### Talks in 2018

#### Mini-Series on Harmonic Analysis and Discrete Geometry. Lecture 2.

**Title:**Energy minimization and discrepancy on the sphere

**Speaker:**Dmitriy Bilyk (University of Minnesota)

**Date:**18.10.2018, 14.00-15.30

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

**Abstract:**

Two most standard ways to measure the quality of a point set on the sphere are discrepancy and energy. In the former, one compares the proportion of points in certain subdomains to their area, while in the latter one views points as electrons that repel according to a certain force. We shall talk about various energy minimization problems on the sphere, when minimal energy induces uniform distribution, how the structure of the function affects minimizers, special point sets that arise as minimizers (tight frames, spherical designs), connections to spherical harmonics, Gegenbauer polynomials, and positive definite functions etc. Then we shall discus discrepancy on the sphere: known bounds, methods, constructions, as well as relations to energy.

This is the second lecture in a mini-series of four lectures. All lectures are independent and fully self-contained. No prior knowledge of any of the subject involved in the talk will be assumed.

#### Vortrag

**Title:**On the Model Selection Properties and Geometry of the Lasso

**Speaker:**Ulrike Schneider (Institut für Stochastik und Wirtschaftsmathematik, TU Wien)

**Date:**18.10.2018, 17.00 Uhr

**Room:**SR für Statistik (NT03098), Kopernikusgasse 24/III, 8010 Graz

**Abstract:**

We investigate the model selection properties of the Lasso estimator in finite samples with no conditions on the regressor matrix X. We show that which covariates the Lasso estimator may potentially choose in high dimensions (where the number of explanatory variables p exceeds sample size n) depends only on X and the given penalization weights. This set of potential covariates can be determined through a geometric condition on X and may be small enough (less than or equal to n in cardinality) so that the Lasso estimator acts as a low-dimensional procedure also in high dimensions. Related to the geometric conditions in our considerations, we also provide a necessary and sufficient condition for uniqueness of the Lasso solutions.

#### Mini-Series on Harmonic Analysis and Discrete Geometry. Lecture 1.

**Title:**Star-discrepancy, Haar functions, and the small ball inequality

**Speaker:**Dmitriy Bilyk (University of Minnesota)

**Date:**16.10.2018, 15.00-16.30

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

**Abstract:**

The discrepancy function is one of the most standard tools designed to measure equidistribution of points in the unit square: it compares the actual vs. expected number of points in axis-parallel rectangles. Various norms of this function provide different information about the uniformity of a point distribution. Lower bounds for the discrepancy, which states that, no matter how one distributes points, their discrepancy cannot be too small, have come to be known collectively as "irregularities of distribution". The methods of proof for such bounds most of the times arrive from harmonic analysis (e.g. Haar wavelets with product structure) and have been actively employed in the past by Roth, Schmidt, Halasz, Beck etc. We shall discuss these methods, known results and techniques, as well as connections of the problem to questions of harmonic analysis, probability, and approximation theory, which have been discovered more recently.

This is the first lecture in a mini-series of four lectures. All lectures are independent and fully self-contained. No prior knowledge of any of the subject involved in the talk will be assumed.

#### Mathematisches Kolloquium

**Title:**Polynomials, rank and cap sets

**Speaker:**P\'{e}ter P\'{a}l Pach (Budapest University of Technology and Economics )

**Date:**22.10.2018, 16.15

**Room:**Hörsaal B (Kopernikusgasse 24, 3. Obergescho\ss)

**Abstract:**

Abstract:

In this talk we will look at a new variant of the polynomial method which

was first used to prove that sets avoiding 3-term arithmetic progressions in groups like $\mathbb{Z}_4^n$ (Croot, Lev and myself) and $\mathbb{Z}_3^n$ (Ellenberg and Gijswijt) are exponentially small (compared to the size of the group).

We will discuss lower and upper bounds for the size of the extremal subsets,

including some recent bounds found by Elsholtz and myself. We will also mention some further applications of the method, for instance, the solution of the Erd\H{o}s-Szemer\'edi sunflower conjecture.

---------

Ab 15.45 Saft, Kekse und Caf\'{e} im Sozialraum des Instituts für Analysis und Zahlentheorie, Kopernikusgasse 24, 2. Stock.

#### Zahlentheoretisches Kolloquium

**Title:**The uniqueness of the extension of infinite two-parameter family of Diophantine triples

**Speaker:**Dr. Alan Filipin (University of Zagreb)

**Date:**Freitag, 21. 9. 2018, 14:00 Uhr

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

**Abstract:**

Abstract: A set of $m$ positive integers is called a Diophantine

$m$-tuple if the product of any two elements in the set increased by 1

is a perfect square. One of the question of interest is how large those

sets can be. Very recently He, Togb\' e and Ziegler proved the folklore

conjecture that there does not exist a Diophantine quintuple.

There is also stronger version of that conjecture which states that

every Diophantine triple can be extended to a quadruple, with a larger

element, in a unique way. That conjecture is still open. In this talk we

study the two families of Diophantine pairs and consider their

extension. More precisely we prove the mentioned conjecture for the

triples $\{a, b, c\}$, where $a$ and $b$ are positive integers defined

by $a=KA^2$, $b=4KA^4+4\varepsilon A$ with $K, A$ positive integers and

$\varepsilon \in \{\pm1\}$ and $c$ is given by $c=c_{\nu}^{\tau}$, where

\begin{align*}

c_{\nu}^{\tau}=\frac{1}{4ab}\left\{(\sqrt{b}+\tau

\sqrt{a})^2(r+\sqrt{ab})^{2\nu}+(\sqrt{b}-\tau

\sqrt{a})^2(r-\sqrt{ab})^{2\nu}-2(a+b)\right\}

\end{align*}

with $\nu$ a positive integer and $\tau \in \{\pm\}$.

This is joint work with Mihai Cipu and Yasutsugu Fujita.

#### Zahlentheoretisches Kolloquium

**Title:**The uniqueness of the extension of infinite two-parameter family of Diophantine triples

**Speaker:**Dr. Alan Filipin (University of Zagreb)

**Date:**Freitag, 21. 9. 2018, 14:00 Uhr

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

**Abstract:**

Abstract: A set of $m$ positive integers is called a Diophantine

$m$-tuple if the product of any two elements in the set increased by 1

is a perfect square. One of the question of interest is how large those

sets can be. Very recently He, Togb\' e and Ziegler proved the folklore

conjecture that there does not exist a Diophantine quintuple.

There is also stronger version of that conjecture which states that

every Diophantine triple can be extended to a quadruple, with a larger

element, in a unique way. That conjecture is still open. In this talk we

study the two families of Diophantine pairs and consider their

extension. More precisely we prove the mentioned conjecture for the

triples $\{a, b, c\}$, where $a$ and $b$ are positive integers defined

by $a=KA^2$, $b=4KA^4+4\varepsilon A$ with $K, A$ positive integers and

$\varepsilon \in \{\pm1\}$ and $c$ is given by $c=c_{\nu}^{\tau}$, where

\begin{align*}

c_{\nu}^{\tau}=\frac{1}{4ab}\left\{(\sqrt{b}+\tau

\sqrt{a})^2(r+\sqrt{ab})^{2\nu}+(\sqrt{b}-\tau

\sqrt{a})^2(r-\sqrt{ab})^{2\nu}-2(a+b)\right\}

\end{align*}

with $\nu$ a positive integer and $\tau \in \{\pm\}$.

This is joint work with Mihai Cipu and Yasutsugu Fujita.

#### Number Theory Mini-Colloquium

**Title:**On Goormaghtigh's equation

**Speaker:**Dijana Kreso (TU Graz)

**Date:**14.9.2018, 11.00

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

**Abstract:**

In my talk I will present results that come from a joint work with Michael Bennett and Adela Gherga from the University of British Columbia. We studied Goormaghtigh's equation:

\begin{equation}\label{eq}

\frac{x^m-1}{x-1} = \frac{y^n-1}{y-1}, \; \; y>x>1, \; m > n > 2.

\end{equation}

There are two known solutions $(x, y,m, n)=(2, 5, 5, 3), (2, 90, 13, 3)$ and it is believed that these are the only solutions. It is not known if this equation has finitely or infinitely many solutions, and not even if that is the case if we fix one of the variables. It is known that there are finitely many solutions if we fix any two variables. Moreover, there are effective results in all cases, except when the two fixed variables are the exponents $m$ and $n$. If the fixed $m$ and $n$ additionally satisfy $\gcd(m-1, n-1)>1$, then there is an effective finiteness result. My co-authors and me showed that if $n \geq 3$ is a fixed integer, then there exists an effectively computable constant $c (n)$ such that $\max \{ x, y, m \} < c (n)$ for all $x, y$ and $m$ that satisfy Goormaghtigh's equation with $\gcd(m-1,n-1)>1$. In case $n \in \{ 3, 4, 5 \}$, we solved the equation completely, subject to this non-coprimality condition.

\vspace{1cm}Remark:} After the talk a light lunch will be served in the social room of the institute.

#### Number Theory Mini-Colloquium

**Title:**Asymptotic distribution of Beurling integers

**Speaker:**Laima Kaziulyte (TU Graz)

**Date:**14.9.2018, 10.00

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

**Abstract:**

We study generalised prime systems $\mathcal{P}$ and generalised integer systems $\mathcal{N}$ obtained from them. The asymptotic distribution of generalised integers is deduced assuming that the generalised prime counting function $\pi_\mathcal{P}(x)$ behaves as $\pi_\mathcal{P}(x)=b ~\textup{li}(x)+O(x^\alpha)$ for some $b>0$ and $\alpha\in(0,1)$.

Remark:} Laima Kaziulyte will work as a guest researcher at the Institute of Analysis and Number Theory from September 2018 -- January 2019. She is a PhD student at Vilnius University.

#### Zahlentheoretisches Kolloquium

**Title:**Average bounds for l-torsion in class groups

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

**Date:**Freitag, 13. 7. 2018, 14:15 Uhr

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

**Abstract:**

Abstract: Let l be a positive integer. We discuss improved average

bounds for the l-torsion of the class groups for some families of number

fields, including degree-d-fields for d between 2 and 5. The

improvements are based on refinements of a technique due to Ellenberg

and Venkatesh. This is joint work with Martin Widmer.

#### Vortrag

**Title:**How Do I Analyze Shapes in My Data? A Review of Some Recent Examples

**Speaker:**Wolfgang JANK (Information Systems and Decision Sciences Department, Muma College of Business/University of South Florida)

**Date:**Donnerstag, 12.07.2018, 15:00 Uhr

**Room:**SR für Statistik (NT03098), Kopernikusgasse 24/III, 8010 Graz

**Abstract:**

Today’s data are increasingly complex and dynamic. Online transactions, financial markets or social media produce information that is highly interconnected and changes instantaneously, resulting in data shapes such as spikes and troughs, or lines or curves, all of which capture the speed – or the dynamics – at which the information is changing. Functional shape analysis takes a collection of curves and attempts to extract common features among the curves with the goal of better explaining – and eventually predicting – the dynamics captured inside the data. In this presentation, we will give an overview of some recent opportunities and challenges of shape analysis. Understanding dynamics has become particularly important in online markets where prices, bids or opinions change continuously. We will provide an overview over techniques that allow us to capture, characterize, segment and predict shapes of online markets. Being able to predict shapes is particularly important in applications where market penetration or diffusion curves are desired and where restrictive parametric approaches are too inaccurate. We will illustrate these techniques on several examples from virtual stock markets, crowd-funding and box office results.

#### Vortrag im Seminar für Kombinatorik und Optimierung

**Title:**Cohomology groups of random simplicial complexes

**Speaker:**Philipp Sprüssel (Technische Universität Graz)

**Date:**Dienstag 10.7.2018, 14:15

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

**Abstract:**

Given a dimension $k \ge 2$ and a probability $p$, define the binomial random $k$-dimensional simplicial complex $\mathcal{G}_p$ as the downward-closure of the binomial random $(k+1)$-uniform hypergraph, in which each hyperedge is present with probability $p$ independently. For each $j \le k$, call a $k$-dimensional simplicial complex $\mathbb{F}_2$-cohomologically $j$-connected} if all cohomology groups for dimensions 1 up to $j$ with coefficients in the two-element field $\mathbb{F}_2$ vanish, and if furthermore the zero-th cohomology group is isomorphic to $\mathbb{F}_2$.

For each $j \le k$, we prove the existence of a sharp threshold for $\mathbb{F}_2$-cohomological $j$-connectedness of $\mathcal{G}_p$. A similar result has been proved for a different model of random complexes by Linial and Meshulam (2006) and by Meshulam and Wallach (2009). In addition, we prove a hitting time result, relating $\mathbb{F}_2$-cohomological $j$-connectedness with the disappearance of the last minimal obstruction. As a corollary, we deduce an analogous hitting time result for the Linial-Meshulam model, a result which has previously only been known for $k = 2$. Finally, we determine the limiting probability for $\mathbb{F}_2$-cohomological $j$-connectedness when $p$ lies in the critical window around the threshold.

In this talk, we focus on the main intuition and proof ideas behind these results.

Joint work with Oliver Cooley, Nicola del Guidice, and Mihyun Kang.

#### Vortrag im Seminar für Kombinatorik und Optimierung

**Title:**Graph limits of random unlabelled k-trees

**Speaker:**Yu Jin (Technische Universität Wien)

**Date:**Dienstag 3.7.2018, 14:15

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

**Abstract:**

We study random unlabelled $k$-trees by combining the colouring approach by Gainer-Dewar and Gessel (2014) with the cycle pointing method by Bodirsky, Fusy, Kang and Vigerske (2011). Our main applications are Gromov--Hausdorff--Prokhorov and Benjamini--Schramm limits that describe their asymptotic geometric shape on a global and local scale as the number of hedra tends to infinity.

This is the joint work with Benedikt Stufler.

#### FWF START Seminar

**Title:**A survey on the evaluation of the values of Dirichlet $L$-functions and of their logarithmic derivatives at $1+it_0$

**Speaker:**Sumaia Saad Eddin (JKU Linz)

**Date:**28.6.2018, 14:00

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

**Abstract:**

Let $q$ be a positive integer $q>1$, and let $\chi$ be a Dirichlet character modulo $q$. Let $L(s, \chi)$ be the attached Dirichlet $L$-functions,

and let $L^\prime(s, \chi)$ denote its derivative with respect to the complex variable $s$. In this talk, we survey certain known results on the evaluation of values of Dirichlet $L$-functions and of their logarithmic derivatives at $1+it_0$ for fixed real number $t_0$.

We also give a new asymptotic formula for the $2k$-th power mean value of $\left|(L^\prime/L)(1+it_0, \chi)\right|$ when $\chi$ runs over all Dirichlet characters modulo $q>1$, for any fixed real number $t_0$. This is joint work with professor Kohji Matsumoto.

#### Vortrag im Seminar für Kombinatorik und Optimierung

**Title:**The genus of the Erd\H{o}s-R\'enyi random graph and the fragile genus property

**Speaker:**Chris Dowden (Technische Universität Graz)

**Date:**Dienstag 19.6.2018, 14:15

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

**Abstract:**

We investigate the genus of the Erd\H{o}s-R\'enyi random graph $G(n,m)$, providing a thorough description of how this relates to the function $m=m(n)$, and finding that there is different behaviour depending on which region $m$ falls into.

We then also show that the genus of a fixed graph can increase dramatically if a small number of random edges are added - we thus call this the ``fragile genus'' property.

This is joint work with Mihyun Kang and Michael Krivelevich.

#### Gastvortrag

**Title:**Computing indecomposable decompositions of persistence modules

**Speaker:**Emerson Escolar (RIKEN (Japan))

**Date:**21.6.2018, 9:00

**Room:**Seminarraum 2, Geometrie

**Abstract:**

The indecomposable decompositions of persistence modules play an important role in understanding their structure. Especially in the case of persistence of a filtration, the indecomposables (the building blocks) are intervals, and have the interpretation as lifespans of topological features. We discuss the computation of indecomposable decompositions of some more general persistence modules. In particular, we focus on the 2 by n commutative grid (called commutative ladders) and provide an algorithm for the representation finite case ($n<5$) via matrix problems, from which persistence diagrams can be obtained. We also describe some problems and complications in the representation infinite case.

Short bio:

Emerson is currently a postdoctoral researcher in the Topological Data Analysis Team of the Center for Advanced Intelligence Project at RIKEN (Japan). Research interests include topological data analysis, representation theory, and computation. Website:

https://emerson-escolar.github.io/index.html

#### Geometrieseminar

**Title:**Discussions on Discrete Stratified Morse Theory

**Speaker:**Bei Wang (University of Utah)

**Date:**19.6.2018, 8:30

**Room:**Seminarraum 2, Geometrie

**Abstract:**

Inspired by the works of Forman on discrete Morse theory, we give a combinatorial adaptation of stratified Morse theory of Goresky and MacPherson, referred to as the discrete Stratified Morse theory. We relate the topology of a finite simplicial complex with the critical simplices of a discrete stratified Morse function on the complex. We provide an algorithm that constructs such a function on any finite simplicial complex equipped with a real-valued function. We also discuss on-going work that study the relation to classical stratified Morse theory. This is joint work with Kevin Knudson (Univ. Florida).

#### ACHTUNG - VORTRAG ABGESAGT! Zahlentheoretisches Kolloquium

**Title:**On Goormaghtigh's equation

**Speaker:**Dr. Dijana Kreso (University of British Columbia)

**Date:**Freitag, 15. 6. 2018, 14:00

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

**Abstract:**

Abstract:

In my talk I will discuss results coming from a joint work with Mike

Bennett and Adela Gherga from the University of British Columbia. We show

that if $n \geq 3$ is a fixed integer, then there exists an effectively

computable constant $c (n)$ such that if $x, y$ and $m$ are integers

satisfying \[ \frac{x^m-1}{x-1} = \frac{y^n-1}{y-1}, \; \; y>x>1, \; m > n,

\] with $\gcd(m-1,n-1)>1$, then $\max \{ x, y, m \} < c (n)$. Prior to our

work, to obtain finiteness results for this equation, two of the variables

$x, y, m$ and $n$ had to be fixed. Furthermore, in case $n \in \{ 3, 4, 5

\}$, we completely solve this equation subject to the same constraint on

the exponents. This extends the result of Yuan who completely solved the

equation in the case $n=3$ and $m$ is odd.

#### Seminar Angewandte Analysis und Numerische Mathematik

**Title:**Trace formulas and $\zeta$-functions for differential operators $-$ or, a spectral theorist computes $\pi$

**Speaker:**Prof. Dr. Fritz Gesztesy (Baylor University, Waco, USA)

**Date:**14.6.2018, 14:15 Uhr

**Room:**Seminarraum AE 02

**Abstract:**

We discuss an effective method of computing traces, determinants, and $\zeta$-functions for some classes of linear operators and apply this to the concrete case of Sturm-Liouville operators.

To illustrate the formalism, we will sketch a spectral theorist's computation of $\pi$, Jacobi's classical transformation formula for one-dimensional theta functions (utilizing the heat equation on the circle), and sketch a derivation of a formula for Ap\'ery's constant, $\zeta(3)$, employing a trace formula.

The talk will minimize technicalities and be accessible to students.

This is based in part on recent joint work with Klaus Kirsten, Lance Littlejohn, and Hagop Tossounian.

#### Strukturtheorie-Seminar / Bachelorpräsentationen

**Title:**Das Cutoff-Phänomen für Markovketten

**Speaker:**Lorenz Frühwirth und Tristan Repolusk (TU Graz)

**Date:**Donnerstag, 7. Juni 11:00 s.t.

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

**Abstract:**

Eine ergodische Markovkette hat 'cutoff', wenn die Aproximation der stationären Verteilung innerhalb eines kurzen Zeitfensters von 'schlecht' auf 'gut' übergeht. Dies geht auf Ideen von Persi Diaconis zurück.

In den beiden Kurzvorträgen werden das Cutoff-Phänomen erklärt und Beispielklassen vorgestellt.

#### Mathematisches Kolloquium / Strukturtheorie-Seminar

**Title:**Restricted Lattice Walks

**Speaker:**Prof. Dr. Manuel KAUERS (Universität Linz)

**Date:**Montag, 4.6.2006, 14:00 s.t.

**Room:**Seminarraum A306, Steyrergasse 30, 3. Stock

**Abstract:**

Manuel Kauers is professor for algebra at Johannes Kepler University. He is working on computer algebra and its applications to discrete mathematics. Together with Doron Zeilberger and Christoph Koutschan, Kauers proved two famous open conjectures in combinatorics using large scale computer algebra calculations. The first concerned Gessel's lattice path conjecture, the second was the so-called q-TSPP conjecture. In 2009, Kauers received the Austrian Start-proze, and in 2016, with Ch. Koutschan and D. Zeilberger he received the David P. Robbins prize of the American Mathematical Society.

Abstract of the talk: We give an overview of some recent developments in the

area of counting restricted lattice walks. This his a hot topic in the combinatorics community which has attracted the interest of a lot of people. We will limit ourselves to aspects in which computer algebra has helped making some progress.

#### FWF START Seminar

**Title:**Some complexity results in the theory of normal numbers

**Speaker:**William Mance (Adam Mickiewicz University Poznan)

**Date:**4.5.2018, 11:00

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

**Abstract:**

Let $N(b)$ be the set of real numbers which are normal to base $b$. A well-known result of H. Ki and T. Linton is that $N(b)$ is $\boldsymbol{\Pi}^0_3$-complete. We show that the set $N^\perp(b)$ of reals which preserve $N(b)$ under addition is also $\boldsymbol{\Pi}^0_3$-complete. We use the characterization of $N^\perp(b)$ given by G. Rauzy in terms of an entropy-like quantity called the noise}. It follows from our results that no further characterization theorems could result in a still better bound on the complexity of $N^\perp(b)$. We compute the exact descriptive complexity of other naturally occurring sets associated with noise. One of these is complete at the $\boldsymbol{\Pi}^0_4$ level. Finally, we get upper and lower bounds on the Hausdorff dimension of the level sets associated with the noise.

#### Kolloquium

**Title:**Festkolloquium

**Speaker:**()

**Date:**4.5.2018

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

**Abstract:**

13:30-13:40 Eröffnung

13:40-14:00 E. STADLOBER: Erinnerungen an Ulrich Dieter

14:00-14:20 S. HÖRMANN: Zur Prognose von funktionalen Zeitreihen

14:20-14:40 S. THONHAUSER: Über die optimale Wahl von Rückversicherung

14:40-15:30 Kaffeepause

15:30-16:00 N. HASELGRUBER: Strategien der Stichprobenplanung zur Verifikation von Zuverlässigkeitszielen

16:00-16:30 Th. MENDLIK: Warum man den Klimawandel nicht genau vorhersagen kann, uns aber die Statistik dennoch hilft

16:30-17:00 M. PIFFL: Modell-Basierte Flotten Validierung

17:00-17:30 P. SCHEIBELHOFER: Statistische Methoden für die fortgeschrittene Prozesskontrolle in der Halbleiterfertigung

#### Seminar Angewandte Analysis und Numerische Mathematik

**Title:**The Inverse Scattering Problem for Singular Perturbations

**Speaker:**Prof. Dr. Andrea Mantile (Universite de Reims)

**Date:**3.5.2018, 15:00 Uhr

**Room:**AE 02

**Abstract:**

We consider the inverse scattering problem of determining a singular perturbation supported on a bounded and closed surface in 3D from the knowledge of the scattering data. Different approaches to this problem and possibly some recent result will be presented.

This talk is based on joint work-in-progress with A. Posilicano.

#### Seminar Angewandte Analysis und Numerische Mathematik

**Title:**Scattering theory for Singular Perturbations

**Speaker:**Prof. Dr. Andrea Posilicano (University of Insubria)

**Date:**3.5.2018, 14:00 Uhr

**Room:**AE 02

**Abstract:**

We consider the direct scattering problem and provide a corresponding representation of the scattering matrix for the couple $(A_{0},A)$, where $A_{0}$ and $A$ are semi-bounded self-adjoint operators in $L^{2}(M,{\mathscr B},m)$ such that the set $\{u\in D(A_{0})\cap D(A):A_{0}u=Au\}$ is dense. Applications to the case in which $A_{0}$ corresponds to the free Laplacian in $L^{2}({\mathbb R}^{n})$ and $A$ describes the Laplacian with self-adjoint boundary conditions on rough compact hypersurfaces are given.

The talk is based on joint work with Andrea Mantile.

#### Seminar Angewandte Analysis und Numerische Mathematik

**Title:**Dirichlet (magnetic) Laplacians on radially symmetric unbounded layers

**Speaker:**Dr. Vladimir Lotoreichik (Nuclear Physics Institute, Rez, Czech Republic)

**Date:**19.4.2018, 15:00 Uhr

**Room:**AE 02

**Abstract:**

The radially symmetric unbounded surface $\Sigma\subset\mathbb{R}^3$ is defined via the mapping $\mathbb{R}^2\ni x\mapsto (x,f(|x|))$, where a sufficiently smooth profile function $f\colon \mathbb{R}_+\rightarrow \mathbb{R}_+$ satisfies $f(0) = 0$. The radially symmetric layer $\Omega := \{x\in\mathbb{R}^3\colon {\rm dist}\,(x,\Sigma) < a\} \subset\mathbb{R}^3$ with $a > 0$ can be viewed as a tubular neighbourhood of $\Sigma$. Under rather non-restrictive extra assumptions on $f$, it was proven by Duclos, Exner, and Krej\v{c}i\v{r}\'{i}k in 2001 that the essential spectrum of the self-adjoint Dirichlet Laplacian on $\Omega$ coincides with $[\frac{\pi^2}{4a^2},\infty)$ and that its discrete spectrum below $\frac{\pi^2}{4a^2}$ is infinite.

First, we will re-visit the case of the conical layer ($f(t) = kt$, $k >0$).

Spectral asymptotics for such a layer was obtained by Dauge, Ourmi\`{e}res-Bonafos, and Raymond in 2015. We will discuss what happens with the spectrum, upon replacement the usual Laplacian by its magnetic counterpart with a magnetic field of an Aharonov-Bohm-type. It turns out that there are two regimes, the transition between which is abrupt. The discrete spectrum remains infinite for the weak magnetic field, but its flux enters the leading term in the spectral asymptotics. In the strong field regime, the discrete spectrum completely disappears.

Second, we will consider generalized parabolic layers

($f(t) = kt^{1+\alpha}$, $\alpha, k > 0$).

In this setting, we, for the first time, compute the spectral asymptotics.

The power $\alpha$ influences the law of accumulation for the eigenvalues. In particular, for the conventional parabolic layer ($\alpha = 1$) the spectral asymptotics is similar to the one for the Hamiltonian of the hydrogen atom.

These results are obtained in collaborations with Pavel Exner,

David Krej\v{c}i\v{r}\'{i}k, Thomas Ourmi\`{e}res-Bonafos.

#### Seminar Angewandte Analysis und Numerische Mathematik

**Title:**Spectral estimates for Dirichlet Laplacians on twisted tubes

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

**Date:**19.4.2018, 14:00 Uhr

**Room:**AE 02

**Abstract:**

In the first part of the talk we investigate the Dirichlet Laplacian in a straight twisted tube of a non-circular cross section with a local perturbation of the twisting velocity. It is known that the essential spectrum covers the half-line. Under some additional assumptions on the twisting velocity perturbation the non-emptiness of the discrete spectrum is also guaranteed (P. Exner and H. Kovarik; 2005). In the second part of the talk we study the Dirichlet Laplacian on a straight tube but already with exploding to infinity twisting velocity. Then the spectrum is becoming purely discrete (D. Krejcirik; 2015). In both cases we investigate the upper bounds for eigenvalue moments.

#### Gastvortrag

**Title:**Multi-Parameter Persistence

**Speaker:**Sara Scaramuccia (Univ. Genova)

**Date:**Dienstag, 17.4.2018, 8:30

**Room:**Seminarraum 2 Geometrie, Kopernikusgasse 24

**Abstract:**

The basic goal of topological data analysis is to apply algebraic topology

tools to understand and describe the shape of data. In this context,

homology is one of the most relevant topological descriptors. Persistent

Homology (PH) tracks homological features along an increasing one-parameter

sequence of spaces and its descriptors are well-appreciated for comparing

and recognizing changes in the shape of data.

The focus of this talk is on a generalization of Persistence Theory to the

analysis of multivariate data, called Multiparameter Persistent Homology

(MPH).

At the moment, MPH requires both computational optimizations towards the

applications to real-world data, and theoretical insights for finding and

interpreting suitable descriptors. We address the problem of reducing

computational costs in MPH by proposing a fully discrete preprocessing

algorithm. In doing so, we propose a new notion of optimal preprocessing

algorithm for MPH and show that our proposed algorithm satisfies it for low

dimensional domains. We provide experimental results in comparing our

approach to other similar approaches, and in evaluating the impact of our

approach on MPH computations.

#### Zahlentheoretisches Kolloquium

**Title:**Reducing integer factorization to modular tetration

**Speaker:**Markus Hittmeir (Universität Salzburg)

**Date:**Freitag, 13. 4. 2018, 14:00 Uhr

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

**Abstract:**

Abstract.}

Let $a,k\in\mathbb{N}$. For the $k-1$-th iterate of the exponential function $x\mapsto a^x$, also known as tetration, we write

\[

^k a:=a^{a^{.^{.^{.^{a}}}}}.

\]

In this talk, we show how an efficient algorithm for tetration modulo natural numbers $N$ may be used to factorize $N$. In particular, we prove that the problem of computing the squarefree part of integers is deterministically polynomial-time reducible to modular tetration.

#### Zahlentheoretisches Kolloquium

**Title:**Elliptic curves and rational sequences

**Speaker:**Márton Szikszai (University of Debrecen, dzt. TU Graz)

**Date:**Donnerstag, 22. 3. 2018, 12:00

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

**Abstract:**

Abstract:

In this talk, we explore the connection between the ranks of elliptic curves and the length of certain rational sequences contained in the coordinates of rational points. The structure of the presentation is the following. First, we give a lazy survey of rank records and the methods used in obtaining them, especially the algebraic approach of Mestre. In the middle part, we talk about the past and recent progress in the construction of elliptic curves admitting long arithmetic and geometric progressions. Finally, we present some related new results as part of my research project at TU Graz.

#### Seminar Applied Analysis and Computational Mathematics

**Title:**Spectral gaps and discrete magnetic Laplacians

**Speaker:**Prof. Dr. Olaf Post (Universität Trier)

**Date:**21.3.2018, 16:15

**Room:**A306

**Abstract:**

The aim of this talk is to give a simple geometric condition

that guarantees the existence of spectral gaps of the discrete Laplacian

on periodic graphs. For proving this, we analyse the discrete magnetic

Laplacian on the finite quotient and interpret the vector potential as a

Floquet parameter. We develop a procedure of virtualising edges and

vertices that produces matrices whose eigenvalues (written in ascending

order and counting multiplicities) specify the bracketing intervals

where the spectrum of the Laplacian is localised.

#### FWF START Seminar

**Title:**From Interval Exchange Transformations to low-discrepancy

**Speaker:**Christian Weiß (Hochschule Ruhr West)

**Date:**20.3.2018, 10.30

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

**Abstract:**

Kronecker sequences build an important class of one dimensional low-discrepancy sequences. They can also be realized as orbits of circle rotations, which are in in turn the simplest class of interval exchange transformations (IETs). Hence, it is natural to ask whether there exist more general IETs which also yield low-discrepancy sequences. In the case of n=3 intervals, the criteria of circle rotation can easily be carried over. For more than three intervals the dynamic of IETs is much more complex. In this talk, it is shown how to obtain low-discprancy sequences involving certain IETs with an arbitrary number of intervals.

#### KOLLOQUIUMSVORTRAG im Vorfeld eines Habilitationsantrags

**Title:**Spectral and asymptotic properties of periodic media

**Speaker:**Andrii KHRABUSTOVSKYI (Institut für Angewandte Mathematik)

**Date:**Donnerstag, 15.3.2018, 14:00 Uhr

**Room:**TU Graz, Seminarraum AE02, Steyrergasse 30, EG

**Abstract:**

It is well-known that the spectrum of self-adjoint periodic differential operators has the form of a locally finite union of compact intervals called bands. In general the bands may overlap. A bounded open interval (a,b) is called a gap in the spectrum σ(H) of the operator H if (a,b)∩σ(H)=Ø with a,b belonging to σ(H).

The presence of gaps in the spectrum is not guaranteed: for example, the spectrum of the Laplacian in ℝn has no gaps. Therefore the natural problem is a construction of periodic operators with non-void spectral gaps. The importance of this problem is caused by various applications, for example in physics of photonic crystals.

Another interesting question arising in this area is how to control the location of spectral gaps via a suitable choice of coefficients of the underlying operators or/and via a suitable choice of geometric parameters of the medium. In the talk we give an overview of the results, where this problem is studied for various classes of periodic differential operators. In a nutshell, our goal is to construct an operator (from some given class of periodic operators) such that its spectral gaps are close (in some natural sense) to predefined intervals.

#### Vortrag im Seminar für Kombinatorik und Optimierung

**Title:**Lambda-terms enumeration and statistics

**Speaker:**Dani\`ele Gardy (Universit\'e de Versailles Saint-Quentin)

**Date:**Dienstag 6.3.2018, 14:15

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

**Abstract:**

Lambda-calculus is a well-known model of computation, whose statistical properties have been recently under investigation. From a syntactic point of view, the terms of lambda-calculus (lambda-terms) are interesting combinatorial structures: the underlying structure is a Motzkin tree to which are added edges from some unary nodes towards leaves, or equivalently whose nodes are colored, according to some specific set of rules. While trees are usually easy to enumerate, this enriched version is actually a class of directed graphs, and their combinatorial study gives rise to many interesting problems.

While the enumeration and statistical analysis of unrestricted lambda-terms still remains an open problem, different models, either redefining the notion of size of the lambda-term or restricting the set of lambda-terms under investigation (such as terms of restricted height or restricted node arity), have proved more amenable to study. We present enumeration and statistical behaviour for these models, using tools from analytic combinatorics.

(No previous knowledge either of lambda-terms or of analytic combinatorics is assumed.)

#### Advanced Topics in Discrete Mathematics

**Title:**Information Dimension of Random Variables and Stochastic Processes

**Speaker:**Dr. Bernhard Geiger (Signal Processing and Speech Communication Laboratory, TU Graz)

**Date:**Freitag 26.1.2018, 11:00 (Kaffee ab 10:30)

**Room:**Seminrraum 2 für Geometrie, Kopernikusgasse 24/4

**Abstract:**

In view of the interplay of the DK `Discrete Mathematics' with the Field of Expertise `Information, Communication & Computing' of TU Graz, the last talk of this semester's seminar comes from a sister institue within the FoE. There are clear relations with the topics of the DK.

Abstract. Information dimension of random variables was introduced by Alfred Renyi in 1959. Only recently, information dimension was shown to be relevant in various areas in information theory. For example, in 2010, Wu and Verdu showed that information dimension is a fundamental limit for lossless analog compression. Recently, Geiger and Koch generalized information dimension from random variables to stochastic processes. They showed connections to the rate-distortion dimension and to the bandwidth of the process. Specifically, if the process is Gaussian, then the information dimension equals the Lebesgue measure of the support of the process' power spectral density. This suggests that information dimension plays a fundamental role in sampling theory.

The first part of the talk reviews the definition and basic properties of information dimension for random variables. The second part treats the information dimension of stochastic processes and sketches the proof that information dimension is linked to the process' bandwidth.

#### Strukturtheorie-Seminar

**Title:**Schreier graphs of spinal groups

**Speaker:**Aitor Perez (Univ. Genf)

**Date:**Donnerstag, 25.1.2018, 11 Uhr c.t.

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

**Abstract:**

Spinal groups form a family of branch groups acting on d-regular rooted trees containing many interesting examples, including Grigorchuk's family of groups of intermediate growth. Because of its natural action on the tree, it is interesting to know how their Schreier graphs look like. In this talk we will define the spinal family, provide some examples, describe the Schreier graphs in general and find some of the properties they exhibit, like how many ends do they have, how do they partition into isomorphism classes and under which conditions is the action linearly repetitive or Boshernitzan, properties related to symbolic dynamics.

#### Zahlentheoretisches Kolloquium

**Title:**$f$-vectors of simplicial and simple polytopes

**Speaker:**Dr. Roswitha Rissner (TU Graz)

**Date:**Freitag, 19. 1. 2018, 14:15 Uhr

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

**Abstract:**

The $f$-vector of a $d$-dimensional (convex) polytope $P$ is defined as

\begin{equation*}

f(P) = (f_0(P), f_1(P), \ldots, f_d(P))

\end{equation*}

where $f_i(P)$ is the number of $i$-dimensional faces of $P$.

The question is whether a given vector of non-negative numbers is the $f$-vector of a polytope.

For $d\ge 4$, finding a complete characterization of $f$-vectors is an open problem.

However, the so-called $g$-Theorem gives a description for $f$-vectors of simplicial and simple polytopes.

The purpose of this talk is to give a summary of the lectures of a summer school at MSRI on the $g$-Theorem and related topics.

#### Strukturtheorie-Seminar

**Title:**Spectral estimates for infinite quantum graphs

**Speaker:**Dr. Norma Nicolussi (Univ. Wien)

**Date:**Donnerstag, 18.1.2018, 11:15

**Room:**Seminar room AE02, Steyrergasse 30, ground floor

**Abstract:**

We investigate the bottom of the spectra of infinite quantum graphs, i.e., Kirchhoff Laplacians on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Moreover, we establish a connection with the combinatorial isoperimetric constant, which enables us to prove a number of criteria for a quantum graph to be uniformly positive or to have purely discrete spectrum. If time permits, we'll demonstrate our findings by considering trees, antitrees and Cayley graphs.

#### Strukturtheorie-Seminar

**Title:**Infinite quantum graphs

**Speaker:**Prof. Aleksey Kostenko (Univ. Wien + Laibach)

**Date:**Donnerstag, 18.1.2018, 10:30

**Room:**Seminar room AE02, Steyrergasse 30, ground floor

**Abstract:**

The notion of quantum graph refers to a graph considered as a one-dimensional simplicial complex and equipped with a differential operator (``Hamiltonian'').

From the mathematical point of view, quantum graphs are interesting because they are a good model to study properties of quantum systems depending on geometry and topology of the configuration space. They exhibit a mixed dimensionality being locally one-dimensional but globally multi-dimensional of many different types.

We will review the basic spectral properties of infinite quantum graphs (graphs having infinitely many vertices and edges). In particular, we will discuss recently discovered, fruitful connections between quantum graphs and discrete Laplacians on graphs.

Based on a joint work with P. Exner, M. Malamud and H. Neidhardt