### Talks in 2017

#### Habilitationsvortrag (Lehrprobe)

**Title:**The Tarski-Seidenberg Principle

**Speaker:**Dr. Christopher Frei (Univ. of Manchester)

**Date:**Freitag, 15. 12. 2017, 14:00 Uhr

**Room:**SR Analysis-Zahlentheorie (NT02008), Kopernikusgasse 24, 2.Stock

**Abstract:**

#### Doktoratskolleg Discrete Mathematics

**Title:**

**Speaker:**Discrete Mathematics Day 2017 ()

**Date:**Thursday, 14.12.2017, 10:30-16:40

**Room:**Hörsaal BE01, Steyregasse 30, EG

**Abstract:**

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.

#### Strukturtheorie-Seminar

**Title:**Reinforced random walk

**Speaker:**Michael Kalab (TU Graz)

**Date:**Donnerstag, 7.12.2017, 11 Uhr c.t.

**Room:**Seminarraum AE02, Steyrergasse 30, Erdgeschoss

**Abstract:**

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

#### Vortrag im Seminar für Kombinatorik und Optimierung

**Title:**Tamari-like intervals and planar maps

**Speaker:**Wenjie Fang (TU Graz)

**Date:**Dienstag 5.12.2017, 14:15

**Room:**Seminarraum AE06, Steyrergasse 30, Erdgeschoss

**Abstract:**

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.

#### Seminar Applied Analysis and Computational Mathematics

**Title:**Eigenvalue bounds for the magnetic Laplacians and Schroedinger operators

**Speaker:**Diana Barseghyan (University of Ostrava)

**Date:**14.12.2017, 14:00

**Room:**Seminarraum STEG006, Steyrergasse 30, Erdgeschoss

**Abstract:**

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.

#### FWF START Seminar (Mini-Colloquium)

**Title:**Inhomogeneous Diophantine Approximation with Restricted Denominators

**Speaker:**Agamemnon Zafeiropoulos (TU Graz)

**Date:**4.12.2017, 16:00

**Room:**Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24, 2.OG

**Abstract:**

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.)

#### FWF START Seminar (Mini-Colloquium)

**Title:**Joint universality for dependent L-functions

**Speaker:**Lukasz Pankowski (Adam Mickiewicz University Poznan)

**Date:**4.12.2017, 15:15

**Room:**Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24, 2.OG

**Abstract:**

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.

#### Reminder: Vortrag im Seminar für Kombinatorik und Optimierung

**Title:**The Partition Adjacency Matrix realization problem

**Speaker:**L\'aszl\'o Sz\'ekely (University of South Carolina)

**Date:**Freitag 1. Dezember, 14:00 Kaffeepause 13:30 (Steyrergasse 30, 2. Stock, Mathematik)

**Room:**Hörsaal BE01, Steyrergasse 30, Erdgeschoss

**Abstract:**

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.

#### Vortrag im Seminar für Kombinatorik und Optimierung

**Title:**Tangle Crossing Numbers and theTanglegram Kuratowski Theorem

**Speaker:**\'Eva Czabarka (University of South Carolina)

**Date:**Freitag 1. Dezember, 15:00

**Room:**Hörsaal BE01, Steyrergasse 30, Erdgeschoss

**Abstract:**

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.

#### Zahlentheoretisches KolloquiumACHTUNG - Die Beginnzeit des Vortrages hat sich geändert!

**Title:**Metric discrepancy results for geometric progressions with small ratios 3/2, 4/3, etc.

**Speaker:**Prof. Dr. Katusi Fukuyama (Kobe University, Japan)

**Date:**Dienstag, 28. 11. 2017, 12:00 Uhr

**Room:**Seminarraum Analysis-Zahlentheorie (NT02008), Kopernikusgasse 24/II

**Abstract:**

#### Sturkturtheorie-Seminar

**Title:**The connective constant

**Speaker:**Christian Lindorfer (TU Graz)

**Date:**Donnerstag, 23.11.2017, 11 Uhr c.t.

**Room:**Seminarraum AE02, Steyrergasse 30, Erdgeschoss

**Abstract:**

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.

#### Vortrag im Seminar für Kombinatorik und Optimierung

**Title:**On the number of arithmetic progressions in sparse random sets

**Speaker:**Christoph Koch (University of Warwick)

**Date:**Dienstag 21.11.2017, 14:15

**Room:**Seminarraum AE06, Steyrergasse 30, Erdgeschoss

**Abstract:**

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).

#### Zahlentheoretisches Kolloquium

**Title:**Sum of elements locating along horizontal rays in Pascal pyramid

**Speaker:**Prof. Dr. László Szalay (Univ. Sopron)

**Date:**Freitag, 17. 11. 2017, 14:00 Uhr

**Room:**Seminarraum Analysis-Zahlentheorie (NT02008), Kopernikusgasse 24/II

**Abstract:**

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.

#### Strukturtheorie-Seminar

**Title:**Decision Problems and Automaton Structures

**Speaker:**Jan Philipp Wächter (Univ. Stuttgart)

**Date:**Monday, 13.11.2017, 11:15

**Room:**Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24/II

**Abstract:**

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.

#### Festkolloquium

**Title:**Festkolloquium zum Anlass des 50. Geburtstages von Herrn Univ.-Prof. Dr. Olaf Steinbach

**Speaker:**()

**Date:**Freitag, 10.11.2017, 14:00 Uhr

**Room:**Hörsaal BE01, Steyrergasse 30, EG, TU Graz

**Abstract:**

\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

#### Sturkturtheorie-Seminar

**Title:**Bachelor thesis: lamplighter random walks on finite graphs

**Speaker:**Eva Hainzl (TU Graz)

**Date:**Donnerstag, 9.11.2017, 11 Uhr c.t.

**Room:**Seminarraum AE02, Steyrergasse 30, Erdgeschoss

**Abstract:**

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.)

#### Seminar Angewandte Analysis und Numerische Mathematik

**Title:**Eigenvalues of Robin Laplacian with strong attractive parameter

**Speaker:**Dr. Nicolas Popoff (Université de Bordeaux)

**Date:**21.11.2017, 10:00 Uhr

**Room:**A 306

**Abstract:**

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.

#### Vortrag im IST Seminar

**Title:**Lower Bounds for Searching Robots, some Faulty

**Speaker:**Emo Welzl (Department of Computer Science, ETH Zürich)

**Date:**23.10. 2017, 16:15

**Room:**Seminarraum IST, Inffeldgasse 16b, 2.Stock

**Abstract:**

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.)

#### Festkolloquium

**Title:**Festkolloquium aus Anlass des 60.Geburtstages von Prof.Dr.Robert Tichy

**Speaker:**()

**Date:**19. - 20. Oktober 2017

**Room:**HS BE01, Steyrergasse 30/EG, TU Graz

**Abstract:**

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}

#### Zahlentheoretisches Kolloquium

**Title:**On the representation of k-free integers by binary forms

**Speaker:**Dr. Stanley Yao Xiao (University of Oxford)

**Date:**6. 10. 2017, 14:15

**Room:**Seminarraum Analysis-Zahlentheorie (NT02008)

**Abstract:**

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.

#### FWF START Seminar

**Title:**Distribution of zeros of the derivatives of the Riemann zeta function and Dirichlet L-functions

**Speaker:**Ade Irma Suriajaya (RIKEN Tokyo)

**Date:**27.9.2017, 13:30

**Room:**Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24, 2.OG

**Abstract:**

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.

#### Zahlentheoretisches Kolloquium

**Title:**Improving Burgess via Polya-Vinogradov

**Speaker:**Dr. Leo Goldmakher (Williams College)

**Date:**Freitag, 15. 9. 2017, 14:15 Uhr

**Room:**Seminarraum Analysis-Zahlentheorie (NT02008), Kopernikusgasse 24/II

**Abstract:**

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.

#### ICG Visual Computing Seminar

**Title:**Discrete Geodesic Paths in the Space of Images

**Speaker:**Martin Rumpf (Univ. Bonn)

**Date:**11.7.2017, 13:00h

**Room:**Seminarraum ICG, Inffeldgasse 16, 2.OG

**Abstract:**

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.

#### Seminar Angewandte Analysis und Numerische Mathematik

**Title:**A shape optimization problem for the relativistic $\delta$-shell interaction in ${\mathbb R}^3$

**Speaker:**Dr. Albert Mas (Universitat de Barcelona)

**Date:**6.7.2017, 15:00 Uhr

**Room:**AE02

**Abstract:**

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.

#### Seminar Angewandte Analysis und Numerische Mathematik

**Title:**Self-adjoint operators of the type div sgn grad

**Speaker:**Dr. Konstantin Pankrashkin (Universite Paris-Sud & ENSTA ParisTech)

**Date:**6.7.2017, 14:00 Uhr

**Room:**AE02

**Abstract:**

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).

#### Zahlentheoretisches Kolloquium

**Title:**Prescribing the binary digits of squarefree numbers and quadratic residues

**Speaker:**Rainer Dietmann (Royal Holloway, London)

**Date:**3.7.2017, 15 ct

**Room:**Seminarraum Analysis und Zahlentheorie, (NT02008), Kopernikusgasse 24/II

**Abstract:**

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.

#### Strukturtheorie-Seminar

**Title:**ON A QUESTION OF YU. MUKHIN

**Speaker:**Prof. Wolfgang Herfort (TU Wien)

**Date:**Monday, 3.7.2017, 14 s.t. (!!!)

**Room:**Seminarraum Analysis und Zahlentheorie, (NT02008), Kopernikusgasse 24/II

**Abstract:**

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.

#### Zahlentheoretisches Kolloquium

**Title:**Smooth numbers with digital restrictions

**Speaker:**Dr. Walid Wannes (TU Graz)

**Date:**Freitag, 30. 6. 2017, 15:00

**Room:**Seminarraum Analysis-Zahlentheorie (NT02008), Kopernikusgasse 24/II

**Abstract:**

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 $m \geq2$.

#### Zahlentheoretisches KolloquiumACHTUNG - Zeit und Ort des Vortrages haben sich geändert!

**Title:**An invitation to spectral spaces

**Speaker:**Dr. Carmelo Finocchiaro (Univ. Roma Tre)

**Date:**Freitag, 30. 6. 2017, 14:00 s.t.

**Room:**Seminarraum Analysis-Zahlentheorie (NT02008), Kopernikusgasse 24/II

**Abstract:**

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.

Math. Soc.

**365**2013, no. 12, 6199--6216.

\end{thebibliography}

#### Vortrag im Seminar für Kombinatorik und Optimierung

**Title:**Supersaturation Problem for the Bowtie

**Speaker:**Tamás Makai (TU Graz)

**Date:**Dienstag 27.6.2017, 14:15

**Room:**Seminarraum AE06, Steyrergasse 30, Erdgeschoss

**Abstract:**

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.