Talks in 2015


Title: Über Automaten-Gruppen
Speaker: Dr. Daniele D'Angeli (Institut für Mathematische Strukturtheorie)
Date: 18.12. 11:30
Room: Seminarraum Geometrie

In diesem Vortrag möchte ich eine grundlegende Präsentation der
Theorie der Automatengruppen, auf die meine Forschung
sich größenteils konzentriert, halten. Ich werde eine historische
Einleitung, die grundlegenden Definitionen und die Motivationen,
die überzeugen sollen, warum es interessant ist, solche Gruppen zu
studieren, diskutieren. In dem zweiten Teil des Vortrags
werde ich einige Forschungsergebnisse, die ich in den letzten Jahren
im Kontext der Automatengruppen bekommen habe, und
mögliche Forschungsprobleme, die ich in der Zukunft behandlen möchte, zeigen.


Title: Asymptotic Entropy of Random Walks on Regular Languages
Speaker: Dr. Lorenz Gilch (Institut für Mathematische Strukturtheorie)
Date: 18.12. 10:15
Room: Seminarraum Geometrie

In this talk I will consider transient random walks on an infinite set
of finite words over a finite alphabet. These random walks are of
interest in an information-theoretic context; important results in this
field have been achieved by S. Lalley and V. Malyshev.
We are interested in the asymptotic entropy, which is a characteristic
invariant of transient random walks measuring the asymptotic amount of
information or uncertainty contained in the random word at time n. While
existence of the asymptotic entropy of random walks on groups is
well-known, existence for random walks on other structures is not known
a priori.
The purpose of this talk is to give a short introduction to entropy, to
present the ideas of my existence proof of asymptotic entropy for random
walks on regular languages and to prove the real-analytic behaviour of
the entropy in terms of probability measures of constant support. The
methods used in my work comprise generating function techniques and the
concept of cones in graphs, which allow to cut the random walk
into pieces such that this sequence of pieces form a hidden Markov chain.

Algebra Kolloquium

Title: Polynomial functions over residue class rings of the integers
Speaker: Ashwin Guha (Indian Institute of Science, Bangalore)
Date: Donnerstag, 17. 12. 2015, 16:00 s.t.
Room: SR A206, Geodäsie, Steyrergasse 30/2.St.

Polynomials and their induced functions over finite fields have been
well-studied. The literature on polynomial functions over finite
commutative rings is considerably smaller. In this talk I will
provide a description of polynomial functions over a specific instance of
finite commutative rings, namely, set of residue classes of integers. This
characterization obtained by considering the set of polynomial
functions as a module and providing a set of generators for it is
particularly useful for computational purposes.

Vortrag im Seminar Diskrete Mathematik und Optimierung

Title: The core in random hypergraphs and local weak convergence
Speaker: Kathrin Skubch (Goethe-Universität Frankfurt)
Date: Dienstag 15.12.2015, 14:15
Room: Seminarraum C208, Steyrergasse 30, 2. Stock

\noindent The degree of a vertex in a hypergraph is defined as the number of edges incident to it.
In this talk we study the $k$-core, defined as the maximal induced subhypergraph of minimum degree at least $k$, of the random $r$-uniform hypergraph $\mathbf{H}_r(n,p)$ for $r\geq 3$.
We consider the case $k\geq 2$ and $p=d/n^{r-1}$ for which every vertex has fixed average degree $d>0$. We derive a multi-type branching process
that describes the local structure of the $k$-core together with the mantle, i.e. the vertices outside the core.

Advanced Topics Seminar

Title: Maximal Persistence
Speaker: Primož Škraba (Univerza v Ljubljani, Slowenien)
Date: Freitag, 11.12.2015, 10:30 Uhr
Room: Seminarraum 2 (Geometrie), 4. Stock, Kopernikusgasse 24

Persistent homology is a central tool in topological data analysis. It describes various structures such as components, holes, voids, etc. via a barcode (or a persistence diagram), with longer bars representing ``real'' structure and shorter bars representing ``noise.'' A natural question is how long are the bars we can expect to see from data with no structure, i.e. noise. In this talk, I will introduce some recent results regarding the persistent homology of random processes, specifically, a homogeneous Poisson process. Only an understanding of basic probability will be assumed, with the required topological and probabilistic methods introduced as needed.

Zahlentheoretisches Kolloquium

Title: Problems with packing periodicity
Speaker: Dr. Wöden Kusner (TU Graz)
Date: Freitag, 11. 12. 2015, 14:00 Uhr, s.t.
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz

When we think about packing problems, our intuition often leads us to
believe that the solutions should exhibit some exceptional
symmetry. I'll survey some interesting problems and examples dealing
with packing and periodicity, particularly some that break more symmetry
than you might think.

In this context, I'll also tell you about recent joint work on packings
of a generic polygon in the plane: By introducing a useful topology on
packings and describing the local optimality of a configuration among
double lattices as a consequence of an algorithm of Mount, linear
programming techniques can often certify the best double lattice packing
as a local maximum for volume fraction among all packings.

Kolloquium: Mathematische Methoden in den Natur- und Ingenieurwissenschaften

Title: Space-time Trefftz discontinuous Galerkin methods for wave problems
Speaker: Univ.-Prof., PhD Ilaria PERUGIA (Universität Wien)
Date: Mittwoch, 9.12.2015, 11:30 Uhr
Room: TU Graz, Steyrergasse 30, 3. Stock, Seminarraum C307

Advanced Topics Seminar

Title: Semidiscrete surfaces with constant mean curvature and their associated families
Speaker: Wolfgang Carl (TU Graz)
Date: Freitag, 4.12.2015, 10:30 Uhr
Room: Seminarraum 2 (Geometrie), 4. Stock, Kopernikusgasse 24


Title: Stable invariants for multidimensional persistence
Speaker: Martina Scolamiero (EPFL - Laboratory for Topology and Neuroscience)
Date: 4.12.2015, 13:00 (ca. 45 min)
Room: Seminarraum 2, Kopernikusgasse 24, 4. Stock

Multidimensional Persistence is a method in topological data analysis
which allows to study several properties of a dataset contemporarily. It
is important to identify discrete invariants for multidimensional
persistence in order to compare properties of different datasets.
Furthermore such invariants should be stable, i.e., data sets which are
considered to be close should give close values of the invariant.

We introduce a framework that allows to compute a new class of stable
discrete invariants for multidimensional persistence. In doing this, we
generalize the notion of interleaving topology on multi-dimensional
persistence modules and consequently the notion of closeness for
datasets. A filter function is usually chosen to highlight properties we
want to examine from a dataset. Similarly, our new topology allows some
features of datasets to be considered as noise.

Zahlentheoretisches Kolloquium

Title: Polynomial overrings of Int($\mathbb{Z}$)
Speaker: J.-L. Chabert (LAMFA, Universit\'e de Picardie, Amiens)
Date: Donnerstag, 3. 12. 2015, 16:00 (s.t.)
Room: SR A206, Geodäsie, Steyrergasse 30/2.St.

Abstract: We show that every polynomial overring of Int($\mathbb{Z}$), the ring of
integer-valued polynomials over $\mathbb{Z}$, may be
considered as the ring of integer-valued polynomials over some subset of
$\hat{\mathbb{Z}}$, the profinite completion of $\mathbb{Z}$ with respect to
the fundamental system of neighbourhoods of 0 consisting of all non-zero
ideals of $\mathbb{Z}$.

Vortrag im Seminar Diskrete Mathematik und Optimierung

Title: Bootstrap percolation in $G(n,p)$
Speaker: Tamas Makai (TU Graz)
Date: Dienstag 1.12.2015, 14:15
Room: Seminarraum C208, Steyrergasse 30, 2. Stock

Bootstrap percolation on a graph $G = G(V;E)$ with activation threshold an integer $r \geq 1$ is a deterministic process which evolves in rounds. Every vertex is in one of two possible states: it is either infected or uninfected.
The set of initially infected vertices is given by $\mathcal{A}(0)$. In each round of the process every uninfected vertex $v$ which has at least $r$ infected neighbours becomes infected. Once a vertex has become infected it remains infected forever. The process stops once no more vertices become infected.

We consider the case when the graph $G$ is a binomial random graph and $\mathcal{A}(0)$ consists of a given number of vertices chosen uniformly at random.

Janson, {\L}uczak, Turova and Vallier (2012) determined a threshold $a_c$ such that for any $\omega(n)\gg \sqrt{a_c}$ if $|\mathcal{A}(0)|<a_c-\omega(n)$ then w.h.p.\ only a few additional vertices become infected. However if $|\mathcal{A}(0)|>a_c+\omega(n)$ then almost every vertex becomes infected. We show that this not only holds with high probability but with probability $1-\exp(-\Omega(\omega^2/a_c))$.

This talk is based on joint work with Mihyun Kang.

Title: Zahlentheoretisches Kolloquium
Speaker: ()
Date: Donnerstag, 26.11.2015 und Freitag, 27.11.2015
Room: 26.11.: SR A206 Geodäsie, Steyrerg.30/2.St., 27.11.:SR 2 Geometrie, Kopernikusg.24/4.St., SR C208, Mathematik, Steyrerg.30/2.St.

Donnerstag: 26. 11. 2015, SR A206, Geodäsie, Steyrergasse 30/2.St.

16:15: Martin Widmer, (Royal Holloway Univ. of London, dzt. TU Graz)
Around the Northcott Property - Many new problems and some new results

17:00: Christopher Frei, (TU Graz)
The Hasse norm principle for Abelian extensions

17:45: Dijana Kreso, (TU Graz)
On common values of lacunary polynomials at integer points

Freitag: 27. 11. 2015, SR 2, Geometrie, Kopernikusgasse 24/4.St.

10:30: Alina Ostafe, (UNSW, Sydney)
On some extensions of the Ailon-Rudnick Theorem

11:15: Arne Winterhof, (RICAM Linz)
Pseudorandom sequences: measures and number-theoretic constructions

SR C208, Mathematik, Steyrergasse 30/2.St.

14:15: Fabrizio Barroero, (Scuola Normale Superiore di Pisa)
Unlikely intersections in certain families of abelian varieties and the polynomial Pell equation

15:00: Volker Ziegler, (Univ. Salzburg)
On the number of integers that are the sum of k units

15:45: Clemens Fuchs, (Univ. Salzburg)
On a family of quartic Thue equations over function fields

Zahlentheoretisches Kolloquium

Title: Intersection of polynomial orbits over finite fields
Speaker: Dr. Alina Ostafe (UNSW, Sydney)
Date: Mittwoch, 25. 11. 2015, 14:15 Uhr
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz


Motivated by results on intersections of orbits of D. Ghioca, T. J. Tucker, and M. E. Zieve in characteristic zero, we answer several natural questions about reductions of orbits modulo primes of polynomial dynamical systems defined over $\mathbb Z$. In particular, under certain restrictions, we give bounds for the frequency of the points in an orbit of the reduction modulo a prime $p$ of an algebraic dynamical system that belong to a given algebraic variety. Our approach is based on recent explicit versions of Hilbert's Nullstellensatz and a new result about the reduction modulo prime numbers of systems of multivariate polynomials over the integers.

Moreover, we establish some links between these problems and the uniform dynamical Mordell-Lang conjecture. In this sense we present some upper bounds on the size of the intersection of an orbit in an algebraic dynamical system with a hypersurface for various cases when it is finite.


Title: Spectra of BBS automata
Speaker: Andrzej Żuk (Université Paris 7 )
Date: Freitag, 20.11.2015, 14 Uhr c.t.
Room: SR C208, Steyrerg. 30/III


Title: Optimales Investment für Versicherer
Speaker: Dr. Stefan Thonhauser (TU Graz)
Date: Donnerstag, 19. 11. 2015, 11:00 Uhr
Room: Seminarraum A306, Geodäsie,

In diesem Vortrag möchte ich auf ein Investmentproblem aus der Sicht
eines Versicherers eingehen. Im Gegensatz zu ähnlichen Problemen aus der
Portfoliooptimierung für welche die Wahl einer Investmentstrategie auf
Profitmaximierung ausgerichtet ist, soll in diesem Fall das Gesamtrisiko
reduziert werden.
Ich möchte dabei vorstellen wie sich die Einführung von
Transaktionskosten auf diese Fragestellung auswirken. Hier bedeuten
Transaktionskosten, Kosten welche bei jeder Adaptierung der Strategie
zusätzlich anfallen. Dieser Punkt ist zwar sehr realistisch, wird aber
in den klassischen Arbeiten zu diesem Thema vernachlässigt. Aus
mathematischer Sicht handelt es sich dabei um ein stochastisches
Impuls-Kontrollproblem dessen Lösung durch iteriertes optimales Stoppen
oder mittels der sogenannten quasi Variations-Ungleichungen
charakterisiert werden kann.

Seminar Angewandte Analysis und Numerische Mathematik

Title: Asymptotics for solutions of second order ODE's in a complex domain
Speaker: Philipp Schmitz (TU Ilmenau)
Date: 26.11.2015, 15:00 Uhr
Room: C307

Motivated by non-Hermitian quantum mechanics (for example $\mathcal{PT}$ symmetric problems) second order linear differential equations in complex domains were studied over the last years. In particular, one is interested in $L^2$-solutions which correspond to eigenvalues of some associated operator.

In this talk the so called WKB approximation is introduced, which describes the asymptomatical behaviour for solutions of these differential equations. For a class of problems in $\mathcal{PT}$ symmetric quantum theory specific solutions are constructed by means of the WKB method. These solutions decay exponentially or diverge exponentially in certain areas of the complex plane, called Stokes wedges.

Seminar Angewandte Analysis und Numerische Mathematik

Title: Eigenvalue estimates for operators with finitely many negative squares
Speaker: Prof. Dr. Carsten Trunk (TU Ilmenau)
Date: 26.11.2015, 14:00 Uhr
Room: C307

Let $A, B$ be selfadjoint operators with resolvent difference of rank one. Then the continuous spectra of the two operators coincide. The number of eigenvalues in a gap of the continuous spectrum is the same or it differs by one. This is a well-known fact for selfadjoint operators in Hilbert spaces.

The same question for selfadjoint operators in Krein spaces is more delicate. It arises naturally in the study of singular indefinite Sturm-Liouville problems which correspond to selfadjoint operators in Krein spaces. Often, a one dimensional perturbation leads to the direct sum of two (somehow 'decoupled') selfadjoint operators in $L^2$-Hilbert spaces} with well-known spectra. As a consequence, such an approach can be utilized to describe the spectrum of the singular indefinite Sturm-Liouville operator. As the continuous spectra of the indefinite Sturm-Liouville operator and its perturbation coincide, it remains to study the point spectrum.
Via the Weyl function, eigenvalue estimates in gaps of the continuous spectrum can be obtained: If the unperturbed operator belongs to one of the well studied
subclasses of selfadjoint operators in Krein spaces like operators with finitely many negative operators, then this is reflected in the properties of the Weyl function which we use to prove eigenvalue estimates. We emphasize that these estimates are sharp.

This is a joint work with J.\ Behrndt (Graz) and R.\ Moews (Berlin).

Title: Zahlentheoretisches Kolloquium
Speaker: Sigrid Grepstad (JKU Linz) Adrian Scheerer (TU Graz) ()
Date: Mittwoch, 18. 11. 2015, ab 14:15 Uhr
Room: Seminarraum C 208, 2. Stock, Steyrergasse 30, TU Graz

14:15: Sigrid Grepstad
Sets of bounded discrepancy for multi-dimensional irrational rotation

15:00: Adrian Scheerer
Algorithms for Absolutely Normal Numbers and Discrepancies


Title: Recent advances in Bayesian spatial prediction and sampling design
Speaker: Jürgen PILZ (Institute of Statistics, University of Klagenfurt)
Date: 27. November 2015, 14:30 h
Room: SR für Statistik (NT03098), Kopernikusgasse 24/III


In my talk, I will report on recent work with my colleagues G. Spoeck and H. Kazianka in the area of Bayesian spatial prediction and design [1]-[5].

The Bayesian approach not only offers more flexibility in modeling but also allows us to deal with uncertain distribution parameters, and it leads to more realistic estimates for the predicted variances. We report on some experiences gained with our approach during a European project on "Automatic mapping of radioactivity in case of emergency".

We then go on and apply copula methodology to Bayesian spatial modeling and derive predictive distributions. Moreover, I report on recent results on finding objective priors for the crucial nugget and range parameters of the widely used Matern-family of covariance functions.
Furtheron, I briefly consider the challenges in stepping from the purely spatial setting to spatio-temporal modeling and prediction.

Finally, I will consider the problem of choosing an "optimal" spatial design, i.e. finding an optimal spatial configuration of the observation sites minimizing the total mean squared error of prediction over an area of interest. Using Bessel-sine/cosine- expansions for random fields we arrive at a design problem which is equivalent to finding optimal Bayes designs for linear regression models with uncorrelated errors, for which powerful methods and algorithms from convex optimization theory are available. I will also indicate problems and challenges with optimal Bayesian design when dealing with more complex design criteria such as minimizing the averaged expected lengths of predictive intervals over the area of interest.

[1] H. Kazianka and J. Pilz: Bayesian spatial modeling and interpolation using copulas. Computers & Geosciences 37(3): 310-319, 2011
[2] H. Kazianka and J. Pilz: Objective Bayesian analysis of spatial data taking account of nugget and range parameters. The Canadian Journal of Statistics 40(2): 304-327, 2012
[3] J. Pilz, H. Kazianka and G. Spoeck: Some advances in Bayesian spatial prediction and sampling design. Spatial Statistics 1: 65-81, 2012
[4] G. Spoeck and J. Pilz: Spatial sampling design based on spectral approximations of the error process. In: Spatio-temporal design: Advances in Efficient Data Acquisition (W.G. Mueller and J. Mateu, Eds.), Wiley, New York 2013, 72-102
[5] G. Spoeck and J. Pilz: Simplifying objective functions and avoiding stochastic search algorithms in spatial sampling design. Front. Environ. Sci. 3:39: 1-22, 2015.

Seminar Angewandte Analysis und Numerische Mathematik

Title: Selfadjoint elliptic operators with boundary conditions on not closed hypersurfaces
Speaker: Prof. Dr. Andrea Mantile (Université de Reims Champagne-Ardenne)
Date: 2.12.2015, 11:00 Uhr
Room: C307

The abstract theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic differential operator on $\mathbb R^n$ with linear boundary conditions on (a relatively open part of) a compact hypersurface. Our approach allows to obtain Krein-like resolvent formulae where the reference operator coincides with the free operator with domain $H^2(\mathbb R^n)$; this provides an useful tool for the scattering problem from a hypersurface. Moreover, Schatten-von Neumann estimates, for the difference of the powers of resolvents of the free and the
perturbed operators, yield the existence and completeness of the wave operators of the associated scattering systems. This is a joint work with A. Posilicano and M. Sini

Vortrag im Seminar Diskrete Mathematik und Optimierung

Title: Homological connectivity of random simplicial 2-complexes
Speaker: Oliver Cooley (TU Graz)
Date: Dienstag 17.11.2015, 14:15
Room: Seminarraum C208, Steyrergasse 30, 2. Stock

Linial and Meshulam introduced a model of random simplicial 2-complexes with n 0-simplices (or vertices) in which all pairs of vertices form 1-simplices (or edges) and each triple of vertices forms a 2-simplex (or face) with probability p independently. They showed that this model undergoes a phase transition with respect to $\mathbb{F}_2$-homological 1-connectivity at around $p= \frac{2\log n}{n}$, and that the critical obstruction to connectivity is the presence of an edge which is in no face.

We consider a similar model, but in which each pair of vertices forms an edge only if it lies in a face. Thus the complex is generated by a random $3$-uniform hypergraph by taking the down-closure. Now by definition the previous critical obstruction to connectivity no longer exists. We show that in this model, the phase transition for $\mathbb{F}_2$-homological 1-connectivity occurs at around $p= \frac{\log n + \frac12 \log \log n}{n}$ and describe what the new critical obstruction is. The arguments are complicated by the fact that in this setting, connectivity is not a monotone property.

This talk is based on joint work with Penny Haxell, Mihyun Kang and Philipp Sprüssel.


Title: On surprising relations between Americans and Europeans
Speaker: Prof. Josef Teichmann (ETH Zürich)
Date: Freitag, 13.11.2015, 14:00
Room: Seminarraum für Statistik (NT03098)

Following work of Jourdain-Martini we shed some light on a surprising relationship between American and European options motivated by questions from Finance, Analysis and Numerics.

Zahlentheoretisches Kolloquium

Title: Parametric geometry of numbers
Speaker: Dr. Antoine Marnat (Université de Strasbourg)
Date: Freitag, 13. November 2015, 9:30 Uhr
Room: Seminarraum 2, Institut f. Geometrie, Kopernikusgasse 24, 4. Stock, TU Graz


Title: Fast factorization of hypergraph products
Speaker: Dr. Florian Lehner (Univ. Hamburg)
Date: Thursday, 12.11.2015, 11:00 c.t.
Room: Seminar room C307, Steyrergasse 30, 3rd floor

The existence of a unique decomposition into prime factors with
respect to the Cartesian product of both graphs and hypergraphs is
known since the 1960s. First polynomial time algorithms for the prime
factor decomposition of graphs were presented in the 1980s and even a
linear time algorithm (due to Imrich and Peterin) is known.

For hypergraphs the situation is different. Until recently no
polynomial time algorithm for the prime factorization was known, the
only such algorithm so far was presented by Bretto, Silvestre, and
Vallée in 2013. Its time complexity is $O(|E| |V| r^6 \Delta ^6)$
time, where $r$ is the rank of the hypergraph and $\Delta$ is the
maximal degree.

In this talk we outline a conceptually simpler an faster algorithm
which runs in $O(|E| |V| r^2)$, and in $O(|E| \log^2 |V|)$ for
bounded rank hypergraphs.


Title: Asymptotic behaviour of transition probabilities for subordinated random walk
Speaker: Dr. Wojciech Cygan (Univ. Wroclaw / TU Graz)
Date: Thursday, 5.11.2015, 11:00 c.t.
Room: Seminar room C307, Steyrergasse 30, 3rd floor

We construct a random walk $S_n^\psi$  in $\mathbb{Z}^d$, obtained by subordinating a strongly aperiodic random walk with finite range according to the concept of discrete subordination. The function $\psi$, which is the Laplace exponent of the subordinator is assumed to be a Bernstein function such that its behaviour at zero is prescribed in the realm of regularly varying functions. We prove a strong version of Tauberian type theorem which allows us to investigate the asymptotic behaviour of the tails of the subordinator. Finally, we find an asymptotic formula for the transition kernel of the subordinated random walk.

Festkolloquium aus Anlass des 60. Geburtstages von Prof. Dr. Maximilian Ganster

Title: Kolloquium aus Topologie
Speaker: Prof. Dr. Michael Kerber (TU Graz) Prof. Dr. Martin Goldstern (TU Wien) Prof. Dr. Jörg Thuswaldner (MU Leoben) ()
Date: Freitag, 30. Oktober 2015, ab 10 Uhr
Room: Vormittag: SR Geometrie 2, Kopernikusgasse 24, 4. Stock (im Rahmen der Advanced Topics in Discrete Mathematics) [2mm]Nachmittag: SR C208, Institut für Mathematik, Steyrergasse 30, 2.Stock

10:00: Kaffee[3mm]
10:30: Begrüßung[3mm]
10:45: Michael Kerber: Topological data analysis[3mm]
12:00: Mittagessen[3mm]
14:00: Martin Goldstern: p-points and ultrafilters without p-point quotients[3mm]
14:45: Musikalische Darbietung von Prof. Dr.Otto Laback[3mm]
15:15: Kaffeepause[3mm]
15:30: Jörg Thuswaldner: Topology of 3-dimensional fractals

Opening of the Second Phase

Title: ... with talks by Prof. Ilse Fischer and Prof. Emo Welzl
Speaker: ()
Date: Dienstag, der 27.10.2015, von 11:00 bis 15:30
Room: HS BE01, Steyrergasse 30, Erdgeschoss

Talks will be given by Prof. Ilse Fischer (University of Vienna) and Prof. Emo Welzl (ETH Zurich). Moreover, there will be two talks by students from the first phase; Mario Weitzer (Project 06) and Christopher Frei (Project 09). Please find further details on our website:

Vortrag im Seminar Diskrete Mathematik und Optimierung

Title: Scaling limits and Benjamini-Schramm limits of some models of random trees, graphs and planar maps
Speaker: Benedikt Stufler (Ludwig-Maximalians-Universität München)
Date: Dienstag 20.10.2015, 14:15
Room: Seminarraum C208, Steyrergasse 30, 2. Stock

We provide an overview of the speakers' scientific work, including the following topics. We establish the Brownian continuum random tree as the scaling limit of random unlabelled unrooted trees, and random graphs from subcritical classes, both in the labelled setting (joint work with K. Panagiotou and K. Weller) and in the unlabelled setting. We provide a new proof for the scaling limit of random P\'olya trees (joint with K. Panagiotou), extending previous results by treating trees with arbitrary vertex-degree restrictions in a unified way. We provide a new proof for the scaling limit of random outerplanar maps, extending previous results by treating maps with independent link weights and obtaining the scaling limit of random bipartite outerplanar maps. We establish Benjamini-Schramm limits of random graphs from subcritical graph classes (in the labelled and unlabelled settings) and classes of outerplanar maps satisfying a subcriticality condition. We use an elegant probabilistic approach in order to obtain scaling limits for the sizes of the $k$-th largest block in random labelled planar graphs, which seems to be a new result for $k \ge 2$.


Title: Boundary preserving transformations of random walks
Speaker: Prof. Vadim A. Kaimanovich (Univ. Ottawa)
Date: Thursday,15.10.2015, 11:00 c.t.
Room: Seminar room C307, Steyrergasse 30, 3rd floor

Given two different random walks on the same group, a priory there is no reason to expect them to have the same Poisson boundary. We shall show that there is a natural class of transformations of random walks (determined by ordinary or, more generally, randomized stopping times) which do not change the Poisson boundary. The related questions and conjectures will also be discussed.