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}

Let $F$ be a binary form of degree $r \geq 3$, integer coefficients, on-zero discriminant, and such that the largest irreducible factor of $F$ has degree $d$. For a positive number $Z$ and a positive integer $k \geq 2$ put R_{F,k}(Z)$ for the number of $k$-free integers in the interval $[-Z,Z]$ which is representable by $F$. We shall give an asymptotic formula for $R_{F,k}(Z)$ when $k > \min\{7d/18, \lceil d/2 \rceil - 2\}$. This is joint work with C.L. Stewart.

Speiser in 1935 showed that the Riemann hypothesis is equivalent to the first derivative of the Riemann zeta function having no zeros on the left-half of the critical strip. This result shows that the distribution of zeros of the Riemann zeta function is related to that of its derivatives. The number of zeros and the distribution of the real part of non-real zeros of the derivatives of the Riemann zeta function have been investigated by Berndt, Levinson, Montgomery, and Akatsuka. Berndt, Levinson, and Montgomery investigated the general case, meanwhile Akatsuka gave sharper estimates under the truth of the Riemann hypothesis. This result is further improved by Ge. In the first half of this talk, we introduce these results and generalize the result of Akatsuka to higher-order derivatives of the Riemann zeta function.

Analogous to the case of the Riemann zeta function, the number of zeros and many other properties of zeros of the derivatives of Dirichlet L-functions associated with primitive Dirichlet characters were studied by Yildirim. In the second-half of this talk, we improve some results shown by Yildirim for the first derivative and show some new results. We also introduce two improved estimates on the distribution of zeros obtained under the truth of the generalized Riemann hypothesis. We also extend the result of Ge to these Dirichlet L-functions when the associated modulo is not small. Finally, we introduce an equivalence condition analogous to that of Speiser’s for the generalized Riemann hypothesis, stated in terms of the distribution of zeros of the first derivative of Dirichlet L-functions associated with primitive Dirichlet characters.

After the talk there will be coffee & cake.

For a typical character $\pmod p$, a classical result of Polya and Vinogradov
implies cancellation for character sums longer than $p^{1/2}$. Burgess' bound
allows us to go further, implying cancellation for character sums longer
than $p^{1/4}$. But in practice, one often needs to bound shorter character
sums, and no such bounds are known for a general character. I will describe
recent work (joint with Elijah Fromm) in which we show that even a mild
improvement of the Polya-Vinogradov inequality would imply cancellation in
character sums as short as $p^{0.00001}$, thus significantly improving the
Burgess bound.

The space of images will be considered as a Riemannian manifold, where the
underlying Riemannian metric simultaneously measures the cost of image
transport and intensity variation, introduced by Trouve and Younes as the
metamorphosis model.
A robust and effective variational time discretization of geodesics paths
will proposed and a variational scheme for a time discrete exponential map
will investigated.
The approach requires the definition of a discrete path energy consisting of
a sum of consecutive image matching functionals over a set of image
intensity maps and pairwise matching deformations.
The talk will present existence and convergence results and discuss
applications in image morphing and image animation.

We will investigate spectral properties of $H+V_\lambda$, where $H=-i\alpha\cdot\nabla+m\beta$ is the free Dirac operator in ${\mathbb R}^3$, $m>0$ denotes the mass and $V_\lambda$ is an electrostatic shell potential (which depends on a parameter $\lambda\in{\mathbb R}$) located on the boundary of a smooth domain in ${\mathbb R}^3$. I will present an isoperimetric-type inequality for the admissible range of $\lambda$s for which the coupling $H+V_\lambda$ generates pure point spectrum in $(-m,m)$. That the ball is the unique optimizer of this inequality will also be discussed. This is a joint work with N. Arrizabalaga and L. Vega.

Being motivated by the study of negative-index metamaterials, we will discuss the definition and the spectral properties of the operators given by the differential expressions $\text{div}\,h\,\text{grad}$ in a bounded domain $U$ with a function $h$ which is equal to $1$ on a part of $U$ and to a constant $b<0$ on the rest of $U$. We will see how the properties of such operators depend on the parameter $b$ and the geometry of $U$. In particular, for the critical case $b=-1$ one can have a non-empty essential spectrum, and our results extend the constructions of Behrndt and Krejcirik for symmetric rectangles (2014) to arbitrary smooth geometries. The proof features an interplay between the machinery of boundary triples and the microlocal analysis. Based on a joint work with Claudio Cacciapuoti and Andrea Posilicano (University of Insubria).

Abstract: In joint work with C. Elsholtz and I. Shparlinski we study the equidistribution of multiplicatively defined sets such as squarefree numbers or
quadratic non-residues in sets which are defined in an additive way, for example sumsets, Hilbert cubes or
sets having digit restrictions. In particular, we show that if one fixes any proportion of less than 2/5 of
the digits of all numbers of a given binary bit length, then the remaining set still has the asymptotically
expected number of squarefree integers.

Yu. N. Mukhin asked in 1984 in the Kourovka Notebook (9.32) to classify all  locally compact groups in which for any two closed subgroups X and Y their set theoretic product XY is a closed subgroup.

In joint work with K. H. Hofmann and F. G. Russo the class of “near abelian” groups has been introduced and extensively discussed. As a byresult we can offer a complete answer to Mukhin’s question.

In this talk I will highlight the concepts and present the classification result.

Abstract:
An integer $n$ is said to be $y-$smooth if its largest prime factor
$P(n)$ is less than $y$. As usual, we denote by $S(x,y)$ the set of
$y-$smooth numbers up to $x$,
$$ S(x,y)= \{ 1\leq n\leq x, P(n)\leq y\}.$$

In this talk, we provide an asymptotic formula for the number of
integers $n$ in $S(x,y)$ such that $S_{q}(n) \equiv l \mod m$ for $l \in
\mathbb{Z}$ and $m \geq2$, where $S_q(n)$ denotes the sum of the digits
in base $q$ of the integer $n$. Also, we show that the sequence $\big(
\alpha S_{q}(n)\big)_{n \in S(x,y)}$ is uniformly distributed modulo 1
if and only if $\alpha $ is irrational.
Furthermore, we study the number of ordered pairs $(a,b) \in A\times B$
such that $P(a+b)\leq y$ and $S_{q}(a+b) \equiv l \mod m$, $(l \in
\mathbb{Z}$ and $m \geq2)$, for a given sets of integers $A$ and $B$.
Finally, we discuss sums of the form
$$ \sum_{n \in S(x,y)\atop{S_{q}(n) \equiv l \mod m}} f(n-1), $$
where $f$ is a multiplicative function, $l \in \mathbb{Z}$ and $m \geq2$.

Prime ideals and prime spectra of rings play a central role both in
Commutative Ring Theory and Algebraic Geometry, being them foundation of
Scheme Theory, for instance. Order properties and topological properties
of spaces of prime ideals have been useful to characterize some classes of
rings. From the topological point of view, since the 60s it has been of
interest to put in evidence conditions that a topological space must
satisfy in order to be homeomorphic to the prime spectrum of some
(commutative) ring (with 1). This was the main subject of the PhD thesis
of M. Hochster, where he proved that the topological spaces that are
homeomorphic to the prime spectrum of a ring - also called spectral
spaces - are precisely the spaces $X$ satisfying the following axioms:

\begin{enumerate}

\item $X$ is compact;

\item $X$ admits a basis of open and compact subspaces that is
closed under finite intersections.

\item $X$ is sober, that is, every irreducible closed
subspace of $X$ has a unique generic point.

\end{enumerate}

While for some class of spectral spaces, like Riemann-Zariski spaces (see
\cite{fi-fo-lo}), a class of rings realizing Hochster corrispondence was
explicitly found, for several other spaces naturally arising in
Commutative Ring Theory it is non trivial to understand if they are
spectral because, in particular, it can be not so easy to verify condition
(3) of Hochster's characterization.

A goal of this survey talk is to present some new perspective about the
study of spectral spaces and, in particular, a criterion, based on
ultrafilters, to decide if a topological space is spectral (see
\cite{fi}). Some recent new examples will be discussed.

\begin{thebibliography}{99}

\bibitem{fi} C. A. Finocchiaro, Spectral spaces and ultrafilters.
Comm. Algebra 42 2014, no. 4, 1496--1508.

\bibitem{fi-fo-lo} C. A. Finocchiaro, M. Fontana, K. A. Loper, The
constructible topology on spaces of valuation domains. Trans. Amer.
Math. Soc. 365 2013, no. 12, 6199--6216.

\end{thebibliography}

The Tur\'an function $ex(n,F)$ denotes the maximal number of edges in an $F$-free graph on $n$ vertices. However once the number of edges in a graph on $n$ vertices exceeds $ex(n,F)$, many copies of $F$ appear. We study the function $h_F(n,q)$, the minimal number of copies of $F$ in a graph on $n$ vertices with $ex(n,F)+q$ edges. The value of $h_F(n,q)$ has been extensively studied when $F$ is colour critical. In this paper we consider a simple non-colour-critical graph, namely the bowtie and establish bounds on $h_F(n,q)$ when $q=o(n^2)$.

This is joint work with Mihyun Kang and Oleg Pikhurko.

Sehr geehrte Damen und Herren,

im Rahmen der Besetzung einer §99 Professoren-Laufbahnstelle aus Finanz- und Versicherungsmathematik am Institut für Statistik finden sechs Bewerbungsvorträge statt, zu denen wir Sie herzlich einladen.


Montag, 26.6.2017

9:00-9:30 Paul Krühner (TU Wien): Density bounds for Ito processes and their applications in acturial and mathematical finance,

10:30-11:00 Julia Eisenberg (Univ. Graz): The effects of exchange rates on some actuarial risk measures,

12:00-12:30 Michaela Szölgyenyi (WU Wien): Numerical methods for SDEs in finance and insurance mathematics

Mittwoch, 28.6.2017

9:00-9:30 Peter Hieber (Univ. Ulm): Retirement products: Recent developments,

10:30-11:00 Stefan Thonhauser (TU Graz): On the solution of optimization problems in insurance

Montag, 3.7.2017

10:30-11:00 Christa Cuchiero (Univ. Wien): Polynomial processes in stochastic portfolio theory

A natural number is called $k$-almost prime if it has exactly $k$ prime factors, counted with multiplicity. In this talk I consider the asymptotic counting of such numbers (mostly with $k\leq 3$) with additional constraints put on the prime factors. E.g., if we take $k=2$ and write $n=pq$ with $p$ and $q$ primes of similar size we obtain the so called RSA-integers that play an important role in cryptography. We also consider some examples with $k=3$ inspired by cryptography and the study of coefficients of cyclotomic polynomials. The talk is based on two papers with coauthors P. Moree (accepted in 2016), and Florian Luca, Robert Osburn and Alisa Sedunova (2017+).

Learning and games comprise some of the most sophisticated behaviors
of agents, and examples of situations combining the two keep coming
up. I will focus on three recent instances in my research: How can
you learn successfully if your data can be of varying quality and is
provided by strategic agents who would rather work less on improving
the data quality? If the points to be classified are strategic, what kind of classifier can best anticipate their gaming of the system? Finally, recent results suggest that learning dynamics in games typically fail to converge to the Nash equilibrium. Is gym this bad news for learning dynamics? Or perhaps for the Nash
equilibrium?

Recently Marcus, Spielman & Srivastava (2015) and Marcus (2016) studied polynomial convolutions and Hadamard products that are inspired by free probability. These convolutions capture the expected characteristic polynomials of random matrices. We explore analogues of these convolutions in the setting of Max-Plus Algebra. In this setting, the max-permanent replaces the determinant and the maximum is the analogue of the expected value. Our results resemble those of Marcus et al. Moreover, whereas in the classical setting Marcus et al provide bounds on the roots of the convolution of polynomials, we get exact description of the roots of the convolution of characteristic polynomials in the Max-Plus setting. A brief introduction to operations in Max-Plus Algebra will be given.

This is a joint work with Franz Lehner and Aljosa Peperko.

In this talk I will give some overview on risk theoretic modeling and recent developments.
Classical approaches try to describe insurance risks by means of a stochastic process and quantify them by a risk functional,
typically a short-fall probability or a function depending on a possible deficit. Since these approaches are based on a static parameter choice,
they are oversimplifying in many situations and are not able to cover many realistic phenomena. We will put a focus on controllable extensions
of classical models, which lead to stochastic optimization problems, and touch some model alternatives.
For example such extensions comprise investments in a financial market, risk control via reinsurance or shareholder participation.
Using concrete examples we can discuss different solution procedures and related mathematical questions.
Finally, some relations to other fields of applied mathematics are highlighted.

Abstract.}
In 1977, Volker Strassen presented a deterministic and rigorous algorithm for solving the problem to compute the prime factorization of natural numbers $N$. His method is based on fast polynomial arithmetic techniques and runs in time $\widetilde{O}(N^{1/4})$, which has been state of the art for the last forty years. In this talk, we discuss the core ideas for improving the bound by a superpolynomial factor. The runtime complexity of our algorithm is of the form
\[
\widetilde{O}\left(N^{1/4}\exp(-C\log N/\log\log N)\right).
\]

The concept of linear representations (aka realizations or linearizations) provides some very powerful tool to deal with non-commutative rational functions, namely elements in the universal field of fractions for the ring of non-commutative polynomials (in finitely many variables). While these methods are mostly of purely algebraic origin, they are also nicely compatible with the analytic machinery of (operator-valued) free probability. This theory and the underlying notion of free independence were invented around 1985 by D. Voiculescu, originally for operator-algebraic purposes. It can be seen as a highly non-commutative analogue of classical probability theory and has deep connections to many other fields of mathematics, especially to random matrix theory. In my talk, I will explain how this fascinating interplay leads to explicit algorithms for the computation of distributions and Brown measures, respectively, of evaluations of non-commutative rational functions in freely independent random variables. As we will see, this can be used to determine the asymptotic eigenvalue distribution of certain random matrix models. Furthermore, some concrete examples will show that these algorithms are easily accessible for numerical computations. This is based on joint works with S. Belinschi, J. W. Helton, and R. Speicher.

Classical Waring--Goldbach problem concerns representability of all large integers satisfying a certain local condition as sums of fixed number of $k$-th powers
of prime numbers where $k \geq 1$. For instance Goldbach's conjecture states that
every even number $\geq 4$ can be expressed as a sum of two primes. Denote by
$H(k)$ the least integer $s$ such that every sufficiently large positive integer satisfying the aforementioned local condition may be expressed as a sum of
$s$ $k$-th powers of primes. Following the pioneering work of Vinogradov (1937) (which
yields $H(1) \leq 3$), Hua (1938-1959) showed that $H(k) \leq 2^k+1$. He then reduced
his bound to $H(k) \leq 4k \log k(1 + o(1))$ for every large $k$. In this talk, we shall
look at Waring--Goldbach problem with primes chosen from Piatetski Shapiro
sequences; sequences of the form $\left\{ \lfloor n^c \rfloor \right\}_{n=1}^\infty$ where $c > 1$. Such sequences are
known to contain infinitely many primes when $1 < c < 1.18$.

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

In this talk, I will explain how we can extend the previous work of Mauduit and Rivat about the Rudin--Shapiro sequence for all sequences which count blocks of digits. At first I will consider the case when the length of the block is constant, and then the case when it is a non decreasing function. If time permits, I will present briefly new results (work in progress with Olivier Robert) about the orthogonality between the Möbius function, and some functions related to polynomials along binary digits (like the Rudin--Shapiro sequence).

In the past decade there has been a lot of progress in analysis on metric measure spaces based on ideas from optimal transport. We discuss how some of these ideas can be developed for Markov chains on discrete spaces, using a discrete analogue of the Kantorovich-Wasserstein metric. In particular we present a discrete notion of Ricci curvature based on geodesic convexity of the entropy, which allows us to obtain discrete functional inequalities, such as spectral gap and logarithmic Sobolev inequalities. We also discuss recent applications to interacting particle systems. This is based on joint works with Matthias Erbar (Bonn), Prasad Tetali (Georgia Tech), and Max Fathi (Toulouse).

Unlike systems of linear equations, systems of multivariate polynomial
equations over the complex numbers or finite fields can be compactly used to
model combinatorial problems. In this way, a problem is feasible (e.g. a
graph is 3-colorable, Hamiltonian, etc.) if and only if a given system of
polynomial equations has a solution. Via Hilbert's Nullstellensatz, we
generate a sequence of large-scale, sparse linear algebra computations from
these non-linear models to describe an algorithm for solving the underlying
combinatorial problem. As a byproduct of this algorithm, we produce algebraic
certificates of the non-existence of a solution (i.e., non-3-colorability,
non-Hamiltonicity, or non-existence of an independent set of size k).


In this talk, we present theoretical and experimental results on the size of
these sequences, and the complexity of the Hilbert's Nullstellensatz
algebraic certificates. For non-3-colorability over a finite field, we
utilize this method to successfully solve graph problem instances having
thousands of nodes and tens of thousands of edges. We also describe methods
of optimizing this method, such as finding alternative forms of the
Nullstellensatz, adding carefully-constructed polynomials to the system,
branching and exploiting symmetry.

Die Vernachlässigung von Termen mit kleinen Parametern in den Differentialgleichun\-gen physikalischer Modelle führt häufig auf stark vereinfachte Modelle und grobe Näherungen der tatsächlichen Lösungen. Die asymptotische Entwicklung bedient sich einer Reihenentwicklung der Lösung und des Originalmodells, um präzisere Approximationen zu ermöglichen.

Die Technik der asymptotischen Entwicklung wird anhand eines Modellbeispiels eingeführt. Dazu wird der senkrechte Wurf mit Luftwiderstand betrachtet, wobei das Stokessche Gesetz zur Beschreibung des Luftwiderstands bei kleinen Geschwindig\-kei\-ten dienen soll. Neben der Lösung des Modells ohne Luftwiderstand wird eine Näherung unter Berücksichtigung der ersten Entwicklungsterme bestimmt. Diese beiden Näherungen werden mit der exakten Lösung für verschiedene Parameter verglichen, insbesondere im Hinblick auf die Wurfhöhe und die Flugdauer.

After their introduction in 1947 by Erd\H{o}s, random graphs had a great impact on discrete mathematics and computer science. Since graphs are one-dimensional simplicial complexes, it is natural to develop an analogous theory for $k$-dimensional random simplicial complexes, for all $k \geq 1$.
In 2006, Linial and Meshulam introduced a first model for the random $2-$dimensional simplicial complex, and since then many scientists focused their attention on this subject, laying the foundations of the application of the probabilistic method in topology.

In this talk I would like to explain the L-M model and the notion of `homological connectivity'. Then, I would like to survey some of the work done in recent years on random simplicial complexes, presenting a different natural model and new results obtained working by analogy with the $G(n,p)$ theory.

\vskip 0.5 cm\noindent
MO 10:00 M. Holzmann, {\it Magnetic Schrödinger operators with electric $\delta$-potentials}
MO 11:00 D. Barseghyan, {\it A regular analogue of the Smilansky model}
MO 14:00 V. Lotoreichik, {\it Optimization of the lowest eigenvalue induced by \hskip 4.95cm singular interactions}


TU 10:00 P. Exner, {\it Leaky graphs and Robin billiards: some open problems}
TU 11:00 M. Khalile, {\it Robin Laplacian on infinite sectors and applications
\hskip 3.9cm
to polygons}


WE 14:00 A. Khrabustovskyi, {\it Periodic Schrödinger operators with $\delta'$-potentials}
WE 15:00 P. Schlosser, {\it A Lieb-Thirring inequality for Schrödinger operators with
\hskip 4.3cm $\delta$-potentials supported on a hyperplane}


TH 10:00 S. Stadler, {\it Eigenvalue inequalities for partial differential operators}
TH 11:00 J. Behrndt, {\it The Birman-Schwinger principle and embedded eigenvalues}





$ $

We consider a resource-constrained project scheduling problem
originating in particle therapy for cancer treatment, in which the
scheduling has to be done in high resolution. Traditional mixed integer
linear programming techniques such as time-indexed formulations or
discrete-event formulations are known to have severe limitations in such
cases, i.e., growing too fast or having weak linear programming relaxations.
We suggest a relaxation based on partitioning time into so-called
time-buckets. This relaxation is iteratively solved and serves as basis for
deriving feasible solutions using heuristics. Based on these primal and dual
bounds the time-buckets are successively refined. Combining these parts we
obtain an algorithm that provides good approximate solutions soon and
eventually converges to an optimal solution. Diverse strategies for doing
the time-bucket refinement are investigated. The approach shows excellent
performance in comparison to the traditional formulations and a GRASP
metaheuristic.

Paul Erd\H{o}s asked the following question in the 60's: How many collinear k-tuples can a planar $n$-point set contain, if it contains no $k+1$ points on a line (where $k>3$ is fixed)?
We will present an elementary construction that significantly improves the previously known lower bound for this value.

Positional Game Theory provides a solid mathematical footing for a variety of two-player games of perfect information, usually played on discrete objects, with a number of applications in other branches of mathematics and computer science. The field is just a few decades old, and it has experienced a considerable growth in recent years. Our goal is to introduce some basic concepts and notions, followed by several recent results and open problems.

The prerequisites include just undergraduate knowledge of discrete mathematics and probability, so the lectures could be of interest for people with a wide range of backgrounds and different levels of seniority.

In 1956 Davenport constructed a symmetrized lattice of 2n points with an irrational slope A in the unit square. He proved that if A is badly approximable by rational numbers, then the L2 discrepancy of this point set is at most constant times the square root of log n. I recently showed that the L2 discrepancy is in fact asymptotically equivalent to an explicitly computable constant times the square root of log n in the special case when A is a quadratic irrational. Moreover, this explicit constant factor has surprising connections to the arithmetic of the quadratic field Q(A). In fact, when A is the golden ratio Davenport's symmetrized lattice has a smaller L2 discrepancy than any other construction of a finite point set in the unit square known to this date.

Understanding high dimensional data remains a challenging problem.
Computational topology, in an effort dubbed Topological Data Analysis (TDA), promises to simplify, characterize and compare such data. However, standard TDA focuses on Euclidean spaces, while many types of high-dimensional data naturally live in non-Euclidean ones. Spaces derived from text, speech, image... data are best characterized by non-metric dissimilarities, many of which are inspired by information-theoretical concepts. Such spaces will be called information spaces.

I will present the theoretical foundations of topological analysis in information spaces. First, intuition behind basic computational topology methods is given. Then, various dissimilarity measures are defined along with information theoretical and geometric interpretation. Finally, I will show how the framework of TDA can be extended to information spaces. In particular, I will explain to what extent existing software packages can be adapted to this new setting.

No previous knowledge about (computational) topology or information theory is required. This is joint work with Herbert Edelsbrunner and Ziga Virk.

In many applications, invariant measures for Markov chains have to be calculated. Most often, this has to be done in an algorithmic way. For Markov chains with a special transition structure (quasi-birth-death processes), efficient algorithms can be written in terms of matrix-valued continued fractions.

In the first part of this talk, we will discuss details concerning this relationship, and in particular, we will see that the convergence of the continued fractions which occur in the context of Markov chains is strongly connected to Pringsheim-type convergence criteria for continued fractions.

In the Markov chain context, the continued fractions and its approximants
have a probabilistic interpretation. In the second part of the talk, we
will consider Markov chains with more general transition structures. Based
on the probabilistic interpretation, we will propose a definition for
generalized continued fractions. Although the primary intention of this
construction is the algorithmic treatment of generally structured Markov
chains, it turns out that this definition enables us to

* define $\mathcal{R}$-valued generalized continued fractions
where $\mathcal{R}$is an arbitrary Banach algebra,

* identify a huge variety of generalizations of continued
fractions found in the literature as special cases,

* prove Pringsheim-type criteria in the same way as for
non-generalized continued fractions.

We consider a variation on the Barab\'asi-Albert random graph process with fixed parameters $m\in \mathbb{N}$ and $1/2 < p < 1$. With probability $p$ a vertex is added along with $m$ edges, randomly chosen proportional to vertex degrees. With probability $1 - p$, the oldest vertex still holding its original $m$ edges loses those edges. It is shown that the degree of any vertex either is zero or follows a geometric distribution. If $p$ is above a certain threshold, this leads to a power law for the degree sequence, while a smaller $p$ gives exponential tails.
It is also shown that the graph contains a unique giant component with high probability if and only if $m \ge 2$.

We shall discuss the influence of Omega bounds on Lebesgue measure of certain sets on oscillation of error terms appearing in various asymptotic formulas.

We study the spectrum of two kinds of operators involving a conical geometry: the Dirichlet Laplacian in conical layers and Schrödinger operators with attractive $\delta$-interactions supported by infinite cones. Under the assumption that the cones have smooth cross-sections, we prove that such operators have infinitely many eigenvalues accumulating below the threshold of the essential spectrum and we express the accumulation rate in terms of the eigenvalues of an auxiliary one-dimensional operator with a curvature-induced potential.

This is joint work with Konstantin Pankrashkin.

A commutative ring embeds into a field if and only if it has no zero divisors; moreover, in this case it admits a unique field of fractions. On the other hand, the problem of localization of noncommutative rings and embeddings into skew fields (that is, division rings) is much more complex. For example, there exists noncommutative rings without zero divisors that do not admit embeddings into a skew field, and rings with several non-isomorphic ``skew fields of fractions''. This lead Paul Moritz Cohn to introduce the notion of the universal skew field of fractions to the general theory of skew fields in the 70's. However, almost all known examples of rings admitting universal skew fields of fractions belong to a relatively narrow family of Sylvester domains. One of the exceptions is the tensor product of free algebras. With the help of matrix evaluations we will construct the skew field of multipartite rational functions, which turns out to be the desired universal skew field of fractions. We will also explain its role in the difference-differential calculus in free analysis.

We proved the absence of extensive long-range correlations throughout the replica symmetric phase}, i.e. below the condensation threshold, for a wide class of random factor graph models, including the p-spin Potts antiferromagnet, random k-NAESAT, random k-XORSAT (for even k), etc. This is done by using Janson’s technique of Small Subgraph Conditioning} to nail down the precise limiting distribution of the free energy in this phase. As an application we show that in the replica symmetric phase the random graph model is statistically indistinguishable from the so-called ``planted model''. This result allows us to verify a general conjecture about the reconstruction phase transition in random factor graph models, which deals with the extent of point-to-set correlations. Additionally, we derive a version of the well-known Kesten-Stigum} bound for general factor graph models.

Joint work with Amin Coja-Oghlan, Charilaos Efthymiou, Nor Jaafari and Mihyun Kang

The $k$-core of a graph is the largest subgraph of minimum degree $k$. It can be determined algorithmically by the peeling process that removes an arbitrary vertex of degree less than $k$ while there is one. In this paper we study the $k$-core of the Erd\H{o}s-R\'enyi random graph $G(n,m)$ for $m=\frac{dn}2$ with some fixed positive constant $d>0$.

We propose a new approach to the $k$-core problem in random graphs. More precisely, we devise a randomised algorithm that produces graphs with a $k$-core of a given order and size.

This algorithm is based on an enhanced ``configuration model'' that explicitly designates which vertices will wind up in the core. As it turns out, the necessary structure to construct such configuration model can be set out by way of Warning Propagation, a message passing scheme that plays an important role in physics work on random constraint satisfaction problems.

This is joint work with Amin Coja-Oghlan, Oliver Cooley and Mihyun Kang.

The problem of representability of a permutation group $A$ as the full automorphism group of a digraph $G = (V, E)$ was first studied for regular permutation groups by many authors, the solution of the problem for undirected graphs was first completed by Godsil in 1979. For digraphs, L. Babai in 1980 proved that, except for the groups $S_2^2$, $S_2^3$ , $S_2^4$, $C_3^2$ and the eight element quaternion group $Q$, each regular permutation group is the automorphism group of a digraph. Later on, the direct product of automorphism groups of graphs was studied by Grech. It was shown that, except for an infinite family of groups $S_n \times S_n$, $n\ge $2, and three other groups $D_4 \times S_2$, $D_4\times D_4$, and $S_4 \times S_2 \times S_2$, the direct product of automorphism groups of two graphs is, itself, an automorphism group of a graph. We study the direct product of automorphism groups of digraphs. We show that, except for the infinite family of permutation groups $S_n \times S_n , n \ge 2$ and four other permutation groups $D_4 \times S_2$, $D_4 \times D_4$, $S_4 \times S_2 \times S_2$, and $C_3 \times C_3$, the direct product of automorphism groups of two digraphs is itself the automorphism group of a digraph.

We suggest and examine an extension of the Traveling Salesperson Problem (TSP)
motivated by an application in mechanical engineering. The TSP with forbidden neighborhoods (TSPFN) with radius $r$ is asking for a shortest Hamiltonian cycle of a given graph G, where vertices traversed successively have a distance larger than $r$.

The TSPFN is motivated by an application in mechanical engineering, more precisely in laser beam melting. This technology is used for building complex workpieces in several layers, similar to 3D printing. For each layer new material has to be heated up at several points. The question is
now how to choose the order of the points to be treated in each layer such that internal stresses are low. Furthermore, one is interested in low cycle times of the workpieces. One idea is to look for
short paths between the points or more precisely between the segments in each layer that do not connect segments that are too close so that the heat quantity in each region is not too high in short
periods. In particular in the instances resulting from this application the layers are rectangular non-regular grids.

In this work we start with the consideration of regular grids, i.e. adjacent vertices in the same row
or column all have the same distance from each other. First we suggest a linear integer
programming formulation of TSPFN. Then we examine TSPFN with $r=0$, $r=1$ and $r= \sqrt 2$. We
determine the length and structure of optimal solutions and show that these problems can be
solved in linear time. After discussing optimal TSPFN tours in the plane we briefly consider the
three dimensional case and determine optimal TSPFN tours for $r=0$ and $r=1$ on regular 3D grids.


(Joint work with Anja Fischer und Anna Jellen)

In almost all areas of modern applied sciences, data analysis is facing the challenge of extracting useful and relevant features from even larger, high-dimensional and noisy data. Topological Data Analysis (TDA) is a new discipline, spanning algebraic topology and computational geometry, aimed to address these needs. One of the claims in TDA is that data has shape and the shape matters. In a nutshell, TDA gives a general framework to analyze data from this new point of view allowing the retrieval of geometrical but coordinate-free information.
Two of the most relevant tools in TDA consist of homology and discrete Morse theory. Specifically, homology and its recent development, called persistent homology, provide the topological information of a shape including connectivity and the classification of loops, handles, and voids within the space. Discrete Morse theory, on the other hand, is a powerful tool to handle shapes by providing a morphology- and homology-consistent model of the space to be analyzed.
In the talk, we will introduce these two tools focusing our attention on their mutual connections and on the possibility to exploit them for efficiently retrieving the core information from large-size and high-dimensional data.

I will give an introduction to Motion Planning, a branch of computational
robotics with many intersections to computational geometry. Recently, the
study of random graphs became an extremely popular topic in this context. I
will give an overview of the seminal paper ``Sampling-based Algorithms for
Optimal Motion Planning'' by Karaman and Frazzoli (2011) and explain the
connection in some detail.

Abstract: A conjecture due to Batyrev and Manin predicts the asymptotic
behaviour of the number of rational points of bounded height on cubic
surfaces. In this talk I will report on some modest progress towards
this conjecture, in joint work with Christopher Frei and Efthymios Sofos.

We discuss some results and unsolved problems related to the famous Minkowski question-mark function $?(x)$,
defined as the limit distribution function for Stern-Brocot sequences $F_n$:
$$
?(x) =\lim_{n\to \infty}\frac{\#\{ \xi \in F_n: \xi\le x\}}{\#F_n},\,\,\,\, x\in [0,1].
$$
In particular we try to give an analog of Franel theorem on distribution of Farey sequences and formulate a question
concerning the number of solutions of the equation $?(x)=x$.

Certain generalizations of Minkowski question-mark function and results due to Denjoy, Tichy and Uitz,
Zhabitskaya and other mathematicians will be considered.

\vspace{15mm}
After the talk there will be a social gathering, where French delicacies are served.

Abstract: We discuss asymptotics and explicit upper bounds of average
divisor sums $\sum_{n\leq N}\tau(f(n))$ where $f(n)$ is monic quadratic
polynomial with integer coefficients.

We then illustrate recent applications of these upper bounds in problems
for $D(n)$- $m$-tuples, which are $m$-tuples of distinct positive
integers $(a_1,\ldots,a_m)$ such that $a_i a_j+n$ is a perfect square
for all $i\neq j$, where $n$ is a nonzero integer.(The conjecture for
nonexistence of $D(n)$- $m$-tuples in the case $(m,n)=(5,1)$ was very
recently resolved by He, Togb\'e and Ziegler with different methods.)

Abstract: A $m$-tuple of distinct positive integers $(a_1,\dots,a_m)$ is called a Diophantine $m$-tuple if $a_ia_j+1$ is a perfect square for all $i\neq j$. It was a long outstanding question whether a Diophantine quintuple exists. In a recent paper joint with Bo He and Alain Togbè we recently proved that none exists. After a short introduction to the problem we present the new ideas
that led to the proof of the so-called Diophantine quintuple conjecture.

I will recall and discuss a notion of heat content related to Levy processes in Euclidean space. To start with, I will present instructive examples including Brownian motion and stable processes, and next I will focus on the study of
the  small time behaviour of the heat content for a rich class of Levy processes. The talk is based on the joint work with  Dr. Tomasz Grzywny (Wroclaw University of Science and Technology).

Abstract: Using data from an experiment by Forsythe, Myerson, Rietz, and Weber (1993), designed for a different purpose, we test the standard theory that players have preferences only over their own monetary payoffs and that play will be in (evolutionary stable) equilibrium. In the experiment each subject is recurrently (24 times) randomly matched with ever changing opponents to play a 14 player game. We find that assuming risk-neutrality for all players leads to a predicted evolutionary stable equilibrium that, while it can be rejected at the 5% level of significance, is nevertheless remarkably close to explaining the data. Moreover, when we assume that players are risk-averse and we calibrate their risk-aversion in one treatment with a simple game, this theory cannot be rejected at the 5% level of significance for another treatment with a more complicated game, despite the fact that we have close to 400 data points.

The biological lens in the eye of a mammal focuses light on the retina. Its shape and size is crucial for that purpose. We base our work on abundance of data collected at Washington University in St Louis, mostly on mice. We provide the first ever growth model of the mouse eye and succeed in capturing a variety of behavior regarding the size of the lens, number of cells in the anterior capsule of the lens (epithelium) and the dynamics of the cell movement between the various zones of the epithelium. The lens grows through the entire life and exhibits significantly different behavior throughout life. Our model is based on branching processes with immigration and emigration.

(This is joint work with Steven Bassnett and members of his lab at Washington University. Research supported by NIH grant R01 EYO9852 and a Marie Curie FP7-PEOPLE-2013-IOF-622890 MoLeGro Fellowship.)