### 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.