zu und selbst Goethes Faust öffnet sich durch eine spieltheoretische Modellierung der Teufelswette einer endgültigen Klärung der paradoxen Rettung Faustens.

#### Advanced Topics in Discrete Mathematics

Title: Diophantine approximation, flows on homogeneous spaces and counting
Speaker: Martin Widmer (Royal Holloway University, London)
Date: Friday 13. Dec. 2013, 10:30
Room: Seminar Room 2 of the Geometry Institute, Kopernikusgasse 23/4
Abstract:

After a very gentle introduction to Diophantine approximation we shall explain
what approximation properties of real numbers by rationals have to do with
flows on homogenous spaces, and how the latter can be used to prove some new
counting results on the number of "good" rational approximations to a given
irrational real number.

#### Zahlentheoretisches Kolloquium

Title: On Stolarsky's second problem : Mean value of the sum of digits of polynomial values
Speaker: Prof.Dr.Thomas Stoll (Université de Lorraine, Nancy, dzt. TU Graz)
Date: Mittwoch, 11. 12. 2013, 16:30 Uhr, s.t.
Room: Seminarraum A306, TU Graz, Steyrerg.30, 3.Stock, Geodäsie
Abstract:

Abstract: In this talk we present our recent resolution of a problem of
Stolarsky (1978) about the mean value of the sum of digits of polynomial
values in simultaneous digital expansions. The proof uses probabilistic
tools such as a result due to Bassily/Katai, and Cesaro means. Joint

#### Doctoral Day

Title: Doctoral School Seminar
Speaker: M. Raseta, A. Bazarova, R. Rissner, C. Kühn ()
Date: 6.12.2013, 10:30-13:00
Room: Seminarraum 2, Kopernikusgasse 24
Abstract:

10:30 M. Raseta

11:00 A. Bazarova

11:30 Lunch Break

12:00 R. Rissner

12:30 C. Kühn

#### Gastvortrag

Title: Direct sums of trace maps and self-adjoint extensions
Speaker: Prof. Dr. Andrea Posilicano (University of Insumbria)
Date: 2.12.2013, 16:00 Uhr
Room: C 307
Abstract:

We give a simple criterion so that a direct sum of trace (evaluation) maps is a trace map.
An application to the theory of self-adjoint extensions of direct sums of symmetric operators is provided;
this gives an alternative approach to results recently obtained by Malamud-Neidhardt and Kostenko-Malamud
using regularized direct sums of boundary triplets. An example regarding the Laplace-Beltrami operator on
conic-type surfaces with singular/degenerate Riemannian metrics is presented.

#### Gastvortrag

Title: No Free Form
Speaker: Michael Eigensatz (Evolute)
Date: 29.11.2013, 14:00
Room: Seminarraum 2, Kopernikusgasse 24, 8010 Graz
Abstract:

Geometry is a central element for architectural expressiveness. Modern computer tools provide the illusion that we are free to design the most extraordinary geometry we can ever imagine. In the real world, however, geometry, like physics, follows rules and is constrained by inevitable mathematical truths. Therefore, a simple lesson from the field: To make extraordinary geometry work, one has to understand geometry. I will show some examples.

#### Zahlentheoretisches Kolloquium

Title: Prime spectra of certain two dimensional integral domains - history and new development
Speaker: Prof. Dr. Aihua Li (Montclair State University, USA)
Date: Freitag, 29. 11. 2013, 14:00, c.t.
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz
Abstract:

Abstract:
For a commutative ring $R$, the spectrum, Spec$(R)$, is the partially ordered set of
the prime ideals of $R$ ordered by inclusion. This talk focuses on prime structure of
$\mathbb Z[x]$, the polynomial ring in one variable over the integers. It is known that several
types of Noetherian integral domains have their prime spectra order-isomorphic
to Spec$(\mathbb Z[x])$. In 1986, Roger Wiegand conjectured that every two-dimensional
integral domain which is a finitely generated $\mathbb Z$-algebra has prime spectrum orderisomorphic
to Spec$(\mathbb Z[x])$. This talk will give a history and recent development
about the conjecture and will introduce some newly developed results on graph
theory properties of Spec$(\mathbb Z[x]) \backslash \{0\}$ as a bipartite graph.

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

Title: Potential methods for Stokes and Brinkman systems of Lipschitz domains
Speaker: Prof. Dr.-Ing. Dr. h.c. Wolfgang L. Wendland (Universität Stuttgart)
Date: 28.11.2013, 16:00 Uhr
Room: TU Graz, Steyrergasse 30, 3. Stock, Seminarraum C307
Abstract:

The lecture consists of two parts. In the firt part we use the method of matched asymptotic expansions for the two-dimensional steady flow of a viscous incompressible fluid at low Reynolds number past a porous body of arbitrary shape. We assume that the flow inside the porous body is modeled by the continuity and the Brinkman equations, and that the velocity and boundary traction fields are weakly continuous across the interface between the fluid and the porous medium. By employing indirect boundary integral representations, the problem is reduced to uniquely solvable systems of Fredholm integral equations of the second kind in some Sobolev spaces. It is shown that the flow and also the force exerted by the exterior flow on the porous body admit an asymptotic expansion with respect to low Reynolds number, whose terms depend on the solutions of the above mentioned systems of boundary integral equations. By using the Oseen flow in the exterior, it can be shown that the Stokes–Brinkman expansion converges in any compact region to the Oseen–Brinkman solution if the Reynolds number tends to zero, in a similar manner as shown by G.C. Hsiao and R.C. MacCamy in 1973 and 1982 for flows around rigid obstacles. In the second part we study boundary value problems of Robin type for the semilinear elliptic Darcy–Forchheimer–Brinkman system on Lipschitz domains. We use a layer potential analysis and Schauder’s fixed point theorem to show the existence and uniquness of the solution on a bounded Lipschitz domain in Rn(n = 2 or 3) with small data in L2–Sobolev spaces.

#### Strukturtheorie-Seminar

Title: From trees to functions, and back
Speaker: Prof. Dr. Rudolf Grübel (Universität Hannover)
Date: Mittwoch, 27.11.2013, 11 Uhr c.t.
Room: SR C307, Steyrergasse 30, 3. Stock
Abstract:

The famous Harris correspondence provides a very useful link between simply generated random trees and random functions on the unit interval. I will

-- describe two attempts (2009, 2014) to obtain an analogue for search trees,

-- discuss some current work, some of it joint with Steve Evans and Anton Wakolbinger, on the relation to ordered ultrametric spaces and IDLA models.

#### Strukturtheorie-Seminar

Title: Rate of convergence in the entropic free Central Limit Theorem
Date: Monday, 25.11.2013, 11 Uhr c.t.
Room: C307, Steyrergasse 30, 3. Stock
Abstract:

We prove an expansion for densities in the free CLT and apply this
result to an expansion in the entropic free central limit theorem assuming a moment condition for the free summands.

#### Advanced Topics in Discrete Mathematics

Title: Counting lattice points, o-minimal structures and applications
Speaker: Fabrizio Barroero (TU Graz)
Date: Friday, 22. Nov. 2013, 10:30
Room: Seminar Room 2 of the Geometry Institute, Kopernikusgasse 23/4
Abstract:

PhD defense

#### Advanced Topics in Discrete Mathematics

Title: Good drawings and rotation systems of complete graphs
Speaker: Prof. Oswin Aichholzer (Institut für Softwaretechnologie, TU Graz)
Date: Freitag, 22. November 2013, 14:15 - 15:00
Room: SR C208, Steyrergasse 30, 2. Stock
Abstract:

In a good drawing of a complete graph the vertices are drawn as distinct points in the plane, edges are drawn as non-self-intersecting continuous arcs connecting its two end points, but not passing through any other point representing a vertex. Moreover, any pair of edges intersects at most once, either in their interior or at a common endpoint, no tangencies are allowed and no three edges pass through a single crossing. These drawings are also called simple topological graphs.

A rotation system (of a good drawing of a complete graph) gives, for each
vertex v of the graph, the circular ordering around v of all edges incident to v. In combinatorics, rotation systems were first used by Hefner in 1891 to encode embeddings of graphs onto orientable surfaces, determining the genus. In the plane (or equivalently on the sphere) the rotation system of a good drawing does not fully determine the drawing, but contains combinatorial information like all pairs of edges which intersect.

We present basic properties of these two concepts, as well as recent progress. This includes results on the number of realizable rotation systems, the crossing number of complete graphs, the number of empty triangles, and relations to other systems like the order type of a point set.

#### Zahlentheoretisches Kolloquium

Title: Number theoretic problems in computer science
Speaker: Prof. Dr. Helmut Prodinger (Univ. of Stellenbosch, dzt. TU Graz)
Date: Mittwoch, 20. 11. 2013, 16:30 Uhr, s.t.
Room: Seminarraum A306, TU Graz, Steyrerg.30, 3.Stock, Geodäsie
Abstract:

Within a period of 35 years, the speaker has encountered numerous
appearences of number theoretic concepts. The simplest is about
Fibonacci numbers, also of graphs. Others include digital problems
which are related to the analysis of the register function, Batcher’s
odd-even merge, the Hamming weight of redundant representations and others.
The fascinating area of q-series is perhaps halfway between Combinatorics and
Number Theory. The talk will highlight a few things from this speaker’s practice;
no special knowledge is required.

#### Gastvortrag

Title: Lattice Polygons and Real Roots
Speaker: Michael Joswig (TU Berlin)
Date: Di 19.11.2013, 16:00
Room: Seminarraum 2, Inst. f. Geometrie, Kopernikusgasse 24
Abstract:

It is known from theorems of Bernstein, Kushnirenko and Khovanskii from the
1970s that the number of complex solutions of a system of multivariate
polynomial equations can be expressed in terms of subdivisions of the Newton
polytopes of the polynomials. For very special systems of polynomials
Soprunova and Sottile (2006) found an analogue for the number of real
solutions. In joint work with Ziegler we could give a simple combinatorial
formula and an elementary proof for the signature of foldable triangulation
of a lattice polygon. Via the Soprunova-Sottile result this translates into
lower bounds for the number of real roots of certain bivariate polynomial
systems.

#### Strukturtheorie-Seminar

Title: Percolation in hyperbolic space: the non-uniqueness phase
Speaker: Jan Czajkowski (Universität Wroclaw/TU Graz)
Date: Monday, 18.11.2013, 11 Uhr c.t.
Room: C307, Steyrergasse 30, 3. Stock
Abstract:

I will talk on one part of my PhD thesis, which I am going to publish. I
consider Cayley graphs of reflection groups of finite-sided Coxeter polyhedra in the 3-dimensional hyperbolic space $H^3$, with the standard sets of generators. As main result, I prove the existence of a non-degenerate non-uniqueness phase of Bernoulli bond and site percolation on such graphs, i.e. that the critical probability is strictly less than the unification probability, for two classes of such polyhedra:

- for any polyhedra as above with at least 13 faces;

- for any compact right-angled polyhedra as above.

I also establish a natural lower bound for the growth rate of such Cayley
graphs (when the number of faces of the polyhedron is at least 6), used to prove the main result.

#### Gastvortrag

Title: Optimal adaptive estimation in nonparametric regression with one-sided errors
Speaker: Moritz Jirak (Institut für Mathematik, Humboldt-Universität zu Berlin)
Date: Freitag, 29. November 2013, 11.00 Uhr
Room: SR für Statistik (NT03098), Kopernikusgasse 24/III
Abstract:

We consider the model of non-regular nonparametric regression where smoothness constraints are imposed on the regression function and the regression errors are assumed to decay with some sharpness level at their endpoints. These conditions allow to improve the regular nonparametric convergence rates by using estimation procedures which are based on local extreme values rather than local averaging. We study this model under the realistic setting in which both the smoothness and the sharpness degree are unknown in advance. We construct adaptation procedures applying a nested version of Lepski's method and the negative Hill estimator which show no loss in the convergence rates with respect to the general $L_q$-risk and a logarithmic loss with respect to the pointwise risk. Optimality of these rates is proved. Some numerical simulations and an application to real data are provided.

#### Strukturtheorie-Seminar

Title: Synchronizing automata and approaches to the Cerny conjecture
Speaker: Dr. Emanuele Rodaro (Univ. Porto, Portugal)
Date: Montag, 11.11.2013, 11 Uhr c.t.
Room: SR C307, Steyrergasse 30, 3. Stock
Abstract:

An automaton is called synchronizing if there is a word w and a state q such that w applied to an arbitrary state p leads to q. This notion naturally appears in different areas of computer science to model systems which are robust against errors. In this seminar we will survey some results on synchronizing automata and different problems related to them, and we will focus in particular on the longstanding Cerny conjecture and some approaches to tackle it.

#### Zahlentheoretisches Kolloquium

Title: Extremes of integral through the copulas
Speaker: Prof. Dr. Oto Strauch (Slovak Academy of Sciences)
Date: Mittwoch, 6.11.2013, 16:30 Uhr, s.t.
Room: Seminarraum A306, TU Graz, Steyrerg.30, 3.Stock, Geodäsie
Abstract:

In this note we discusse maximum or minimum
of the integral $\int_0^1\int_0^1F(x,y)\dd g(x,y)$,
where $g(x,y)$ goes through the copulas, i.e.,
distribution functions $g(x,y)$ satisfying $g(x,1)=x$ and $g(1,y)=y$.
$F(x,y)$ is an arbitrary continuous function on $[0,1]^2$.
The solution is known, if partial derivative
$\frac{\partial^2 F(x,y)}{\partial x\partial y}$ on $[0,1]^2$ has a constant
signum. For maximum we have $g(x,y)=\min(x,y)$ and for minimum
$g(x,y)=\max(x+y-1,0)$. Also there is known a method to compute
$g(x,y)$, if $[0,1]^2$ is divided on two parts $[0,1]\times[0,Y]$
and $[0,1]\times[Y,1]$ containing positive and negative signum
of $\frac{\partial^2F(x,y)}{\partial x\partial y}$, respectively.
All others is open, for example, if $[0,1]^2$ is divided on two
triangles by diagonal.

#### Zahlentheoretisches Kolloquium

Title: Monte Carlo integration in Hilbert space with reproducing kernel
Speaker: Prof. Dr. Vladimír Baláž (Slovak University of Technology)
Date: Mittwoch, 6.11.2013, 17:00 Uhr
Room: Seminarraum A306, TU Graz, Steyrerg.30, 3.Stock, Geodäsie
Abstract:

#### Strukturtheorie-Seminar

Title: Random walks with exotic spectral measure
Speaker: Łukasz Grabowski (University of Oxford)
Date: Thursday, 24.10.2013, 11:00 Uhr c.t.
Room: SR A306, Steyrergasse 30, 3. Stock, Geodäsietrakt
Abstract:

I will describe my recent work with B. Virag in which we exhibit random walk operators on various wreath products with interesting spectral properties: operators with singularly continuous spectra and operators with very large densities, but without atoms.

#### Gastvortrag

Title: Visualisierung -- Simulation -- Animation in der Mathematik
Speaker: Georg Glaeser (Univ. f. Angewandte Kunst, Wien)
Date: 22.10.2013, 17:00 Uhr
Room: Seminarraum 2, Inst. f. Geometrie, Kopernikusgasse 24
Abstract:

Das Erstellen hochwertiger Grafiken -- auch im mathematischen Bereich -- ist heute Standard. Der Computer erlaubt darüber hinaus, durch interaktive Variation verschiedenster Parameter anschauliche Momentaufnahmen von früher nur schwer visualisierbaren Prozessen ,,in Echtzeit darzustellen. Es werden eine Reihe solch interaktiver Programme vorgestellt, z.B. die Kontraktion von beliebig vorgebbaren Flächen durch Oberflächen-Verringerung, Partikelsimulationen, Gleichverteilung von Punkten auf Oberflächen, Optimierung ebener und räumlicher Voronoi-Diagramme, Faltung von beweglichen Polyedern usw.

#### Mathematisches Kolloquium

Title: From Irrational Numbers to Perfect Matchings - 100 Years of Markov´s Uniqueness Conjecture
Speaker: Martin Aigner (Freie Universität Berlin)
Date: Freitag 18.10.2013 16:00 Kaffeepause 16:30 Vortrag
Room: Steyrergasse 30 Kaffeepause: C208, 2. Stock Vortrag: Hörsaal BE01, Parterre
Abstract:

A celebrated result in number theory is the Theorem of Markov which relates two seemingly totally different subjects: approximations of irrational numbers and the solutions of a certain equation. The proof, which Markov only sketched, was studied in great detail by Frobenius precisely 100 years ago. In his paper Frobenius mentioned a problem, now known as the uniqueness conjecture, which has remained unsolved to this day. I will tell you about the theorem and the conjecture and discuss, in particular, the amazing connections to trees, groups, combinatorics of words, lattice paths, and perfect matchings of plane graphs.

#### Gastvortrag

Title: Informational Divergence and Entropy on Rooted Trees with Probabilities
Speaker: Georg Böcherer (Technische Universität München)
Date: Mi 16.10.2013, 15:00 Uhr
Room: SR IDEG134, Inffeldgasse 16c, ground floor
Abstract:

Rooted trees with probabilities are used to analyze properties of a variable length code. A bound is derived on the difference between the entropy rates of the code and a memoryless source. The bound is in terms of normalized informational divergence. The bound is used to derive converses for exact random number generation, resolution coding, and distribution matching.

#### Vortrag

Title: Log-Ratio Analyse von Bieren, Whiskies und Kaffee's und eine praktische Anwendung an Doktoratsstudenten
Speaker: Matthias Templ (Vienna University of Technology, Statistics Austria)
Date: 16.10.2013, 15.00 Uhr
Room: SR für Statistik (NT03098), Kopernikusgasse 24/III
Abstract:

#### Advanced Topics in Discrete Mathematics

Title: Automorphisms of generalised polygons
Speaker: Dr. James Parkinson (University of Sydney)
Date: Friday, 11 October 2013, 10:30 (Coffee at 10:00)
Room: Seminar Room 2 of the Geometry Institute, Kopernikusgasse 23/4
Abstract:

Abstract: Generalised polygons are certain graphs that play an important role in Lie theory. In this talk we give an introduction to this area of discrete mathematics, and discuss some recent results on automorphisms of generalised polygons. Our main aim is to provide some restrictions on how an arbitrary automorphism of a generalised polygon can act, particularly with respect to the important opposition relation in the polygon. In particular, we give a complete classification of automorphisms of finite generalised polygons which map at least one point and at least one line to an opposite, but map no chamber to an opposite chamber. One motivation for this investigation is towards the old conjectures surrounding the classification of flag-transitive finite polygons.

This is joint work with Beukje Temmermans and Hendrik Van Maldeghem.

#### Vortrag

Title: Fast Boundary Element Methods: Coupling with Finite Element Methods and Applications
Speaker: Dr. Günther Of (Institut für Numerische Mathematik, TU Graz)
Date: Montag, 7.10.2013, 11:15 Uhr s.t.
Room: Seminarraum für Statistik, Kopernikusgasse 24/3, TU Graz
Abstract:

The coupling of finite and boundary element methods has been attractive for the numerical solution of second order boundary value problems for decades. In particular, so-called non-symmetric formulations have been very popular in applications for a long time. But the results on the stability of related discrete systems were quite unsatisfying. In the last few years significant progress has been made in the analysis of non-symmetric formulations.
In this talk, recent results on the stability of these formulations are presented and supported by numerical examples. The use of fast boundary element methods for the coupling is demonstrated for fluid-structure interaction problems within the design of ships.

While boundary element methods are advantageous in various situations, the use of fast, data-sparse methods is a postulate for the application of boundary element methods to real world problems. In this talk, a bundle of applications of fast boundary element methods is discussed. These are adaptive boundary element methods, industrial applications in the context of electrostatics and magnetostatics, the Gauss problem, X-ray tomography, and shape optimization.

#### Vortrag

Title: Uniform distribution theory, almost everywhere convergence, and GCD sums
Speaker: Dipl.Ing. Dr. Christoph Aistleitner (TU Graz)
Date: Montag, 7.Oktober 2013, 10.00 Uhr s.t.
Room: Seminarraum f. Statistik, Kopernikusgasse 24/3, TU Graz
Abstract:

Abstract: The theory of uniform distribution modulo 1 is intimately
connected with the theory of trigonometric sums. In particular, results
concerning the uniform distribution modulo 1 of parametric sequences are
often proved using similar results for Fourier series or series of
dilated functions, both in the $L^2$ and in the almost everywhere sense.
In this talk, a brief account of the historical development of the
connection between uniform distribution theory, discrepancy theory,
metric number theory, Fourier analysis and probability theory is given.
It is shown how in the investigation of these problems certain sums
involving greatest common divisors arise in a natural way, and how
recent estimates for such GCD sums can be used to obtain a Carleson-type
maximal inequality for sums of dilated functions, which led to the
solution of an a.e. convergence problem which has been open for several
decades. The talk is based on joint work with Istvan Berkes and Kristian
Seip.

#### Vortrag

Title: Examples of 3D Self-Affine Tiles with Simple Topology
Speaker: Jun LUO (Sun Yat Sen University, Guangzhou, China)
Date: 4.10.2013, 11:00
Room: Montanuniversität Leoben
Abstract:

We construct self-affine tiles of dimension $n\geq 3$ and show
that they are homeomorphic with the cube $[0,1]^n$. None of those
tiles are a self-affine polytope or the product of an $n-1$
dimensional self-affine tile with an interval. We also explain
why we are interested in such examples.

#### Mittagsseminar

Title: Estimating the Number of Triangulations of a Planar Point Set
Speaker: Raimund Seidl (Univ. Saarland)
Date: Friday 04.10.2013 12:15
Room: IST Seminarraum, Inffeldgasse 16b, 2nd floor
Abstract:

Like all problems in \#P, determining the number of straight edge triangulations of a planar point set admits an unbiased estimator that can be computed in polynomial expected time. I will discuss the practicability and usefulness of such estimators.

#### Mittagsseminar

Title: Compatible bichromatic matchings
Speaker: Luis Felipe Barba (Université libre de Bruxelles)
Date: 03.10.2013 12:15
Room: IST Seminarraum, Inffeldgasse 16b, 2nd floor
Abstract:

For a set $R$ of $n$ red points and a set $B$ of $n$ blue points, a BR-matching is a non-crossing geometric perfect matching where each segment has one endpoint in $B$ and one in $R$. Two BR-matchings are compatible if their union is also non-crossing. We prove that, for any two distinct BR-matchings $M$ and $M'$, there exists a sequence of BR-matchings $M = M_1 ,\dots, M_k = M'$ such that $M_{i-1}$ is compatible with $M_i$. This implies the connectivity of the compatible bichromatic matching graph containing one node for each BR-matching and an edge joining each pair of compatible BR-matchings.

#### Seminarvortrag

Title: Projektive Strukturen, Traktoren und invariante Differentialoperatoren
Speaker: Caroline Moosmüller (Univ. Wien)
Date: 1.10.2013, 15:00 Uhr
Room: Seminarraum 2, Inst. f. Geometrie, Kopernikusgasse 24
Abstract:

Im Rahmen einer Masterarbeit mit dem Titel Projective structures, tractors and invariant differential operators'' beschäftige ich mich mit Traktorbündeln und Traktorkonnexionen im Fall von projektiven Strukturen auf Mannigfaltigkeiten. Das Ziel dieser Arbeit ist es, den damit verbundenen Traktorkalkül ausführlich aufzubereiten und invariante Differentialoperatoren zu konstruieren. Desweiteren werden die Ergebnisse auf dem flachen Modell (der Sphäre) interpretiert.

#### Strukturtheorie-Seminar

Title: Enumeration of clique trees of chordal graphs
Speaker: Christoph Temmel (VU Amsterdam)
Date: Tuesday 1.10.2013, 11 Uhr c.t.
Room: A206, Steyrergasse 30, 2. Stock, Geodäsietrakt
Abstract:

Abstract: (joint with Florian Lehner) A chordal graph is a graph containing no induced cycle of length greater than three. A famous result by Gavril states, that a graph is chordal, iff it can be represented by a family of subtrees of a tree. The natural representations of a chordal graph are given by a subclass of the spanning trees of its clique graph, called the clique trees. We present a novel, local condition on edges of the clique graph to be a member of a clique tree. This allows us to enumerate all clique trees of a given chordal graph.

#### Miniworkshop

Title: Schrödinger operators with Delta-interactions on manifolds
Speaker: ()
Date: 24.09.2013, 10:00 - 17:00 Uhr
Room: TU Graz, Steyrergasse 30, 3. Stock, Seminarraum C307
Abstract:

Program
10:00-11:00 I. Popov (St. Petersburg)
Laplacian perturbation supported by line and regular approximation
11:00-12:00 M. Jex (Prague)
Discrete spectrum of a strong 0-interaction supported by a planar loop
14:00-15:00 C. Kühn (Graz)
Schrödinger operators with -potentials on manifolds
15:00-16:00 M. Holzmann (Graz)
Approximate solutions for evolution equations for weighted Laplacians
16:00-17:00 V. Lotoreichik (Graz)
Schrödinger operators with - and 0-interactions on Lipschitz surfaces
and chromatic numbers of associated partitions

#### Mini-Workshop

Title: Phase Transitions in Random Graphs
Speaker: ()
Date: Friday 20.9.2013, 08:50-16:00
Room: Lecture Theatre BE01, Steyrergasse 30
Abstract:

Plenary Speaker:
\hspace{0.5cm} Oliver Riordan (University of Oxford)

Invited Speakers:
\hspace{0.5cm} Oliver Cooley (Graz University of Technology)
\hspace{0.5cm} Charilaos Efthymiou (Goethe University Frankfurt am Main)

Further details can be found at

Title: Leoben-Ljubljana Seminar on Graph Theory
Speaker: ()
Date: Montag, 16.9. + Dienstag 17.9.2013
Room: Graz, Bildungshaus Mariatrost, Seminarraum 5
Abstract:

For the programme and other information, see

www.math.tugraz.at/discrete/ll13/

Daytime visitors who wish to attend some talks are kindly asked to inform the organisers via

discrete@math.tugraz.at

#### Strukturtheorie-Seminar

Title: Martin boundary forRandom walks with unbounded jumps on hyperbolic groups
Speaker: Sebastien Gouezel (Univ. Rennes I)
Date: Wednesday, 11.9.2013, 11 Uhr c.t.
Room: SR A306, Steyrergasse 30, 3. Stock, Geodäsietrakt
Abstract:

The identification of the Martin boundary of random walks with bounded jumps on hyperbolic groups dates back to Ancona in the 80's. It is a crucial tool to understand the properties of such random walks. We will explain how to obtain similar results for random walks with possibly unbounded jumps (under a necessary
condition of super-exponential moment). Applications to the local limit theorem will also be discussed.

#### Zahlentheoretisches Kolloquium

Title: INTERACTIONS BETWEEN RAMSEY THEORY AND COMBINATORICS ON WORDS
Speaker: Dr. Michelangelo Bucci (University of Turku, Finland)
Date: Montag, 19. 8. 2013, 14:00 Uhr, s.t.
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz
Abstract:

According to Wikipedia, Ramsey Theory is a branch of mathematics that
studies conditions under which order must appear. Coherently with this
definition, one of the most common studied topics in Ramsey Theory is
formed by Partition Regular properties, that is properties of sets
(usually of nonnegative integers) that cannot be destroyed by finite
partitioning.
In this talk we will examine some interactions between Ramsey Theory and
Combinatorics on Words, observing how some key objects of Ramsey Theory
can be used to gain a new understanding on the structure of wide classes
of words and how, vice-versa, a good knowledge of the combinatorial
properties of such words can be exploited to obtain new results into the
apparently distant domain of Additive Number Theory (with N. Hindman, S. Puzyinina and L. Zamboni).

#### Zahlentheoretisches Kolloquium

Title: Torsion-anomalous Intersections
Speaker: Dr. Francesco Veneziano (Georg-August-Universität Göttingen)
Date: Freitag, 19.Juli 2013, 10:00 Uhr s.t.
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz
Abstract:

Abstract:
Anomalous Intersections are a fairly recent framework introduced by
Bombieri, Masser and Zannier, which comprises and generalises a vast body
of problems and conjectures in Arithmetic Geometry.
Let $V$ be a variety contained in a group variety $G$, which is usually
taken to be an abelian variety or a torus.
When intersecting $V$ with an algebraic subgroup $B$, if the intersection
$V\cap B$ has a component of dimension strictly greater than "expected",
then such a component is said to be torsion-anomalous.
In analogy with many fundamental results in the field, there are conjectures
giving geometrical conditions for the variety $V$ to have only finitely
many (maximal) torsion-anomalous subvarieties.
The formulation of these conjectures generalises famous problems such as
the Manin-Mumford Conjecture and is related to the Mordell-Lang problem.

#### Zahlentheoretisches Kolloquium

Title: On the failure of Kronecker's density theorem for powers of an algebraic number
Speaker: Dr. Maurizio Monge (Scuola Normale Superiore di Pisa)
Date: Freitag, 19.Juli 2013, 11:30 Uhr
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz
Abstract:

Abstract:
We will present a quantitative estimate on the failure of Kronecker's
density theorem for the subgroup of the torus generated by the vector
formed by m powers of an algebraic number, when m is big. We prove
that the resulting subgroup is epsilon-dense, where epsilon is related
to the Mahler measure of the algebraic number. The problem is
motivated by a problem in control theory, where we assume that only
the integral part of the behaviour is known. The estimate on the
density is proved to be best-possible up to a constant, for m big
enough; this optimality is proved by means of a result on linear
recurrences of finite length, and estimates on the determinant of
Toeplitz matrices. We formulate a conjecture on the constant
provinding the best possible estimate, relating our problem to
algebraic dynamical systems on the torus. (joint work with N. Dubbini)

#### Zahlentheoretisches Kolloquium

Title: On the Diophantine equation $f(x)=g(y)$
Speaker: Prof. Dr. Michael Zieve (University of Michigan)
Date: Donnerstag, 18. Juli 2013, 16:00 Uhr s.t.
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz
Abstract:

Abstract: I will explain a recent joint result classifying all
polynomials $f(x)$ and $g(x)$ for which there is an algebraic number
field $K$ such that the image sets $f(K)$ and $g(K)$ have infinite
intersection. This involves a new approach to computing the genus of an
irreducible curve of the form $f(x)=g(y)$, as well as a novel
application of the classification of finite simple groups.

#### Vortrag im Seminar Diskrete Mathematik und Optimierung

Title: Connectedness, Sperner's Lemma, and combinatorial problems
Speaker: Penny Haxell (University of Waterloo)
Date: Dienstag 16.07.2013, 14:15
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

Let $G$ be a graph whose vertex set is partitioned into classes $V_1\cup\ldots\cup V_t$. An {\it independent transversal} in $G$ is an independent set $\{v_1,\ldots,v_t\}$ in $G$ such that $v_i\in V_i$ for each $i$. Many combinatorial problems can be formulated by asking whether a certain vertex-partitioned graph has an independent transversal, for example various colouring, hypergraph matching and covering problems. We discuss how the topological connectedness of the independence complex of $G$ can be used to show the existence of independent transversals, and hence give solutions to some of these problems.

#### Zahlentheoretisches Kolloquium

Title: Approximate Counting“ mit m „Countern“
Speaker: Prof. Dr. Helmut Prodinger (University of Stellenbosch, South Africa)
Date: Montag, 8.Juli 2013, 11:00 s.t.
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz
Abstract:

Einer Idee von Cichon folgend wird der Algorithmus im Titel analysiert.
Die eintreffenden n Daten werden zufaellig auf die m Zaehler verteilt,
und das Ergebnis ist die Summe der Einzelergebnisse. Die Analyse greift
auf alte Rechnungen ueber die Pfadlaenge digitaler Suchbaeume zurueck.
Andere Zugaenge wurden von Louchard und auch von M. Fuchs vorgeschlagen.
Die Uebereinstimmung der auftretenden Konstanten direkt zu zeigen ist
ueberraschend schwierig und benuetzt Elemente der q-Analysis.
Es wird auch ueber die „m-ifizierung“ weiterer Strukturen berichtet:
m-digitale Suchbaeume, m-binare Suchbaeume, m-PORTs. (PORT = plane
oriented recursive trees)

#### Zahlentheoretisches Kolloquium

Title: Central limit theorems for the number of parts in a partition
Speaker: Prof.Dr.Stephan Wagner (University of Stellenbosch, South Africa)
Date: Montag, 8.Juli 2013, 10:00 s.t.
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz
Abstract:

We consider the length (number of parts) and the number of distinct parts in a
random partition of an integer $n$ into elements of a sequence $\Lambda$. Under
very mild technical assumptions, we determine asymptotic formulae for mean and
variance and prove a central limit theorem. The limiting distribution turns out
to be Gaussian for the number of distinct parts, while one observes three
different phases for the total number of parts. In the borderline'' case of
Mahler partitions (partitions into powers of 2), we observe an oscillating
behaviour. This talk is based on joint work with Hsien-Kuei Hwang and Dimbinaina
Ralaivaosaona.

Title: Random walks on (mapping class) groups
Speaker: Giulio Tiozzo (Harvard University)
Date: 5.7.2013, 10:30
Room: Seminarraum 2 (Geometrie), Kopernikusgasse 24
Abstract:

Let G be a group of isometries of a metric space X. A random walk is defined by acting on some base point by a randomly chosen group element.

Starting from the 'multiplicative ergodic theorems' of Oseledets and Furstenberg, several questions have been addressed about the asymptotic behavior of such walks, for instance whether a generic sample path escapes to the boundary of X, and if it converges to a particular direction.

We will show that, under relatively mild assumptions on the geometry of X, a typical sample path lies within sublinear distance of some geodesic. We will then apply the result to the case of the mapping class group of a surface acting on Teichmueller space, answering a question of Kaimanovich. We will also see its consequences in terms of the singularity of the harmonic measure on the boundary of X.

#### Workshop "Groups, graphs, random processes"

Title: Recurrence of rotor-router walks
Speaker: Wilfried Huss (TU Graz)
Date: Tuesday, 2.7.2013, 16:00-16:40
Room: Seminar Room of the Statistics Institute, Neue Technik/III
Abstract:

#### Workshop "Groups, graphs, random processes"

Title: Centralizers and dynamics in Thompson's group V
Speaker: Francesco Matucci (Univ. Paris-Sud)
Date: Tuesday, 2.7.2013, 13:50--14:30
Room: Seminar Room of the Statistics Institute, Neue Technik/III
Abstract:

#### Workshop "Groups, graphs, random processes"

Title: Combinatorial, probabilistic and analytical aspects in the theory of Automata Groups
Speaker: Daniele d'Angeli (TU Graz)
Date: Tuesday, 2.7.2013, 11:30--12:10
Room: Seminar Room of the Statistics Institute, Neue Technik/III
Abstract:

#### Workshop "Groups, graphs, random processes"

Title: Rotor-router models
Speaker: Ecaterina Sava-Huss (TU Graz)
Date: Monday, 1.7.2013, 13:20--14:00
Room: SR C307, Steyrergasse 30, 3. Stock
Abstract:

#### Workshop "Groups, graphs, random processes"

Title: Clustering in random geometric graphs on hyperbolic spaces
Speaker: Elisabetta Candellero (University of Birmingham)
Date: Monday, 1.7.2013, 11:00--11:40
Room: SR C307, Steyrergasse 30, 3. Stock
Abstract:

#### Workshop "Groups, graphs, random processes"

Title: Groups of measurable currents and relative property (T)
Speaker: Markus Neuhauser (TU Aachen)
Date: Monday, 1.7.2013, 9:30-10:10
Room: SR C307, Steyrergasse 30, 3. Stock
Abstract:

Title: An introduction to wild topology
Speaker: Greg Conner (Brigham Young University)
Date: 28.6.2013, 10:15
Room: Seminarraum 2 (Geometrie), Kopernikusgasse 24
Abstract:

We will discuss the basic tools of fundamental groups of locally complicated spaces including the study of infinite words. We will discuss some of the standard examples, such as the Hawaiian earring, and what makes them interesting. We will talk about a number of classical and recent results about these groups due to the speaker, Cannon, Eda, Kent and others.

#### Zahlentheoretisches Kolloquium

Title: Normal and non-normal numbers with respect to Markov partitions
Speaker: Manfred Madritsch (Université de Lorraine, Frankreich)
Date: Freitag, 28. 6. 2013, 14.15 Uhr
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz
Abstract:

A real number in $[0,1]$ is called normal, if for any positive integer
$k$ and any block $B\in\{0,1,\ldots,q-1\}^k$ of digits of length $k$
the number of occurrences of this block within the $q$-ary expansion
is equal to the expected limiting frequency, namely $q^{-k}$. In the
first part of the talk we want to start with various equivalent
definitions of normal numbers. Then we provide constructions of normal
numbers by different methods. We will end the first part considering
normal numbers to different bases. In the second part we switch to
non-normal numbers. Constructing sets of essentially and extremely
non-normal numbers we show that these numbers are interesting from a
topological point of view. Finally in part three we focus on symbolic
dynamical systems and, in particular, on Markov partitions. The aim of
this part is to show, that one can generalize the above constructions
to a certain extend also to these numeration systems.

#### Strukturtheorie-Seminar

Title: Isotropic Markov semigroups on ultra-metric spaces
Speaker: Wolfgang Woess (TU Graz)
Date: Donnerstag, 27.6.2013, 15:00 s.t. (!!)
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

This will be a very informal and introductory talk about the recent collaboration of Bendikov, Girgor'yan, Pittet and Woess.

In the past 30 years, ultra-metric Markov processes, the associated Laplacians'' and their analysis have been considered by various authors, with some emphasis on the sitation where the ultrametric space is the p-adic field. The recent approach initiated by Bendikov and Grigory'an is particularly attractive because of its conceptual clarity. It leads to very complete results on recurrence/transience, transition kernel estimates and the spectrum of the Markov operator. The results are strongly linked with random walks on infinite trees.

#### Workshop Groups, graphs, random processes''

Title: Random walks and random graphs
Speaker: Angelica Pachon Pinzon (TU Graz)
Date: Wednesday, 26.6.2013, 11:00-11:40
Room: Attention: Seminar Room of the Statistics Institute, Neue Technik/III
Abstract:

Change of Seminar room !!

#### Vortrag im Seminar Diskrete Mathematik und Optimierung

Title: Cop numbers and capture times on the $n$-dimensional torus
Speaker: Dominik Vu (University of Memphis)
Date: Dienstag 25.06.2013, 15:20-15:50
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

The pursuit-evasion game Cops and Robbers} has enjoyed some attention in both discrete mathematics and theoretical computer science. It concerns a set of cops chasing one or more robbers on a fixed graph. Natural questions to ask are those for the number of cops required to ensure capture in finite time, and for the number of steps required in this case. Previous work both asked these questions for all graphs of fixed order, as well as for certain classes of graphs where bounds may be better due to the underlying structure of the graph. Considering the $n$-dimensional torus, we determine the number of cops needed in order to capture a single robber and give bounds on the capture time. This is joint work with Sebastian Koch (University of Cambridge).

#### Vortrag im Seminar Diskrete Mathematik und Optimierung

Title: Random hyperbolic graphs: degree distribution, clustering and component structure
Speaker: Nikolaos Fountoulakis (University of Birmingham)
Date: Dienstag 25.06.2013, 14:15
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

Random geometric graphs have been studied over the last 50 years in great detail. These are graphs that are formed between points randomly allocated on a Euclidean space and any two of them are joined if they are close enough. However, all this theory has been developed when the underlying space is equipped with the Euclidean metric. But, what if the underlying space is curved? Our focus will be on the case where the underlying space is a hyperbolic space. We will discuss the typical degree distribution of these random graphs as well as triangle counts and global clustering. Furthermore, we will give a critical condition on the parameters of the model that determines the existence of a giant component.

This is joint work with E. Candellero, M. Bode and T. Mueller.

#### Strukturtheorie-Seminar

Title: Noncommutative characterization of free Meixner processes
Speaker: Wiktor Ejsmont (Universität Wrocław)
Date: 20.6.2013, 16:00 s.t.
Room: Ort: SR C208, Steyrergasse 30, 2. Stock
Abstract:

#### Strukturtheorie-Seminar

Title: Classical and free mixture of Boolean stable law
Speaker: Takahiro Hasebe (Université de Franche-Comté, Besançon)
Date: 20.6.2013, 15:00 s.t.
Room: SR C208, Steyrergasse 30, 2. Stock
Abstract:

#### Vortrag im Seminar Diskrete Mathematik und Optimierung

Title: Extremal parameters in critical and subcritical graph classes
Speaker: Michael Drmota (TU Wien)
Date: Dienstag 18.06.2013, 14:15
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

In recent years there has been increasing interest in the asymptotic
analysis of (several classes of) planar maps and planar graphs. This was
initiated by bijective methods (e.g. the Shaeffer bijection), generating
function methods (e.g. Gimenez and Noy's result on the asymptotics of
number of planar graphs) and the search for probabilistic limiting
objects (e.g. the Brownian map by Le Gall). In particular in the
discussion of several planar graph classes (like series-parallel graphs
or labelled planar graphs) a dichotomy between a critical'' and
subcritical'' behaviour between 2-connected and connected graphs was
observed. Informally a graph class is subcritical when all 2-connected
components are small (i.e., at most of log n - size) and one observes a
treelike structure''. Conversely a graph class is critical when the
largest 2-connected component is comparable to the size of the whole graph.

#### Vortrag im Rahmen des Seminars TM (Angewandte Analysis und Numerische Mathematik)

Title: Mathematical challenges of Zero-Range Physics: a survey of old and new results
Speaker: Alessandro Michelangeli, Ph.D. (Mathematisches Institut der Universität München)
Date: Dienstag, 18.6.2013, 11:00 Uhr
Room: TU Graz, Steyrergasse 30, 3. Stock, Seminarraum A306
Abstract:

#### Zahlentheoretisches Kolloquium

Title: Rational points on some del Pezzo surfaces over imaginary quadratic fields
Speaker: Christopher Frei (TU Graz)
Date: Freitag, 14.6.2013, 14.15 Uhr
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz
Abstract:

Abstract: A conjecture of Manin predicts an asymptotic formula for the
number of rational points of bounded height on certain projective
varieties in terms of their geometric structure. We discuss Manin's
conjecture and recent proofs for some del Pezzo surfaces over imaginary

#### Strukturtheorie-Seminar

Title: Generalized Brownian motion and geometry of Cayley graphs of Coxeter groups
Speaker: Prof. Marek Bożejko (Universität Wrocław)
Date: 13.6.2013, 15:00 s.t.
Room: SR C208, Steyrergasse 30, 2. Stock
Abstract:

Plan:
1.1. $q$-CCR relations $a(f) \,a^*(g) -q\,a^*(g) \,a(f) = \langle f,g\rangle$, $f,g$ from a Hilbert space.

1.2. $q$-Gaussian law, Theta function of Jacobi, $q$-Hermite polynomials
$$x H(x,n) = H(x,n+1) + (q^{n} -1)/ (q-1) H(x,n-1) .$$

1.3. $q$-Brownian motion, for $q\in [-1,1]$

2. Length functions on permutations groups, Coxeter groups and free product
groups.

3. $q$-determinant and realization of $q$-CCR relations and $q$-Brownian motion on
$q$-Fock space.

3. Applications to free probability and von Neumann algebras.

#### Vortragsabsage

Title: Vortragsabsage
Speaker: Winfried Hochstättler (FernUniversität in Hagen)
Date: Dienstag 11.06.2013, 14:15
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

Der Vortrag von Winfried Hochstättler muss leider abgesagt werden.

#### Zahlentheoretisches Kolloquium

Speaker: Prof. Dr. Rainer Dietmann (Royal Holloway, University of London)
Date: Montag, 10.Juni 2013, 15:00 s.t.
Room: Seminarraum C307, TU Graz, Steyrergasse 30, 3.Stock
Abstract:

Abstract: It is a classical problem in Analytic Number Theory to obtain
asymptotic formulas for the number of representations of positive integers
by positive definite quadratic forms. One can generalise this question
to the representation of positive definite quadratic forms B by positive definite
quadratic forms A. Here one has the famous Siegel mass formula, which averages
representation numbers over all forms in the genus of A, and Raghavan, using Siegel
modular forms, obtained results for representations by individual forms A, given
that all successive minima of B are of comparable size. In this talk we
report on joint work with Michael Harvey, where we use an approach based
on the circle method. This yields asymptotic formulas for representations
by individual A without assumptions on the minima of B, at the expense of
needing bigger dimension of A in terms of B.

Title: Discrete Mathematics Day
Speaker: E. Szemeredi / I. Baranyi / U. Neugebauer / DK students ()
Date: Freitag, 7. Juni 2013, 09:15 - 16:30
Room: Hörsaal P2, Petersgasse 16 (Physikgäude, EG)
Abstract:

09:15 Opening
09:30 E. Szemeredi: Tight bound for embedding large maximum degree tree
10:30 Break
11:00 M. Weitzer: TBA
11:20 M. R. Iaco: A dynamical system approach to the Kakutani-Fibonacci sequence
11:40 Lunch Break
14:00 I. Baranyi: Extremal problems for convex lattice polytopes
15:00 A. Bazarova: Extremal theory of dependent processes
15:20: Break
15:30: U. Neugebauer: The Greatest Happiness Imaginable (movie & discussion).

#### Strukturtheorie-Seminar

Title: Size of the largest component in a multi-type generalization of Erdös-Rényi random graphs
Speaker: Christoph Koch (TU Graz)
Date: Do, 6.6.2013, 15:00 s.t. (!!)
Room: SR C208, Steyrergasse 30, 2. Stock
Abstract:

Galton-Watson branching processes are a very efficient tool to establish
the size of the giant component in the supercritical phase of Erdös-Rényi random graphs. Recently Béla Bollobás and Oliver Riordan showed that a similar approach works also in the weakly supercritical region providing a new short proof of the size of the largest component.
In the talk we will discuss how this approach can be adapted to a generalized random graph model containing different types of vertices. This involves a multi-type branching process, a notion of a dual branching process, and the width of a rooted tree associated with a branching process as well as the second moment method.

This is joint work with Mihyun Kang and Angélica Páchon.

#### Vortrag im Seminar Diskrete Mathematik und Optimierung

Title: Flip Distances in Triangulations
Speaker: Oswin Aichholzer (Institute for Software Technology, TU Graz)
Date: 4.6. 2013, 14:15
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

A flip in a (geometric) triangulation $T$ is the operation of replacing
an edge of $T$ by a different edge, such that the resulting graph is
again a triangulation. The flip distance between two triangulations of
a given domain (point set, polygon, convex polygon, ...) is the smallest
number of flips that is necessary to transform one triangulation into
the other.

Computing the flip distance is a challenging research problem with
quite some history and several recent developments. We will survey
relations of flip sequences to other combinatorial structures
and present latest results on the hardness of computing the flip
distance in the case of point sets and simple polygons.
\vspace{3cm}

Title: Kolloquium aus Anlass des 50.Geburtstages von Frau Prof.Dr.Sophie Frisch
Speaker: ()
Date: Montag, 3.Juni 2013, ab 16:00 s.t.
Room: Seminarraum A306, TU Graz, Steyrerg.30, 3.Stock, Geodäsie
Abstract:

16:00: Prof. Dr. Paul-Jean Cahen (Université d'Aix Marseille)
Some new results about integer-valued polynomials

Abstract: Let $D$ be a Noetherian domain and $Int(D)$ be the corresponding ring of integer-valued polynomials. We consider the prime ideals of $Int(D)$ above an height one maximal ideal $M$ of $D$ with finite residue field. Without loss of generality we may assume $D$ to be local with maximal ideal $M.$ In case $D$ is unibranch, it is known by a topological argument that these prime ideals (all maximal) are not finitely generated, using the fact there are infinitely many such primes. In case $D$ is not unibranch, there are only finitely many such primes, but we prove however here, by a similar topological argument, that these primes are again not finitely generated. It follows that $D$ is not almost strong Skolem in this case (whereas it is known to be so, in case $D$ is analytically irreducible).

16:45: Prof.Dr.Reinhard Winkler (TU Wien)
Alte Hüte immer noch frisch:
Kulinarisches zu den Nahtstellen von Algebra, Topologie und Maßtheorie

Abstract:
Jeden Mathematiker lockt die Frage, was denn nun seine Welt
im Innersten zusammenhält. Darf er ihren Lockungen haltlos nachgeben?
Bei mir brach der Damm vor etwa 15 Jahren rund um eine gemeinsame
Arbeit mit Sophie Frisch, Robert Tichy und Milan Pasteka über
endlich additive Maße auf Gruppen und Ringen.
Meine aus Dissertations- und Habilitationszeiten ererbte Welt
war vor allem die der Verteilung von Folgen und gewisser
begleitender algebraischer und topologischer Aspekte.
Die Frage nach dem innersten Zusammenhalt all dessen führt
tatsächlich ziemlich direkt zu Maßen auf
topologisch-algebraischen Strukturen.
Spürt man von ihnen ausgehend den Verflechtungen von Algebra,
Topologie und Maßtheorie noch systematischer nach, so stößt man auf
vorwiegend wohlbekannte Zusammenhänge, die wichtige Teile der
Mathematik des 20.Jahrhunderts maßgeblich vorangetrieben haben
und weiterhin wirken. In meinem Vortrag möchte ich das Bewusstsein
für diese Zusammenhänge durch einzelne möglicherweise weniger bekannte
Aspekte ergänzen, vor allem aber generell auffrischen.

R.Tichy

18:30: Nachsitzung mit Musik

#### Finanz- und Versicherungsmathematisches Kolloquium

Title:
Speaker: ()
Date: Freitag, 24.Mai 2013, 10:30 Uhr
Room: Seminarraum 2, Institut f. Geometrie, Kopernikusg.24/4
Abstract:

10:30: Eröffnung durch Univ.-Prof.Dr.Robert Tichy
10:45: Univ.-Prof.Dr.Gerhard Larcher (JKU Linz)
11:30: Kaffeepause
11:45: Univ.-Prof.Dr.Friedrich Hubalek (TU Wien)
Joint analysis and estimation of stock prices and trading volume in stochastic volatility models jump
12:30: Mittagspause
14:00: Univ.-Prof.Dr. Stefan Gerhold (TU Wien)
Portfoliooptimierung unter Transaktionskosten
14:45: Kaffeepause
15:00: DI Viola Schmied (ecofinance, Graz)
Impact of basis spreads on valuation of interest rate derivatives
15:45: DI Markus Zahrnhofer (Merkur Versicherung AG, Graz)
Die Herausforderung von ''Marktvolatilität'' in einem marktkonsistenten Umfeld
16:30: Ende

#### Strukturtheorie-Seminar

Title: Asymptotic Entropy of Random Walks on Regular Languages
Speaker: Dr. Lorenz Gilch (TU Graz)
Date: Do, 23.5.2013, 15:00 s.t. (!)
Room: SR C208, Steyrergasse 30, 2. Stock
Abstract:

In this talk I will present my recent results about asymptotic entropy
and its properties of random walks on regular languages over a finite
alphabet. In particular, this setting applies to the case of random walks on virtually free groups. Existence of the asymptotic entropy is shown and formulas for it are presented. Moreover, I will show that the entropy is the rate of escape with respect to the Greenian metric and that it varies analytically in terms of probability measures of fixed support.

#### Zahlentheoretisches Kolloquium

Title: Integration and approximation of analytic functions in Korobov spaces
Speaker: Peter Kritzer (Johannes Kepler Universität Linz)
Date: Freitag, 17.5. 2013, 14:00 Uhr s.t.
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz
Abstract:

We study multivariate integration and approximation using quasi-Monte
Carlo rules for a weighted Korobov space of analytic periodic functions
for which the Fourier coefficients decay exponentially fast. The weights
in the function space are defined in terms of two non-decreasing sequences
\bfseries a} and \bfseries b}, and a parameter $\omega \in (0, 1)$.
We check conditions on \bfseries a}, \bfseries b}, and
$\omega$ such that the integration/approximation error decays
exponentially fast. Furthermore, we discuss how the errors of our
algorithms depend on the dimension of the problems. For this purpose, we
introduce the concepts of weak, polynomial, and strong polynomial
tractability. We study how they are related to each other, and which
properties of the weights are necessary and sufficient for these concepts
to hold.
Regarding the choice of points employed in our algorithms,
special types of regular grids turn out to be very useful. This is rather
surprising, as these regular grids seem to be better to handle than common
classes of quasi-Monte Carlo point sets such as lattice points or digital
nets.

The talk is based on joint work with J. Dick (Sydney), F. Pillichshammer
(Linz), and H. Wo\'zniakowski (New York/Warsaw).

#### Zahlentheoretisches Kolloquium

Title: An example of Unlikely Intersections in the multiplicative group
Speaker: Laura Capuano (Scuola Normale superiore di Pisa)
Date: Freitag, 17. 5. 2013, 15:30 Uhr
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz
Abstract:

Abstract: In this seminar we are going to give a different proof of the
theorem of Bombieri, Masser and Zannier of 1999 about intersecting a
curve with algebraic subgroups of the multiplicative group. To do that
we use mainly Pila-Zannier method and some estimates about rational
points of bounded height of Pila-Wilkie type. This method was used for
the first time in 2008 by Pila and Zannier to give a new proof of
Manin-Mumford conjecture but is very general and can be used to prove
other cases of “Unlikely Intersections” problems in many different contexts.

#### Zahlentheoretisches Kolloquium

Title: Constructions of Generating Matrices for Digital (t, s)-Sequences
Speaker: Roswitha Hofer (TU Graz)
Date: Freitag, 17.5. 2013, 14:45 Uhr
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz
Abstract:

Most of the well-known $s$-dimensional low-discrepancy sequences can be constructed via the digital method, which was introduced by Niederreiter and generalized earlier forms by Sobol and Faure.
The digital methods constructs a sequence $(\boldsymbol{x}_n)_{n\geq 0}$ in $[0,1]^s$ as follows.:
Choose a finite field ${\mathbb F}_q$ with cardinality $q$, and put $Z_q=\{0,1,\ldots,q-1\} \subset {\mathbb Z}$.
Choose
\begin{enumerate}
\item[(i)] bijections $\psi_r: Z_q \to {\mathbb F}_q$ for all integers $r\geq 0$, satisfying $\psi_r(0)=0$ for all sufficiently large $r$;
\item[(ii)] generating matrices} $C^{(i)}:=(c^{(i)}_{j,r})_{j\geq 1,r\geq 0} \in{\mathbb F}_q^{{\mathbb N}\times {\mathbb N}_0}$ for $1 \le i \le s$;
\item[(iii)] bijections $\lambda_{i,j}: {\mathbb F}_q\to Z_q$ for $1\leq i\leq s$ and $j\geq 1$.
\end{enumerate}
The $i$th coordinate $x_n^{(i)}$ of the $n$th point $\boldsymbol{x}_n=(x^{(1)}_n,\ldots,x^{(s)}_n)$
of the sequence is computed as follows. Given an integer $n \ge 0$, let $n=\sum_{r=0}^{\infty} z_r(n)q^r$ be the digit expansion of $n$ in base
$q$, with all $z_r(n) \in Z_q$ and $z_r(n)=0$ for all sufficiently large $r$. Carry out the matrix-vector product
$$C^{(i)}\cdot\left(\begin{matrix}\psi_0(z_0(n))\psi_1(z_1(n))\vdots \end{matrix}\right)=:\left(\begin{matrix}y_{n,1}^{(i)}y_{n,2}^{(i)} \vdots \end{matrix}\right)\quad\mbox{and put}\quad x^{(i)}_n=\sum_{j=1}^{\infty}\lambda_{i,j}(y^{(i)}_{n,j})q^{-j}\in [0,1].$$
Note that the distribution of the generated sequence mainly depends on the choice of the generating matrices.
%To obtain digital $(t,s)$-sequences with low discrepancy, the challange is to find generating matrices with in a certain sense linearly independent rows. The quality parameter $t$ describes the quality of the linear independence of the matrices''.

In this talk we discuss different methods to construct matrices that are qualified to generate low-discrepancy sequences.

#### Seminar of the Doctoral School

Title: Doctoral Day
Speaker: ()
Date: 17.5.2013, 10:30-13:00
Room: Seminarraum 2, Institut für Geometrie, Kopernikusgasse 24
Abstract:

#### Strukturtheorie-Seminar

Title: Interacting growth processes and invariant percolation
Speaker: Dr. Sebastian Müller (Univ. Marseille / TU Graz)
Date: Korrektes Datum: Donnerstag, 16.5.2013, 15:00 s.t. (!!)
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

The aim of this talk is to underline the relation between reversible growth processes and invariant percolation. We present two models of interacting branching random walks (BRWs), truncated BRWs and competing BRWs, where survival of the growth process can be formulated as the existence of an infinite cluster in an invariant percolation on a tree. Our approach is fairly conceptual and allows generalizations to a wider set of reversible''
growth processes.

#### Vortrag im Seminar Diskrete Mathematik und Optimierung

Title: Even cycle decompositions of graphs with no odd-$K_4$-minor
Speaker: Sang-il Oum (KAIST)
Date: Dienstag 14.05.2013, 14:15
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

An even cycle decomposition of a graph G is a partition of E(G) into cycles of even length. Evidently, every Eulerian bipartite graph has an even cycle decomposition. Seymour [circuits in planar graphs. J. Combin. Theory Ser. B, 31(3):327–338, 1981] proved that every 2-connected loopless Eulerian planar graph with an even number of edges also admits an even cycle decomposition. Later, Zhang [On even circuit decompositions of Eulerian graphs. J. Graph Theory, 18(1):51–57, 1994] generalized this to graphs with no $K_5$-minor. We propose a conjecture involving signed graphs which contains all of these results. Our main result is a weakened form of this conjecture. Namely, we prove that every 2-connected loopless Eulerian odd-$K_4$-minor free signed graph with an even number of odd edges has an even cycle decomposition. This is a joint work with Tony Huynh and Maryam Verdian-Rizi.

#### Vortragsabsage

Title: Vortragsabsage
Speaker: Christian Krattenthaler (Universität Wien)
Date: Dienstag 07.05.2013, 14:15
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

Aus gesundheitlichen Gründen muss der Vortrag von Christian Krattenthaler leider abgesagt werden.

#### Vortrag

Title: A Projective Framework for Polyhedral Mesh Modeling
Speaker: Amir Vaxman (TU Wien)
Date: Fr. 26.4.2013, 15:30
Room: Seminarraum 2, Kopernikusgasse 24, 4. Stock
Abstract:

I present a novel framework for polyhedral mesh editing with face-based projective maps, that preserves planarity by definition. Such meshes are essential in the fields of construction and architectural design. By using homogeneous coordinates to describe vertices, we gain a rich and linear shape space of meshes with planar faces. The generality of this space allows for polyhedral geometric processing methods to be conducted with ease. We demonstrate its usefulness in polyhedral mesh subdivision, a resulting multi-resolution editing algorithm, and novel shape space exploration possibilities. Furthermore, we show that our shape space is a discretization of a continuous space of conjugate-preserving projective transformation fields on surfaces.

#### Zahlentheoretisches Kolloquium

Title: On generalisations of Selmer's continued fraction algorithm
Speaker: Prof. Dr. Henk Bruin (Universität Wien)
Date: Freitag, 26.April 2013, 14:15 Uhr
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz
Abstract:

Abstract: A higher-dimensional continued fraction algorithm
is meant to find rational approximations of a higher-dimensional
vectors: each component of the vector is supposed to be approximated
by rationals with a common denominator.
Subtractive algorithms form a large class of such algorithm,
of which Jacobi-Perron is probably the best known. Selmer's
algorithm is another. In this joint paper with Fokkink & Kraaikamp
we are finding invariant measures for generalisations
of Selmer's, and apply this to estimate the quality of
approximations.

#### Strukturtheorie-Seminar

Title: Zig-zag product of Schreier graphs: examples and open problems
Speaker: Dr. Daniele D'Angeli (TU Graz)
Date: Donnerstag, 25.4.2013, 15:00 s.t. (!!)
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

The Zig-zag product of graphs was introduced by Reingold, Vadhan & Wigderson (2002) to construct new sequences of expanders. In this talk we apply this construction to sequences of graphs associated with the action of a self-similar group on a rooted tree. We will show some interesting examples and discuss a list of open questions.

(In collaboration with A. Donno and E. Sava-Huss)

#### Gastvortrag

Title: Contour method for parameterizing canal surfaces
Speaker: Miroslav Lav\'\i\v cka (Univ. Plze\v n)
Date: 25.4.2013, 15:00
Room: Seminarraum 2, Kopernikusgasse 24
Abstract:

A canal surface is the envelope of a 1-parameter set of spheres centered at the spine curve $m(t)$ and with the radii described by the function $r(t)$. Any canal surface given by rational $m(t)$ and $r(t)$ possesses a rational parameterization, and most of (exact or approximate) parameterization methods are based on a construction of a rational unit normal vector field guaranteeing rational offsets. We will study a condition which guarantees that a given canal surface has rational generalized contour curves (i.e., contour curves with respect to a given direction), which are later used for a straightforward computation of rational parameterizations of canal surfaces providing rational offsets. Our approach follows a construction of rational spatial MPH curves from the associated planar PH curves and gives it to the relation with the contour curves of canal surfaces given by their medial axis transforms. We also present a simple method for computing rational offset blends between two canal surfaces based on the contour method.

#### Vortrag im Seminar Diskrete Mathematik und Optimierung

Title: Finding Large Planar Subgraphs
Speaker: Oliver Cooley (TU Graz)
Date: Dienstag 23.04.2013, 14:15
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

Given a graph $G$, we consider the problem of finding a planar subgraph $H$ of $G$ with many edges. Define the planarity} pl$(G)$ of $G$ to be $\max \{e(H)\}$ over all planar subgraphs $H\subseteq G$. Given integers $n$ and $d$, let pl$(n,d)$ be $\min\{$pl$(G)\}$ over all graphs $G$ on $n$ vertices with minimum degree $d$.

In this talk we will examine the curious behaviour of pl$(n,d)$ when $n$ is approximately $n/2$. Kühn, Osthus and Taraz showed that for $\Theta(n)=d\leq n/2$ we have pl$(n,d)= (2+o(1))$. In this talk we will outline a proof that

\center{
\hspace{0.5cm}pl$(n,(n+1)/2)$ $=(2.25+o(1))n$ \hspace{0.5cm} for $n$ even and
\hspace{0.3cm}pl$(n,n/2+1)$ $=(2.5+o(1))n$ \hspace{0.7cm} for $n$ odd.
}

\vspace{0.2cm}

\raggedright
Thus the asymptotic behaviour of the parameter pl$(n,d)/n$ is to remain constant at $2$ for some time before exhibiting two discrete jumps at $d=(n+1)/2$ and $d=n/2+1$.

This is based on joint work with Tomasz \L uczak, Anusch Taraz and Andreas Würfl.

#### Zahlentheoretisches Kolloquium

Title: Explicit functions with small Gowers norms
Speaker: Prof. Dr. Emmanuel Kowalski (ETH Zürich)
Date: Freitag, 19.April 2013, 14:15 Uhr
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz
Abstract:

Abstract: The Gowers uniformity norms were introduced by T. Gowers in his proof of Szemeredi's
Theorem and have played an important role since then in additive combinatorics. It is easy to estimate
the Gowers norms of certain "random" functions, and after recalling the general context of these norms, we will show how to construct many examples of explicit functions which achieve similar estimates, using the Riemann Hypothesis over finite fields.

(Joint work with É. Fouvry and Ph. Michel)

#### Zahlentheoretisches Kolloquium

Title: Invariants of polynomial and rational function decomposition
Speaker: Dijana Kreso (TU Graz)
Date: Freitag, 22.März 2013, 15:00 Uhr
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz
Abstract:

In the 1920's J. F. Ritt studied the question of non-uniqueness of the 'prime factorization' of polynomials over complex numbers under the operation of functional composition. Ritt's results have been completed, generalized and applied to a variety of topics. One such topic, studied by several authors, is exhibiting invariants of polynomial and rational function decomposition. In this talk an overview of existing results will be given. These include very recent results which are an extension of known invariants to wider classes of polynomials. Furthermore, rational function decomposition will be discussed and it will be shown by concrete counterexamples that analogous results do not hold in this case. These recent results are a joint work with M. Zieve from University of Michigan.

#### Zahlentheoretisches Kolloquium

Title: Properties of the Group of Polynomial Permutations modulo $p^n$
Speaker: Daniel Krenn (TU Graz)
Date: Freitag, 22.März 2013, 14:15 Uhr
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz
Abstract:

Abstract

Fix a prime $p$ and a positive integer $n$. A polynomial
permutation is a bijective function over the integers modulo $p^n$
which can be represented as a polynomial. This talk is about the
group (with respect to composition) of those polynomial
permutations.

While the order of the group of polynomial permutations modulo
$p^n$ is known for about a hundred years, its structure seems to
be complicated. The presented results contain an enumeration of
the Sylow $p$-subgroups and a precise description of those, and,
further, some non-trivial normal subgroups will be shown.

#### Zahlentheoretisches Kolloquium

Title: Elliptic curves over number fields
Speaker: Dr.Filip Najman (University of Zagreb)
Date: Freitag, 15.März 2013, 14.15 Uhr
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz
Abstract:

Abstract: We will survey the most important classical results about elliptic curves over number fields, as well as recent developments in the subject.

#### Vortrag im Seminar Diskrete Mathematik und Optimierung

Title: Ramsey properties of random discrete structures
Speaker: Yury Person (Freie Universität Berlin)
Date: Dienstag 12.03.2013, 14:15
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

An exciting research direction in combinatorics in the last twenty years has been to transfer classical results such as Ramsey's theorem, van der Waerden's theorem, Szemerédi's and Turán's theorems and their generalizations to random and pseudorandom settings.

In my talk I will discuss some recent and not so recent Ramsey-type theorems for random graphs, hypergraphs and random subsets of integers and some sharp threshold phenomena that occur.

Parts of my talk are based on joint works with Luca Gugelmann, Angelika Steger and Henning Thomas, and with Ehud Friedgut, Hi\d{\^{e}}p H\an and Mathias Schacht.

Title: The Current State of the Foundations of Set Theory
Speaker: Prof. Dr. Sy David Friedman (Universität Wien)
Date: Freitag, 8.März 2013, 14:15
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz
Abstract:

Set-theorists have for many years had a pretty good system
of axioms for mathematics, the ZFC axioms. Nearly all of the
theorems of mathematics can be translated into
set theory and then shown to follow from the ZFC axioms. But
Goedel's incompleteness theorem tells us that no system of
axioms, not even ZFC, is really complete: there
always are statements that can be neither proved or
disproved in any formal system. The most famous example for
ZFC is Cantor's continuum hypothesis (CH), a statement
about abitrary sets of reals, but there are many more
examples, even about nicely definable sets of reals.

Goedel conjectured that one might resolve this
incompleteness problem by adding axioms of large infinity to
ZFC, now called large cardinal axioms, in order to resolve
many of the natural problems of set theory like CH.
Goedel was only partly right: Many natural questions
concerning nicely definable sets of reals are resolved by
large cardinal axioms as well as virtually any question
statements of set theory. But CH remains untouched by large
cardinal axioms.

Recently people have investigated a different kind of axiom,
those which assert not the existence of large infinities,
but rather the strength of the power set operation. These
axioms have a better chance of resolving CH, and there are
good candidates for doing that, but much work remains to be done.
I will also discuss the interesting possibility that the
"correct" choice of axioms to be added to ZFC will come not
from within set theory itself, but from other areas of
mathematics which also suffer from Goedel incompleteness.

Title: Kardinalzahlen zwischen abzählbar und Kontinuum
Speaker: Prof. Dr. Martin Goldstern (Technische Universität Wien)
Date: Freitag, 8.März 2013, 15:30 Uhr
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz
Abstract:

In den letzten Jahrzehnten wurde eine große Anzahl von Kardinalzahlen
gefunden, die alle im abgeschlossenen Intervall zwischen aleph1
(der kleinsten überabzählbaren Kardinalzahl) und c (der Kardinalität
der reellen Zahlen) liegen, wie zB die Antwort auf die Frage
''wie viele Nullmengen braucht man, um die reellen Zahlen
zu überdecken''?

Ich möchte ein paar dieser Kardinalzahlen vorstellen, und ausführen,
dass sowohl beweisbare Ungleichungen zwischen manchen dieser Zahlen
als auch die Unbeweisbarkeit von gewissen Gleichungen zwischen diesen
Kardinalzahlen unser Verständnis der Teilmengen der reellen Zahlen
erweitern.

#### Vortrag im Seminar Diskrete Mathematik und Optimierung

Title: Catching the k-NAESAT Threshold
Speaker: Konstantinos Panagiotou (University of Munich)
Date: 05.03.2013, 14:15
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

The best current estimates of the thresholds for the existence of
solutions in random constraint satisfaction problems ('CSPs') mostly derive
from the first and the second moment method. Yet apart from a very few
exceptional cases these methods do not quite yield matching upper and lower
bounds. According to deep but non-rigorous arguments from statistical
mechanics, this discrepancy is due to a change in the geometry of the set of
solutions called condensation that occurs shortly before the actual
threshold for the existence of solutions. To cope with condensation,
physicists have developed a sophisticated but non-rigorous formalism called
Survey Propagation, which yields precise conjectures on the threshold values
of many random CSPs. In this talk I will discuss a new Survey Propagation
inspired method for the random k-NAESAT problem, which is one of the
standard benchmark problems in the theory of random CSPs. This new technique
allows us to overcome the barrier posed by condensation rigorously, and
prove very accurate estimates for the k-NAESAT threshold; in particular, we
verify the statistical mechanics conjecture for this problem. This is joint
work with Amin Coja-Oghlan.

#### Colloquium

Title: Discrete Optimization
Speaker: ()
Date: 1.3.2013, 8:45-14:00
Room: Hörsaal BE01, Steyrergasse 30, Parterre
Abstract:

Colloquium Discrete Optimization
Dedicated to Rainer E. Burkard's 70-th Birthday

Program

8:30-8:45 Bettina Klinz, Mihyun Kang (TU Graz)

Opening
_________________________________________________________

8:45-9:45 Silvano Martello (University of Bologna)

Assignment problems: At the roots of combinatorial optimization

9:50-10:35 Franz Rendl (University of Klagenfurt)

A new hierarchy of bounds for Max-Cut and related problems
___________________________________________________________

10:40-11:10 Coffee break
___________________________________________________________

11:10-12:10 Gerhard Woeginger (TU Eindhoven, TU Berlin)

Optimization at the second level

Polynomially solvable cases of NP-hard problems
____________________________________________________________

13:00- Lunch buffet

#### Vortrag im Seminar Diskrete Mathematik und Optimierung

Title: Computer Vision and Optimization
Speaker: Horst Bischof und Thomas Pock (Institute for Computer Graphics and Vision, TU Graz)
Date: 26.2. 2013, 14:15
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

This talk will first present an overview of ongoing activities
and research projects at the Inst. for Computer Graphics and Vision
at TU Graz. Many of these projects require a significant amount of
optimization (discrete or continous) methods. In the second part of
the talk we highlight some recent results in convex optimization and
we show how these results can be efficiently implemented on
modern graphics hardware.

#### Strukturtheorie-Seminar

Title: Soft local times and decoupling of random interlacements
Speaker: Prof. Serguei Popov (Univ. Campinas, Brasilien)
Date: Do, 14.2.2013, 16 Uhr c.t.
Room: SR C307, Steyrergasse 30, 3. Stock
Abstract:

We establish a decoupling feature of the random interlacement process
$I^u$ in $Z^d$, at level u, in dimension 3 or bigger. Roughly speaking, we
show that observations of $I^u$ restricted to two disjoint subsets $A_1$ and $A_2$ of
$Z^d$ are approximately independent, once we add a sprinkling to the
process $I^u$ by slightly increasing the parameter u. Our results differ
from previous ones in that we allow the mutual distance between the
sets $A_1$ and $A_2$ to be much smaller than their diameters. We then
provide an important application of this decoupling for which such
flexibility is crucial. More precisely, we prove that, above a certain
critical threshold u**, the probability of having long paths that
avoid $I^u$ is exponentially small, with logarithmic corrections for
d=3. To obtain the above decoupling, we first develop a general method
for comparing the trace left by two Markov chains on the same state
space. This method is based on what we call the soft local time of a
chain. In another crucial step towards our main result, we also prove
that any discrete set can be "smoothened" into a slightly enlarged
discrete set, for which its equilibrium measure behaves in a regular
way. This is a joint work with Augusto Teixeira.

#### Minikurs/Intensivseminar

Title: Self-adjoint realizations of elliptic differential operators in smooth and non-smooth domains
Speaker: Till Micheler (Institut für Mathematik, TU Berlin)
Date: 11.2. 11:30-12:30; 12.2. 13:30-16:30; 13.2. 9:30-12:00; 14.2. 9:30-12:00
Room: C307
Abstract:

This seminar consists of two parts. I: Self-adjoint realizations of the Laplacian on Lipschitz- and quasi-convex domains, and II: Self-adjoint realizations of 2m-th order elliptic differential operators on smooth domains

We discuss the characterization of all self-adjoint
(symmetric, dissipative, accumulative, closed)
realizations of the Laplacian (of a 2m-th order elliptic differential
operator) on Lipschitz- and quasi-convex
(smooth) domains in terms of general boundary conditions. In addition we
provide regularity statements and Krein type resolvent formulas.
Moreover we describe the ranges of the Dirichlet and Neumann trace
extended to the maximal domain of the Laplacian.

#### Strukturtheorie-Seminar

Title: On the edge expansion of infinite Cayley graphs
Speaker: Dr. Amnon Rosenmann (Ruppin Academic Center and Tel Aviv University)
Date: Dienstag, 5.2.2013, 11 Uhr c.t.
Room: SR C307, Steyrergasse 30, 3. Stock
Abstract:

The edge expansion (or isoperimetric number) of a finite graph is a well-known and studied object with diverse applications, the analog of Cheeger constant in Riemannian Geometry. It is defined to be the minimum of the ratio of the edge boundary by the (vertex) cardinality of a subgraph, where the minimum is taken over all finite subgraphs of cardinality at most half the cardinality of the whole graph. It was thoroughly investigated, in particular with regard to expander graphs, by algebraic means like the spectrum of the Laplace operator or by Kazhdan constant (for Cayley graphs), by probabilistic means like random walks, and more. In the case of infinite graphs which are Cayley graphs of finitely generated infinite groups, the asymptotic invariant which was mainly studied was the isoperimetric profile of amenable groups. The edge expansion, i.e. the infimum, over all finite subgraphs, of the ratio of the edge boundary by the cardinality of the subgraph, was studied for specific groups. In the talk we will describe our work of obtaining formulas and bounds for the edge expansion constant of infinite Cayley graphs with respect to basic onstructions of the underlying groups. These will also be compared to the behavior of the spectral radius of a symmetric random walk and to the Euler characteristic.

#### Miniworkshop

Title: Spectral and Perturbation Theory of Selfadjoint Operators in Krein Spaces
Speaker: ()
Date: 30.-31.1.2013 jeweils 9-17 Uhr
Room: Seminarraum C307 (30.1.) und A 306 (31.1.)
Abstract:

\vglue 0.5cm

{\bf January 30, C307}

{\bf K. Veselic}, {\it Spectral theory of the Klein-Gordon equation}
\vskip -0.01cm
{\bf V. Kostrykin}, {\it The spectrum of block operator matrices}
\vskip -0.01cm
{\bf C. Trunk}, {\it Sharp eigenvalue estimates for nonnegative operators in Krein spaces}
\vskip -0.01cm
{\bf F. Philipp}, {\it Bounded and relatively bounded $J$-selfadjoint perturbations}
\hglue 2.4cm {\it of $J$-positive operators}

{\bf January 31, A306}

{\bf A. Motovilov}, {\it Sharp norm bounds on variation of spectral subspaces}
\vskip -0.01cm
{\bf H. Woracek}, {\it Almost Pontryagin space completions}
\vskip -0.01cm
{\bf A. Kostenko}, {\it The HELP inequality and the similarity problem for}
\hglue 2.87cm {\it indefinite Sturm-Liouville operators}
\vskip -0.01cm
{\bf J. Behrndt}, {\it Spectral theory of elliptic differential operators with indefinite weights}

#### Zahlentheoretisches Kolloquium

Title: An application of bounded Harman variation
Speaker: Florian Pausinger (IST Austria)
Date: Freitag, 25.1.2013, 14:15 Uhr
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz
Abstract:

Abstract. We discuss a measure of variation of multi-dimensional functions
due to Harman, which has applications in numerical integration and
allows to prove a version of the classical Theorem of Koksma-Hlawka. We
investigate how it relates to the classical Vitali and Hardy-Krause variations
and report on recent results concerning closure properties for the family of
functions of bounded Harman variation. Furthermore, we sketch an application
of this notion to a concrete problem in biological image analysis, which
is the main reason for our interest in this topic.
This is a joint work with Herbert Edelsbrunner.

#### Strukturtheorie-Seminar

Title: The Dirichlet Series of a Profinite Group
Speaker: Dung Hoang Duong (Universität Leiden, Niederlande & Universität Padua, Italien)
Date: Do, 24. 1. 2013, 16:30 (Ersatz für abgesagten Termin vom 6.12.2012))
Room: SR C307, Steyrergasse 30, 3. Stock
Abstract:

Given a finitely generated profinite group $G$, one may associate to $G$ a Dirichlet series $P_{G}(s)$ interpolating the probability that a random $s$-tuple generates the group $G$ topologically. The reciprocal of $P_{G}(s)$ is usually called the Probabilistic Zeta Function of the group $G$. In this talk, I will first present a brief survey of the subject, then I will present some results concerning the finiteness of the group $G$ from the rationality of $P_{G}(s)$. I will then end up with some open problems.

#### Gastvortrag

Title: Porisms
Speaker: Boris Odehnal (University of Applied Arts, Vienna)
Date: 22.01.2013, 14:30 Uhr
Room: SR D, Franz-Josef-Str. 18, EG, Montanuniv. Leoben
Abstract:

This talk gives an overview on porisms, i.e., geometric figures and configurations of geometric objects that close in some sense. The most famous example of a porism is that of Poncelet: Given two conic sections (in general position). If an $n$-gon with vertices on one conic and edges tangent to the other closes for one certain starting point, then it closes for any choice of the starting point. A well-known example is given by a triangle with its incircle and circumcircle. (The triangle can be rotated freely such that its vertices trace the circumcircle and its edges are tangent to the incircle.) Note that in the general case we don't have a rigid body motion.

There are many such closing theorems in geometry. We give some examples of Poncelet like porisms and others. Further we want to collect some more or less known results from elementary and algebraic geometry dealing with the case of two circles. Finally we want to gain insight into the mathematics behind the proof of Poncelet's theorem.

#### Gastvortrag

Title: Discrete linear Weingarten surfaces
Speaker: Udo Hertrich-Jeromin (TU Wien)
Date: Freitag 18.1.2013, 14:00 Uhr
Room: Seminarraum 2, Institut f. Geometrie, Kopernikusgasse 24
Abstract:

We shall investigate discrete quadrilateral linear Weingarten'' meshes in spaces of constant curvature. A Lie geometric approach reveals a surprising relation with the theory of discrete isothermic nets.

#### Strukturtheorie-Seminar

Title: Dismantlability of graphs and applications to fixed point theorems
Speaker: Dr. Damian Osajda (Fakultät für Mathematik, Universität Wien)
Date: Do., 17.1.2013, 15:00 s.t.
Room: Seminarraum C208, Steyrergasse 30, 2. Stock
Abstract:

This is joint work with Sebastian Hensel and Piotr Przytycki.
Dismantlability is a property of graphs implying e.g. collapsibility of canonically associated simplicial complexes. Moreover, it follows that a finite group of automorphisms of a dismantlable graph fixes a clique. We use this fact to show some fixed point theorems for complexes canonically associated with various moduli spaces. In particular, for the arc complex, sphere complex and disc complex. This implies some realisation results, like e.g. a solution to the well-known Nielsen Realisation Problem for punctured surfaces. As further corollaries we describe classifying spaces for proper actions for corresponding groups.

Title: Extremal lattices
Speaker: Prof. Dr. Gabriele Nebe (RWTH Aachen)
Date: Freitag, 11.1.2013, 10:30 Uhr
Room: Seminarraum 2, Institut f. Geometrie, Kopernikusg.24/4
Abstract:

\footnotesize The classification of the densest sphere packings in Euclidean space is a very old and difficult problem. In dimension 3 this was the famous Kepler conjecture, proved by Thomas Hales only a few years ago. The problem becomes much easier if one restricts to lattice sphere packings, where the centers of the spheres form a group. The density function has only finitly many local extrema on the ($n(n+1)/2-1$)-dimensional space of similarity classes of $n$-dimensional lattices and Korkine, Zolotareff and Voronoi ($\sim$ 1900) developed an algorithm to enumerate all of them. This has been done up to dimension 8. Dimension 24 is the only other dimension where one knows the densest lattice, due to the existence of the famous Leech lattice. The Leech lattice is one example of an extremal even unimodular lattice: The theta series of an even unimodular lattice of dimension $n$ is a modular form of weight $n/2$ for the full modular group PSL$_2({\bf Z})$. It has already been observed by Siegel that the theory of modular forms allows to explicitly upperbound the
density of an even unimodular lattice of dimension $n$: $$\min (L) \leq \lfloor \frac{n}{24} \rfloor +1 .$$ Lattices achieving equality are called {\bf extremal}. Of particular interest are extremal lattices and codes in the jump dimensions'' - the multiples of 24.
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I will give a construction of the extremal even unimodular lattice $\Gamma$
of dimension 72 I discovered in summer 2010. The existence of such a lattice was a longstanding open problem. The construction that allows to obtain the minimum by computer is similar to the one of the Leech lattice from $E_8$ and of the Golay code from the Hamming code (Turyn 1967). $\Gamma$ can also be obtained as a tensor product of the Leech lattice (realised over the ring of integers $R$ in the imaginary quadratic number field of discriminant $-7$) and the 3-dimensional Hermitian unimodular $R$-lattice of minimum 2, usually known as the Barnes lattice. This Hermitian tensor product construction shows that the automorphism group of $\Gamma$ contains the absolutely irreducible rational matrix group $($SL$_2(25) \times$PSL$_2(7)):2$.

#### Strukturtheorie-Seminar

Title: Random walk on (random) graphs and random interlacements
Speaker: Jiří Černý (Uni Wien)
Date: Do, 10 Jän, 15:00
Room: SR C208
Abstract:

How long does it take for a simple random walk on a large
d-dimensional discrete torus to disconnect the torus? Despite a sizeable
effort, the answer is still We do not know.'' In my talk I will survey the recent development in answering this question, evolving around the random interlacement model, a dependent percolation on $Z^d$, which describes the microscopic texture left by the random walk on the torus. I will also answer the question for some classes of simpler' graphs.

#### Vortrag

Title: Quasi Monte Carlo (QMC) designs and weighted QMC designs on the sphere
Speaker: Johann S. Brauchart (University of New South Wales, Sydney, Australien)
Date: 9.1.2012, 11:15
Room: Seminarraum A206
Abstract:

A Quasi Monte Carlo rule approximates the integral of a continuous function with respect to the uniform measure using the average of function values at well-chosen nodes. For example, on the unit sphere $\mathbb{S}^d$ in $\mathbb{R}^{d+1}$ such nodes may form spherical $t$-designs and thus integrate exactly all polynomials of degree $t$ or less.

The quality of a sequence of node sets can be measured using test functions from a smooth enough Sobolev space $\mathbb{H}^s( \mathbb{S}^d )$ over $\mathbb{S}^d$ -- which intentionally is a reproducing kernel Hilbert space -- by means of the worst-case error (WCE) of the associated QMC rules. A perhaps amazing fact is that there is a natural setting where the WCE can be (almost) expressed in terms of the sum of certain powers of all mutual Euclidean distances of the integration nodes and has a geometric interpretation as a discrepancy that generalizes the concept of the spherical cap $\mathbb{L}_2$-discrepancy.

Good point configurations have small WCE (for a given $s$). This leads to the concept of QMC design sequences for $\mathbb{H}^s( \mathbb{S}^d )$}, where the node sets, by definition, have optimal order of WCE (arXiv:1208.3267v1 [math.NA]). In fact, it is known that a sequence of spherical $t$-designs with optimal(!) order of the number of points achieves optimal order of convergence of WCE for any(!) $s > d/2$. The introduction of weights (weighted QMC rules) and their adjustment by minimizing the associated WCE improves the performance of the rules in certain cases when equal weights would give considerably worse results (say, e.g., random points).

Interestingly, the usual low-discrepancy sequences on $\mathbb{S}^d$ almost satisfy the QMC design property for $s \in (d/2, (d+1)/2)$ and it is an open question if the smoothness $s$ can be higher.

The talk is rounded off by discussing a spatial extension of QMC designs.