Discrete Mathematics 2010-2022

10:30 opening
10:40-11:30 main talk 1: Christopher Frei (Manchester)
11:30-10:40 Math.Video
10:45-12:15 PhD talk 1: JunSeok Oh (KFU Graz)
12:15-12:25 Math.Video
12:25-13:30 Lunch buffet
13:45-14:15 PhD talk 2: Shu-Qin Zhang (MU Leoben)
14:15-14:25 Math.Video
14:30-15:00 PhD talk 3: Irene de Parada (TU Graz)
15:00-15:10 Math.Video
15:10-15:40 Coffee break
15:40-16:30 main talk 2: Silke Rolles (TU München)
16:30-16:40 Math.Video

A more detailed programme will follow.

In this master-seminar, linearly reinforced random walks are explained and some results are presented.

Tamari lattice is a partial order defined on objects counted by Catalan numbers, such as binary trees and Dyck paths. It is a well-studied object in combinatorics, with several generalizations. In this talk, we will see how intervals in these Tamari-like lattices are related to planar maps. More precisely, we discovered a bijection between non-separable planar maps and intervals in the generalized Tamari lattice, which naturally extends to a bijection between bridgeless planar maps and intervals in the original Tamari lattice. We will also discuss the consequences of these bijections in enumerative and structural studies of the related objects.

We are going to derive spectral estimates for several classes of magnetic Laplacians. They include the magnetic Laplacian on three-dimensional regions with Dirichlet boundary conditions as well as the magnetic Laplacian defined in $\mathbb{R}^3$ with the local change of the magnetic field. We establish two-dimensional Berezin-Li-Yau and Lieb-Thirring-type bounds in the presence of constant magnetic fields and, using them, get three-dimensional estimates for the eigenvalue moments of the corresponding magnetic Laplacians. Also we derive separately the Lieb-Thirring bounds for the magnetic Schroeodinger operators defined on two dimensional circle with radially symmetric magnetic field and electric potentials.

We formulate and prove a Khintchine-type law for inhomogeneous Diophantine approximation. The denominators form a lacunary sequence of integers, while the size of the set of well-approximable numbers is given with respect to a probability measure with Fourier coefficients of a prescribed logarithmic decay rate.

(Remark: Agamemnon Zafeiropoulos is a new member of the Institute of Analysis and Number Theory, who started here as a Postdoc researcher in November 2017.)

We prove that, for arbitrary Dirichlet $L$-functions $L(s;\chi_1),\ldots, L(s;\chi_n)$ (including the case when $\chi_j$ is equivalent to $\chi_l$ for $j\ne k$), suitable shifts of type $L(s+i\alpha_jt^{a_j}\log^{b_j}t;\chi_j)$ can simultaneously approximate any given analytic functions on a simply connected compact subset of the right open half of the critical strip, provided the pairs $(a_j,b_j)$ are distinct and satisfy certain conditions. Moreover, we consider a discrete analogue of this problem where $t$ runs over the set of positive integers.

On Facebook, people with high number of connections tend to be connected more likely than randomness would suggest, while in biological networks vertices with high number of connections tend to be connected less likely than randomness would suggest. In terms of network science, the first network is assortative, while the second is disassortative.

Degrees (number of connections) do not tell if a network is assortative or disassortative. The Joint Degree Matrix (JDM) of a network (graph) counts number of edges between the sets of degree $i$ and degree $j$ vertices, for any $i,j$. The JDM realization problem asks whether a graph exists with prescribed number of connections (degree) at the vertices, and with prescribed number of edges between the sets of degree $i$ and degree $j$ vertices, for any $i,j$. The JDM realization problem is well understood. The usual measure for assortativity, the assortativity coefficient, the Pearson correlation coefficient of degree between pairs of linked nodes, is computable from the JDM.

A further generalization of the JDM is the following.
Given a set $W$ and numbers $d(w)$ associated with $w\in W$, and a $W_i:i\in I$ partition of $W$, with numbers $c(W_i,W_j)$ associated with unordered pairs of partition classes, the Partition Adjacency Matrix (PAM) realization problem asks whether there is a simple graph $G$ on the vertex set $W$ with degrees $d_G(w)=d(w)$ for $w\in W$, with exactly $c(W_i,W_j)$ edges with endpoints in $W_i$ and $W_j$; and the PAM construction problem asks for such a graph, if they exist. (These problems are conjectured to be NP-hard.) The bipartite version of these problems are more restricted: $I=I_1\cup I_2$ and $c(W_i,W_j)=0$ whenever $i,j\in I_1$ or $i,j\in I_2$.

We provide algebraic Monte-Carlo algorithms for the bipartite Partition Adjacency Matrix realization and construction problems, which run in polynomial time, say, when $|I|$ is fixed. When the algorithms provide a positive answer, they are always correct, and when the truth is positive, the algorithms fail to report it with small probability.

A tanglegram of size $n$ is a triplet $(L,R,M)$ where $L$ and $R$ are rooted binary trees with $n$ leaves each, and $M$ is a perfect matching between the two sets of leaves. Two tanglegrams $(L_1,R_1,M_1)$ and $(L_2,R_2,M_2)$ are the same if there is a pair of tree-isomorphisms $(\phi,\psi)$ mapping $L_1$ to $L_2$ and $R_1$ to $R_2$ such that matched pairs of leaves get paired to matched pairs of leaves. Tanglegrams are used in phylogenetics, where for example they can represent the phylogenetic trees of parasites and hosts, where the matching gives which parasite infects which host.

A tanglegram layout (i.e. the way tanglegrams are usually drawn) is as follows: draw the two rooted binary trees in the plane with straight lines and without crossing edges such that the leaves of $L$ are on the line $x=0$ and $L$ is drawn in the semi-plane $x\le 0$, the leaves of $R$ are drawn on the line $x=1$ and $R$ is drawn in the semi-plane $x\ge 1$, and the edges of the matching are drawn with straight line. The crossing number of a layout is the number of unordered pairs of matching edges that cross and the tangle crossing number of a tanglegram is the minimum crossing number over all of its layouts. The tangle crossing number is related to a number of biologically important quantities, e.g. the number of times parasites switched hosts. I will present some results about the tangle crossing number, including a Kuratowski type theorem.

In this master seminar, self-avoiding walks on infinite graphs are discussed,
with focus on Cayley graphs and quasi-transitive graphs.
The connective constant is the exponential growth rate of the number of self-avoiding walks of length n. Its computation for lattices is a difficult problem coming from Statistical Physics. In the talk, some basic properties, recent results, and computations are presented.

We study arithmetic progressions $\{a,a+b,a+2b,\dots,a+(\ell-1) b\}$, with $\ell\ge 3$, in random subsets of the initial segment of natural numbers
[$n$]$:=\{1,2,\dots, n\}$. Given $p\in[0,1]$ we denote by [$n$]$_p$ the random subset of [$n$] which includes every number with probability $p$, independently of one another. The focus lies on sparse random subsets, i.e.\ when $p=p(n)=o(1)$ with respect to $n\to\infty$.

Let $X_\ell$ denote the number of distinct arithmetic progressions of length $\ell$ which are contained in [$n$]$_p$. We determine the limiting distribution for $X_\ell$ not only for fixed $\ell\ge 3$ but also when $\ell=\ell(n)\to\infty$ sufficiently slowly. Moreover, we prove a central limit theorem for the joint distribution of the pair $(X_{\ell},X_{\ell'})$ for a wide range of $p$. Our proofs are based on the method of moments and combinatorial arguments, such as an algorithmic enumeration of collections of arithmetic progressions.

These results are joint work with Y.~Barhoumi-Andr\'eani and H.~Liu (Warwick).

Abstract.
After surveying the related results in Pascal triangle we turn
our attention to Pascal pyramid to consider horizontal rays. The newest
result desribes linearly recurrent sequences with rational function coefficients as the sum of elements located along some specific rays. In this way we obtain, for instance, the central Delannoy numbers. The application of the theorem proves many recurrence relations conjectured in The On-Line Encyclopedia of Integer Sequences of Sloane.

Traditionally, algebraic structures are presented by stating generators and relations between words over these generators. There are, however, alternatives to this way of presentation. One of these is the use of automata. Although this approach does not work for every group, the class of groups admitting automaton presentations, the so-called automaton groups, have received quite some attention in research since many groups answering important questions in group theory (such as the Milnor Problem and the Burnside Problem) turn out to be automaton groups. Starting with groups, this interest seems to extend more and
more also to automaton semigroups as it often turns out to be much easier to obtain undecidability results for automaton semigroups than it is for automaton groups.

In this talk, we are going to introduce automaton semigroups and groups, and look into known results as well as open problems concerning decision problems in this area.

\hskip 5pt 14:00 Wolfgang L. Wendland (Universität Stuttgart)
\hskip 1.1cm On Neumann's series and the double layer potential

14:45 Martin Neumüller (Johannes Kepler Universität Linz)
\hskip 1.1cm On Space Time Methods

15:30 Kaffeepause

16:00 Matthias Taus (MIT)
\hskip 1.1cm Fast and accurate methods for wave propagation

16:45 Sergej Rjasanow (Universität des Saarlandes)
\hskip 1.1cm Alternative effective numerical methods for partial differential
equations:
\hskip 1.1cm Differences and Bridges

In this report on the bachelor thesis, we present results on the convergence
to stationarity of lamplighter random walks on some finite graphs.

(Due to a master thesis defense, the talk might start with a small delay.)

I will present recent results on the asymptotics in singular limits of low-lying eigenvalues of self-adjoint operators defined in corner domains. As a model case, I will present the Robin Laplacian with a large Dirichlet parameter.

Firstly, I will give results for the regular case. The asymptotics is given through an effective semi-classical Hamiltonian, defined on the boundary, involving the mean curvature. We deduce from these results a Faber-Krahn inequality for the regular case, rising the question of optimizing the mean curvature of an open set of fixed volume.

Secondly, I will focus on the analysis in n-dimensional corner domains, in which the the singularities of the boundary modify the asymptotics. I will present the recursive class of corner domains and associated singular chains. The asymptotics of the first eigenvalues is obtained through a minimization process over the tangent geometries, and a multi-scale analysis provides an estimate of the remainder.

We consider the following generalization of the classical ``cow path problem''. Suppose we are sending out $k$ robots from $0$ to search the real line at constant speed (with turns) to find a target at an unknown location; $f$ of the robots are faulty (of so-called crash type), meaning that they fail to report the target although visiting its location. The goal is to find the target in time at most $\lambda |d|$, if the target is located at $d$, $|d|\ge 1$, for $\lambda$ as small as possible. We show that it cannot be achieved for $\lambda < 2\frac{(1+\rho)^{1+\rho}}{\rho^\rho} + 1$ where
$\rho := 2\frac{f+1}{k}-1$, which is tight due to earlier work. This also gives some better than previously known lower bounds for so-called Byzantine-type faulty robots (that may, deceitfully, actually wrongly report a target).

(Joint work with Andrey Kupavskii.)

Vortragende:[5mm]

Donnerstag, 19.10.2017
09:00-09:30: Eröffnung durch VR Bischof und Dekan Ernst
09:30-10:00: Harald Niederreiter
Donald Knuth’s problem and Robert Tichy’s solution
10:00-10:45 János Pintz
Some conjectures of Erdös and Turán on consecutive
primegaps
10:45-11:15 Kaffeepause
11:15-12:00 Kálmán Györy
S-parts of values of binary forms and decomposable forms
Mittagspause
14:15-15:00 Yuri Bilu
Effective bounds for singular units
15:00- 15:45 Pietro Corvaja
The Hilbert Property for algebraic varieties
15:45-16:15 Kaffeepause
16:15-17:00 Clemens Fuchs
Diophantine triples and linear recurrences of Pisot type
17:15-18:00 A. V.[5mm]

Freitag, 20.10.2017
09:00-09:45 Klaus Schmidt
Entropy and periodic points of algebraic actions of discrete groups
09:45-10:30 Vitaly Bergelson
Ramsey Theory at the Junction of Additive and Multiplicative Combinatorics
10:30-11:00 Kaffeepause
11:00-11:45 István Berkes
On the uniform theory of lacunary series
Mittagspause
14:00-14:45 Michael Drmota
Digital Expansions and Uniform Distribution
14:45-15:15 Kaffeepause
15:15-16:00 Gerhard Larcher
On Weyl Products and Irregularities of Distribution[5mm]
Aktuelle Informationen unter:
http://www.math.tugraz.at/Tichy60}