Discrete Mathematics 2010-2022

The notion of density of a subgraph is usually defined as the number of edges in the subgraph divided by the
number of its vertices. However, this definition is too permissive for the characterization of communities in graphs
where each vertex should be densely connected within the community, with regard to its degree. Motivated by the idea of
defining a notion of density more suitable for the study and identification of communities in graphs, we will be
interested in the property of ``proportional density''. Specifically, we define a ``proportionally dense subgraph'' (PDS) as
an induced subgraph such that each vertex in the PDS has proportionally more neighbors inside the PDS than in the whole
graph. We will focus on the problem of finding a PDS of maximum size. Some hardness results on restricted classes of
graphs will be presented (NP-hardness and APX-hardness), as well as a very simple polynomial-time 2-approximation
algorithm. In the last part of the talk, we will give an upper bound on the size of a PDS in a graph based on its
maximum degree and prove that all Hamiltonian cubic graphs have a PDS that reaches this bound.

The approximation of probability measures by atomic measures (measures supported on a finite set of points) is a classical task in approximation theory with a wide range of applications. Here, we replace atomic measures by measures supported on curves of finite length. For discrepancy as distance between measures, we prove optimal approximation rates in terms of the curve's length. We also develop numerical schemes that achieve the optimal rates, and we provide several numerical examples.

celebrating the beginning of the scientific
activities of the 3rd official funding phase
(01/2019 - 12/2022) of the doctoral program.

10:00 Opening

10:10 Talk by Prof. Klavdija Kutnar
(Rector, University of Primorska, Slovenia):
Lovasz's Hamiltonicity Problem

11:00 Musical interlude by Julian Zalla (e-piano)

11:15 Talk by Mahadi Ddamulira (DK-Project 09):
Repdigits as sums of three Padovan numbers

11:45-13:15 Lunch break

13:15 Talk by Irene Parada (DK-Project 11):
Extending drawings and arrangements

13:45 Musical interlude by Julian Zalla

14:00 Talk by Prof. Stephan Wagner
(Stellenbosch University & Uppsala University):
Coefficients of graph polynomials associated
with random trees and graphs

14:50 Musical finale by Julian Zalla

Deligne, Knudsen and Mumford developed the concept of stable curve in
order to find a geometric compactification of curves of genus g with n
marked points on them. They used quite a bit of heavy machinery to carry
out their constructions and proofs. In the talk, we will restrict to
genus zero, i.e., the projective line, and propose an elementary
approach, using graph theory and the concept of phylogenetic trees. The
lecture addresses a general audience.

We consider a bilevel knapsack problem, in which one player, the follower,
decides on the optimal utilization of a bounded resource. The second player,
the leader, can offer incentives so that the follower chooses options
attractive also for the leader. Formally, each of the two players is
associated with a subset of the knapsack items. The follower selects a
subset of all items in order to maximize its overall profit. The leader
receives as pay-off only the values from those of its items that are
included by the follower in the overall knapsack solution. We consider the
case where the leader can offer part of the profit of every item to the
follower, and the case where the leader can set the weights of items for the
leader, aiming at a maximum weight of the selected leader's item.

In both cases the resulting setting is a Stackelberg strategic game. The
leader has to resolve the trade-off between offering highly attractive
incentives to the follower and thereby lowering its own pay-offs. We analyze
the problem for the case in which the follower solves the resulting knapsack
problem to optimality and obtain a number of negative complexity results.
Then we invoke a common assumption of the literature, namely that the
follower's computing power is bounded. Under this condition, we study
several natural Greedy-type heuristics applied by the follower.

Abstract:
Stochastic impulse control problems are continuous-time optimization problems in which a stochastic system is controlled through nitely many impulses causing a discontinuous displacement of the state process. The objective is to construct impulses which optimize a given performance functional of the state process. This type of optimization problem arises in many branches of applied probability and economics such as optimal portfolio management under transaction costs, optimal forest harvesting, inventory control, and valuation of real options.
In this talk, I will give an introduction to stochastic impulse control and discuss classical solution techniques. I will then introduce a new method to solve impulse control problems based on superharmonic functions and a stochastic analogue of Perron’s method, which allows to construct optimal impulse controls under a very general set of assumptions.

Finally, I will show how the general results can be applied to optimal investment problems in the presence of transaction costs.
This talk is based on joint work with Sören Christensen (Christian-Albrechts-University Kiel),Lukas Mich (Trier University), and Frank T. Seifried (Trier University).

References
C. Belak, S. Christensen, F. T. Seifried: A General Verication Result for Stochastic Impulse Control Problems. SIAM Journal on Control and Optimization, Vol. 55, No. 2, pp. 627–649, 2017.

C. Belak, S. Christensen: Utility Maximisation in a Factor Model with Constant and Proportional Transaction Costs. Finance and Stochastics, Vol. 23, No. 1, pp. 29–96, 2019.

C. Belak, L. Mich, F. T. Seifried: Optimal Investment for Retail Investors with Floored and Capped Costs. Preprint, available at http://ssrn.com/abstract=3447346, 2019.

Abstract:
We study linear subset regression in the context of a high-dimensional linear
model. Consider y = a + b'z + e with univariate response y and a d-vector of random regressors z, and a submodel where y is regressed on a set of p explanatory variables that are given by x = M'z, for some d x p matrix M.

Here, `high-dimensional' means that the number d of available explanatory variables in the overall model is much larger than the number p of variables in the submodel. In this paper, we present Pinsker-type results for prediction of y given x.

In particular, we show that the mean squared prediction error of the best linear predictor of y given x is close to the mean squared prediction error of the corresponding Bayes predictor E[y|x], provided only that p/log(d) is small.

We also show that the mean squared prediction error of the (feasible) least-squares predictor computed from n independent observations of (y,x) is close to that of the Bayes predictor, provided only that both p/log(d) and p/n are small. Our results hold uniformly in the regression parameters and over large collections of distributions for the design variables z.

By a classical theorem of Dirichlet, any real number can be approximated by rational numbers with error at most $1/q^2$, where $q$ is the maximal allowed denominator. Numbers for which this theorem cannot be improved by more than a constant factor are called badly approximable. It is not difficult to see that the set of badly approximable numbers has Lebesgue measure zero. Matters get more complicated when asking if the same holds for other natural measures, e.g. Hausdorff measures on fractal sets like the middle-third Cantor set. For (strictly) self-similar sets, this question was only recently answered in the positive by Simmons--Weiss, by reducing it to an equidistribution problem for random walks on homogeneous spaces. Using the same reduction, our work on random walks with Markovian dependence shows that badly approximable numbers are of Hausdorff measure zero also inside graph-directed self-similar sets.

Liebe Kolleginnen und Kollegen!

Am Mittwoch, dem 11. Dezember 2019, findet im Seminarraum für Analysis und Zahlentheorie ein Probevortrag von Frau Dr. Ecaterina Sava-Huss im Rahmen ihres Habilitationsverfahrens zum Thema „Markovketten und elektrische Netzwerke“ statt.

Der Vortrag ist so gestaltet, dass er auch für Studierende des 5. und 6. Semesters geeignet ist.

Ziel des Vortrages ist es, einen Zusammenhang zwischen gewissen Markovketten und elektrischen Netzwerken herzustellen, der es in manchen Fällen erlaubt, zwischen Rekurrenz und Transienz anhand von leicht berechenbaren physikalischen Größen (wie z.B effektive Leitfähigkeit und effektiver Widerstand) zu entscheiden.

Liebe Kolleginnen und Kollegen!


Am Mittwoch, dem 11. Dezember 2019 findet im Seminarraum für Analysis und Zahlentheorie ein Probevortrag von Frau Dr. Ecaterina Sava-Huss im Rahmen ihres Habilitationsverfahrens zum Thema „Markovketten und elektrische Netzwerke“ statt.


Der Vortrag ist so gestaltet, dass er auch für Studierende des 5. und 6. Semesters geeignet ist.


Ziel des Vortrages ist es, einen Zusammenhang zwischen gewissen Markovketten und elektrischen Netzwerken herzustellen, der es in manchen Fällen erlaubt, zwischen Rekurrenz und Transienz anhand von leicht berechenbaren physikalischen Größen (wie z.B effektive Leitfähigkeit und effektiver Widerstand) zu entscheiden.

Abstract:
In insurance practice, claims often occur in clusters and their arrivals may depend on various external and time-dependent factors. In this paper, we propose a statistical approach for modelling claim arrivals by considering clustered arrivals and non-stationarity simultaneously.

To this end we extend the Cox process methodology with Levy subordinators presented in Selch and Scherer [1] relaxing the stationarity of increments assumption. A particular special case of the proposed approach is a dynamic and flexible model of negative binomially distributed claim numbers with trends and seasonal variations of the parameters.

For illustration purposes we fit the model to a fire insurance portfolio, and show that it allows the modelling of cluster occurrences in a seasonal pattern while preserving overdispersion, which is frequently observed in claim count data. We illustrate its use in forecasting and Value-at-Risk and Expected Shortfall computations of the aggregate insurance risk.

Finally, we provide a multivariate extension of the model, where simultaneous cluster arrivals in different components are generated by a non-stationary common subordinator.

(Joint work with Hansjörg Albrecher (UNIL) and Jan Beirlant (KU Leuven))

[1] Selch, D. and Scherer, M. "A Multivariate Claim Count Model for Applications in Insurance." Springer, 2018.

Abstract: The optimal control of some systems of nonlinear reaction-diffusion equations is considered including several important equations of mathematical physics. In particular, equations are covered that develop traveling wave fronts, spiral waves, scroll rings, or propagating spot solutions. Well-posedness of the system, differentiability of the control-to-state mapping, and optimality conditions of first and second order are briey sketched. In particular, the case of sparse optimal control is addressed. A novel application of pointwise state constraints is presented that prevent a propagating spot from hitting the boundary of the spatial domain. Finally, the optimization of time-delays in local and nonlocal Pyragas type feedback control is briey discussed. Various numerical examples illustrate a great diversity of geometrical patterns and their control.

Vorträge im Rahmen des ÖMG-Tags der Mathematik,

Verleihung von ÖMG-Preisen,

ÖMG-Generalversammlung

plus: Mathematisches Kolloquium

A family of lines through the origin in Euclidean space $\mathbb{R}^n$ is called equiangular if any pair of lines defines the same angle. In this talk we discuss recent progress on the longstanding open problem: what is the maximum number of such lines one can have with a fixed angle theta?

Abstract: Structured additive distributional regression models allow for flexible modeling by specifying additive predictors with various types of regression effects, e.g. non-linear effects of continuous covariates, spatial effects or random effects for all parameters of the response distribution (e.g. location, scale or shape parameters). Specification of these models requires to determine not only the most appropriate subset of covariates but also their exact modeling alternative for the multiple regression predictors.

To perform general effect selection in structured additive distributional regression models, a novel spike and slab prior is proposed. Thus, effects on all distributional parameters for arbitrary parametric distributions and various effect types such as non-linear or spatial effects as well as hierarchical regression structures can be included in the model specification and important effects are automatically selected.

The specification of the prior relies on a parameter expansion that separates blocks of regression coefficients into vectors of standardized coefficients and overall scalar importance parameters. Selection of an effect is based on this scalar quantity for which a spike and slab prior with scaled beta prime marginals is specified.

In this talk I will discuss shrinkage properties, elicitation of prior parameters as well as Markov Chain Monte Carlo sampling and illustrate the usefulness of the approach for models with a potentially large number of covariates for simulated data and in a bivariate analysis of undernutrition in Nigeria

Im Rahmen des Auswahlverfahrens für die Professoren-Laufbahnstelle “Angewandte Geometrie” finden am Donnerstag den 14.11.2019 5 Kurzvorträge (20min) mit anschließender Lehr-Demonstration (20 min). statt. Weitere Vorträge gibt es am Diens\-tag und Mittwoch davor. Der Zeitplan für den 14.11 ist der folgende:

{\bf 10:00 Cesar Ceballos} (Univ. Wien): Applied Geometry - Inspiring Results in Algebra and Combinatorics

\par{\bf 11:30 Philipp Harms} (Univ. Freiburg): Riemannian Shape Analysis

\par{\bf 14:00 Felix Günther} (TU Berlin): Smooth polyhedral surfaces

\par{\bf 15:30 Zuzana Patáková} (IST Austria): Partitioning techniques in discrete and computational geometry

\par{\bf 17:00 Mickaël Buchet} (TU Graz): Declutter and resample: Towards parameter-free denoising

Biological data sets, such as gene expressions or protein levels, are
often high-dimensional and thus difficult to interpret.
Finding important structural features and identifying clusters in an
unbiased fashion is a core issue for understanding biological phenomena.
In this talk, we describe an unsupervised data analysis methodology
based on network analysis via Wasserstein optimal transport, dimension
reduction with diffusion maps, and clustering with Mapper (TDA).
Applied to gene expression profiles of the sarcomas in the Cancer Genome Atlas, we are able to recover the known subtypes. In addition, we find a
new signature, mainly described by inactivation of the tumor suppressor
genes p53 and p73, and discuss possible treatment based on its genetic
profile.

This is joint work with J. C. Mathews, M. Pouryahya, I. G. Kevrekidis,
J. O. Deasy, and A. Tannenbaum.

Im Rahmen des Auswahlverfahrens für die Professoren-Laufbahnstelle “Angewandte Geometrie” finden am Mittwoch den 13.11.2019 zwei Kurzvorträge (20min) mit anschließender Lehr-Demonstration (20 min). statt. Weitere Vorträge gibt es am Dienstag und Donnerstag. Der Zeitplan für den 13.11 ist der folgende:

{\bf 15:00 Zijia Li} (RICAM, Linz): Bond Theory and Factorization of Motion Polynomials

\par{\bf 16:30 Josef Schadlbauer} (Graz): Kinematics - a geometers paradise

Im Rahmen des Auswahlverfahrens für die Professoren-Laufbahnstelle ``Angewandte Geometrie'' finden am Dienstag den 12.11.2019 drei Kurzvorträge (20min) mit anschließender Lehr-Demonstration (20 min). statt. Weitere Vorträge gibt es am Mitt\-woch und Donnerstag. Der Zeitplan für den 12.11 ist der folgende:

{\bf 10:00 Arseniy Akopyan} (IST Austria): Circle patterns
and confocal conics

\par{\bf 11:30 Caroline Moosmüller} (Univ. California, San Diego): Manifold learning applied: How geometric
clustering identifies cancerous cells

\par{\bf 14:00 Olga Diamanti} (TU Berlin): Geometric Algorithms for Vector Field Processing and Surface Maps

In 2017 Ghosh and Sarnak predicted how often the integral Hasse principle should fail in the family of Markoff surfaces. They were able to obtain a lower bound for this quantity but its magnitude is still far away from the expectation. In this talk we will explain what can be achieved for this family using tools from Arithmetic geometry developed for studying failures of local-to-global principles. In particular, we will explain how to get sharp upper and lower bounds for the number of failures of the Hasse principle for integral points explained by the Brauer-Manin obstruction. Moreover, we will give a lower bound for the number of failures which are not explained by the Brauer-Manin obstruction.
This talk is based on a join work with Dan Loughran.

Electrical networks with resistors can be considered as weighted graphs. Similarly, infinite electrical networks with resistors can be introduced. In this case the effective resistance of a network can be defined, and it is related to Laplace operator and random walk on graphs.

A natural generalization of a network with resistors is given by an electrical network with resistors, capacitors, and inductors (so-called network with impedances). It is more convenient for us to work with the admittance (the inverse of impedance). We define the mathematical notion of effective admittance $P$ of a finite network and consider it as a rational function of $\lambda$. Although initially $\lambda = i\omega$, where $\omega$ is the frequency of an alternating current, we consider more generally $\lambda$ as taking arbitrary complex values. We present some estimates of the function $P(\lambda)$. Moreover, we discuss the possibility to define an effective admittance of an infinite electrical network. The idea is to exhaust an infinite network by a sequence of finite networks.

Cichoń's Diagram describes a partial order between 10 uncountable cardinals, among them cov(null), the smallest number of Lebesgue null sets needed to cover all real numbers, or non(meager), the smallest cardinality of a non-meager set (=set of second category), as well as aleph1 (smallest uncountable cardinal) and c (the cardinality of R, the set of all real numbers).
In 1963, Paul Cohen invented the method of "forcing" and used it to show the unprovability of Cantor's Continuum Hypothesis, or in other
words:
that the continuum does not necessarily have cardinality aleph1.
It is still open which values the other cardinal's in Cichoń's Diagram may take, but in a recent paper (with Jakob Kellner and Saharon Shelah) we could for the first time construct a set-theoretic universe in which all cardinals in Cichoń's Diagram have different values.
I will talk a bit about the cardinals in Cichoń's Diagram and hint at the methods which allow us to control/manipulate their values.

The main focus is on the relationship between graph ends of an infinite metric graph and the space of self-adjoint extensions of the corresponding minimal Kirchhoff Laplacian. First, we introduce the notion of finite volume for (topological) ends of a metric graph and then investigate their relationship
with the deficiency indices of the Kirchhoff Laplacian. Moreover, it turns out that finite volume graph ends play a crucial role in the study of Markovian extensions. In particular, in the case of finitely many finite volume ends we are even able to provide a complete description of all Markovian extensions.

Based on joint work with D. Mugnolo (Hagen) and N. Nicolussi (Wien - Potsdam)

A graph $G$ on $n$ vertices is called an $\alpha$-expander if the external neighborhood of every vertex subset $U$ of size $|U|\leq n/2$ in $G$ has size at least $\alpha |U|$.

Extending and improving the results of Plotkin, Rao and Smith, and of Kleinberg and Rubinfeld from the 90s, we prove that for every $\alpha>0$,
an $\alpha$-expander $G$ on $n$ vertices contains every graph $H$ with at most $cn/\log n$ vertices and edges as a minor, for $c=c(\alpha)>0$.
Alternatively, every $n$-vertex graph $G$ without sublinear separators contains all graphs with $cn/\log n$ vertices and edges as minors.
Consequently, a supercritical random graph $G(n,(1+\epsilon)/n)$ is typically minor-universal for the class of graphs with $cn/\log n$ vertices and edges.
The order of magnitude $n/\log n$ in the above results is optimal.

A joint work with Rajko Nenadov.

Many applications involving shapes and data not only require analyzing and processing their geometries, but also associated topologies. In the past two decades, computational topology, an area rekindled by computational geometry has emphasized processing and exploiting topological structures of shapes and data. The understanding of topological structures in the context of computations has resulted into sound algorithms and has also put a thrust in developing further synergy between mathematics and computations in general.

This talk aims to delineate this perspective by considering some applications in shape and data analysis, namely, (i) surface/manifold reconstructions, (ii) mesh generation, and (iii) topological data analysis for which computational topology has played a crucial role. For each of the three topics, we will emphasize the role of topology, state some of the key results, and indicate open questions/problems. The hope is that the talk will further stimulate the interest in tying topology and computation together in the larger context of data science.

The height and fill-up-level of trees are important and well-studied parameters. Depending on the underlying stochastic model these parameters behave differently, however, in many cases they are highly concentrated. The purpose of this talk is to discuss symmetric and asymmetric digital search trees, where height and fill-up-level are concentrated (with high probability) at one or two levels, that is, they behave (almost) deterministically. The mathematical analysis for obtaining such stong concentration results is highly involved and makes use of a concatenation of three transforms: Poisson transform, Mellin transform, and a (second) power series expansion - and each of them have to be inverted with complex analytic methods.

The analogue of Hilbert’s 10th problem over the rational numbers is wide open. It asks whether or not there exists an algorithm for checking the solubility of a given homogeneous polynomial Diophantine equation over the integers. What about if you are allowed to pick a Diophantine equation at random? Assuming that the number of variables exceeds the degree it has been conjectured by Poonen and Voloch that 100 percent of these equations satisfy the local-global principle, which in turn gives an algorithm for checking solubility. I shall report on recent work with Pierre Le Boudec and Will Sawin that comes within a whisker of establishing this conjecture by using techniques from the geometry of numbers.

I split my scientific research since 1990 in the field of theoretical analysis, applied and numerical mathematics, in the following core parts.

1. Constrained and numerical optimization: For unilaterally constrained (non-convex) minimization problems described by (hemi-) variational inequalities, generalized Newton methods and associated primal-dual active-set strategies are developed, with application to cohesional contact and crack problems.

2. Inverse problems and generalized FEM: For the acoustic scattering problem described by Helmholtz operator, identification of arbitrary geometric objects is endowed with a zero-order necessary optimality condition in minimizing the misfit of measurements, and supported by the Petrov-Galerkin FEM.

3. Singular perturbations and homogenization: Using asymptotic methods, the shape and topology differentiability of state-dependent objectives is proved for a class of partial differential operators for solids and fluids; the periodic homogenization is extended to nonlinear transmission problems.

4. Nonlinear models in mechanics and electro-kinetics: In the field of non-linear elasticity, well-posedness of implicitly constituted models is established; hypo-plastic models are studied with respect to dynamic behavior; the Poisson-Nernst-Planck diffusion system describing solid electrolyte, and the membrane degradation in fuel cells are researched from a mathematical point of view.

In this talk I will present some of my scientific results on different types of differential operators with singular interactions. These singular perturbations are used in mathematical physics as idealized replacements for more regular strongly localized potentials.

The main part of the talk is devoted to Dirac operators with $\delta$-shell interactions in $\mathbb{R}^2$ and $\mathbb{R}^3$. Such operators are used to describe the propagation of spin-$\frac{1}{2}$ particles (like electrons) taking relativistic effects into account. I discuss results on the self-adjointness and the spectral properties of these operators. An interesting effect is the existence of critical parameters for which the resulting self-adjoint operators have significantly different spectral properties.

Furthermore, I present results on Schrödinger operators with $\delta$-potentials supported on hypersurfaces. In particular, I show how these operators can be approximated by Hamiltonians with regular potentials. This gives a justification for the usage of the singular $\delta$-potentials as an idealized replacement for classical potentials.

Guest-Lecture of Jeremi MIZERSKI
Mathematical Modelling in Medicine –
The Medical Perspective
WS 2019/20120
● Previous knowledge expected:
There is no previous knowledge in physiology or medicine expected from the attendees.

● Content & Objective:
Course in mathematical modeling in medicine should typically strive to:
1. Introduce students to the elements of the morphological bases of physiological processes and their modelling for the clinical applications.
2. Present application-driven mathematics motivated by problems from within and outside medical specialties.
3. Exemplify the value of mathematics in clinical problem solving.
4. Demonstrate connections among different medical fields of application.
On successful completion of this unit, students will be able to:
1. Demonstrate understanding of tools used in the field of clinical medicine such as PICO, VULCAN, FFR, CARTO, CT, MRI etc.
2. Present enhanced knowledge and understanding in the analysis of biological systems.
3. Recognize the power of mathematical modelling in real applications and be able to apply their understanding to their further studies.
4. Understand the basic human anatomy and physiology for everyday life.
5. Get ready to collaborate in the interdisciplinary teams.
Team work on:
- short subject inspired by the lecture
- project of clinical study enhancing the cooperation between mathematics, engineering and medicine.


● Language of instruction:
English

● Scheduled dates:
Monday and wednesday 14:15 – 16:00
Room AE02 (STEG006), Steyrergasse 30, ground floor

● Course Registration:
Via TUGonline
Course Numbers 504.704 Elective subject mathematics / 504.705 Grundthemen Numerik

In this talk I will sketch an elementary proof of the fact that any pointwise finite-dimensional persistence module over a small category decomposes into a direct sum of indecomposables with local endomorphism rings. This result will then be applied to give a short proof of the well-known fact that a pointwise finite-dimensional persistence module over a totally ordered set is interval decomposable. A similar structure theorem for middle exact persistence module over a product of totally ordered sets will also be discussed. This is joint work with W. Crawley-Boevey.

We study the equidistribution properties of random walks on the unit circle. The uniformity of a sequence on the circle can be measured in many different ways; in this talk we shall consider exponential sums, equidistribution in a fixed interval, and discrepancy. We show that under certain conditions these quantities satisfy the law of the iterated logarithm, and we also find their limit distribution. In particular, we give a generalization of the Chung-Smirnov law of the iterated logarithm to random walks.

As an application, we consider subsequences $\{n_k x\}$ of the Kronecker sequence $\{n x\}$. By classical results of Erdös, Gal and Philipp such subsequences are well understood along a fixed lacunary sequence $n_k$, when $x$ is chosen randomly from the interval [0,1]. We give the counterparts of these results when $x$ is a fixed irrational, and the sequence of positive integers $n_k$ is chosen randomly.

We also mention some results about strong uniform distribution, a theory related to the famous Khintchine's Conjecture. We prove, for instance, that a random walk equidistributes in any given Borel subset of the circle with probability 1 if and only if the random walk has an absolutely continuous component. Time permitting, we also discuss generalizations to random walks on compact metrizable groups. Joint work with Istvan Berkes.

Remark: Bence Borda is a new member of the Institute of Analysis and Number Theory. He will work here as a PostDoc researcher for the next 2 years.

The Target Set Selection problem takes as an input a graph $G$ and a
non-negative integer threshold \( thr(v) \) for every vertex $v$.
A vertex $v$ can get active as soon as at least \( thr(v) \) of its
neighbors have been activated. The objective is to select a smallest
possible initial set of vertices, the target set, whose activation
eventually leads
to the activation of all vertices in the graph.

We show that Target Set Selection is in FPT when parameterized
with the combined parameters clique-width of the graph and the
maximum threshold value.
This generalizes all previous FPT-membership results for the
parameterization by maximum threshold,
and thereby solves an open question from the literature.
We stress that the time complexity of our algorithm is surprisingly
well-behaved and grows only single-exponentially in the parameters.

10.00 Uhr: Ass.-Prof. Dr. Kazuki Niino (Kyoto University)
A finite element method for the 1D heat equation with the Hilbert transform

10.30 Uhr: Dr. Marco Zank (Universität Wien)
Numerical integration for the modified Hilbert transformation

11.00 Uhr: Ass.-Prof. Dr. Kazuki Niino (Kyoto University)
The Galerkin method with the Hdiv inner product for the electric field integral equation

11.30 Uhr: Univ.-Prof. Dr. Olaf Steinbach
On the mapping properties of boundary integral operators for the heat equation

We discuss upper bounds for the length of arithmetic progressions
represented by irreducible integral binary quadratic forms (or, more
generally, arbitrary norm forms), which depend only on the form and the
progression's common difference. This is joint work with C. Elsholtz.

Abstract: For given rational functions $f_1,\ldots,f_s$ defined over a number field $\K$, Bombieri, Masser and Zannier (1999) proved that the algebraic numbers $\alpha$ for which the values $f_1(\alpha),\ldots,f_s(\alpha)$ are multiplicatively dependent are of bounded height (unless this is false for an obvious reason).
Motivated by this, we present recent finiteness results on multiplicative relations of values of rational functions at arguments restricted to the maximal abelian extension of $\K$. We go even further and discuss our work in progress on the presence of multiplicative relations modulo finitely generated groups, posing some open questions. If time allows, we will present some finiteness results regarding the presence of powers of S-integers in orbits of polynomial dynamical systems.

Gallagher's theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. This talk is about joint work with Sam Chow where we provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of Diophantine approximation.

Holomorphic eta quotients are certain explicit classical modular forms on suitable Hecke subgroups of the full modular group. We call a holomorphic eta quotient $f$ 'reducible' if for some holomorphic eta quotient $g$ (other than 1 and $f$), the eta quotient $f/g$ is holomorphic. An eta quotient or a modular form in general has two parameters: Weight and level.

We shall show that for any positive integer $N$, there are only finitely many irreducible holomorphic eta quotients of level $N$. In particular, the weights of such eta quotients are bounded above by a function of $N$. We shall provide such an explicit upper bound. This is an analog of a conjecture of Zagier which says that for any positive integer $k$, there are only finitely many irreducible holomorphic eta quotients of weight $k/2$ which are not integral rescalings of some other eta quotients.

This conjecture was established in 1991 by Mersmann. We shall sketch a short proof of Mersmann's theorem and we shall show that these results have their applications in factorizing holomorphic eta quotient. In particular, due to Zagier and Mersmann's work, holomorphic eta quotients of weight $1/2$ have been completely classified. We shall see some applications of this classification and we shall discuss a few seemingly accessible yet longstanding open problems about eta quotients.

This talk will be suitable also for non-experts: We shall define all the relevant terms and we shall clearly state the classical results which we use.

Remark: Soumya Bhattachary is a new member of the Institute of Analysis and Number Theory. He will stay in Graz for one year as a PostDoc researcher.

Abstract: In this talk, I will discuss about the asymptotic formula of the
correlation functions over polynomial ring $\mathbb{F}_q[x]$ in large
degree limit. As a consequences, we get that the correlation of truncated
Liouville function over shifted polynomials is small and also the distribution
of the sum of additive function over $\mathbb{F}_q[x]$.

Abstract: Let $X=\mbox{Sym}^d \mathbf{P}^n:=\mathbf{P}^n \times \cdots \times \mathbf{P}^n/\mathfrak{S}_d$ where the symmetric $d$-group acts by permuting the $d$ copies of $\mathbf{P}^n$. Manin's conjecture gives a precise prediction for the number of rational points on $X$ of bounded height in terms of geometric invariants of a resolution of $X$ and the study of Manin's conjecture for $X$ can be derived from the geometry of numbers in the cases $n>d$ and for $n=d=2$. In this talk, I will explain how one can use the fact that $\mathbf{P}^n$ is an equivariant compactification of an algebraic group and the height zeta function machinery in order to study the rational points of bounded height on $X$ in new cases that are not covered by the geometry of numbers techniques. This might in particular be an interesting testing ground for the latest refinements of Manin's conjecture.

Abstract: The Galois group of a homogeneous linear differential equation is a linear algebraic group that governs the symmetries among the solutions. I will explain progress towards understanding these Galois groups in the case when the linear differential equations have rational function coefficients. Joint work with Anette Bachmayr, Julia Hartmann and David Harbater.

Abstract: Arboreal Galois representations are central objects in modern arithmetic dynamics. They are defined as continuous homomorphisms, associated to rational maps over algebraic varieties, from the absolute Galois group of a field to the automorphism group of a special graph, and they are considered to be the dynamical avatars of Galois representations attached to Tate modules of abelian varieties. Due to their nature, they combine in a beautiful way several combinatorial, arithmetic and group-theoretic information. In this talk I will introduce them, showing peculiar examples and the most important conjectures around the topic. Afterwards, I will explain the recent developments due to my research: our proof of Jones' conjecture (joint with G. Micheli) and our work around the inverse problem (joint with D. Casazza and C. Pagano).

Abstract: We show several instances of machine learning technology in
Finance like deep hedging, deep portfolio optimization, deep
calibration or deep simulation. In return several stochastic methods
from mathematical finance might shed some new light on machine
learning methods: we prove a version of Chow's theorem to underline
that randomness matters in training networks and we apply the
Johnson-Lindenstrauss Lemma to construct tractable approximations of signatures.

Let S(t) be the argument of the Riemann zeta function on the critical line. This function plays an important role to estimate the number of zeros of Riemann zeta function in the critical strip up to a height t. In this talk we will estimate large positive and negative values of S(t) using the resonance method.

Analytic combinatorics is the study of asymptotic enumeration of combinatorial objects through analytical aspects of their corresponding generating functions, especially their singularities. In this series of lectures, we will give an introduction to more advanced methods in this domain, such as saddle point method and Mellin transform, which are applicable to some problems that are out of reach for standard methods such as transfer theorems. This introduction will be given in the context of asymptotic enumeration of variants of integer partitions and plane partitions, including recent work of the lecturer in collaboration with Hsien-Kwei Hwang and Mihyun Kang.

The classical boundary values for regular Sturm-Liouville operators associated with a three-coefficient differential expression on a compact interval $[a,b]$, is extended in a natural manner to the case where the differential expression is singular on an arbitrary open interval $(a,b)$ of the real line under the assumption that the associated minimal operator is bounded from below. The notion of (non)principal solutions of the associated differential equation plays a key role in this analysis.

We briefly discuss the singular Weyl-Titchmarsh-Kodaira m-function and illustrate the theory with the special case of Bessel and Legendre operators.

This is based on joint work with Lance Littlejohn and R. Nichols.

The extremal number ex$(n,F)$ of a graph $F$ is the maximum number of edges in an $n$-vertex graph not containing $F$ as a subgraph. A real number $r \in [1,2]$ is realisable if there exists a graph $F$ with ex$(n,F) = \Theta(n^r)$. Erd\H{o}s and Simonovits conjectured that every rational number in $[1,2]$ is realisable. We show that $2 - \frac{a}{b}$ is realisable for any integers $a,b \geq 1$ with $b>a$ and $b = \pm 1$ (mod $a$). This includes all previously known realisable numbers. This is joint work with Dong Yeap Kang and Hong Liu.

The Markov numbers are the solutions $(x,y,z) \in \mathbb{Z}^3_{+}$  to the Markov equation $x^2+y^2+z^2=3xyz$. Markov (1879) showed that all possible solutions are generated from $ (1,1,1)$ by a simple algorithm. Does this algorithm generate each solution in a unique way? More precisely, Frobenius $(1913)$ asked whether it is true that for all $z> 0$, there exists at most one pair $(x, y)$ such that $x <y <z$ and $(x, y, z)$ is a solution. This conjecture remains open to this day, despite the simplicity of its statement.

Markov numbers arise in many different contexts such as binary quadratic forms, hyperbolic geometry, combinatorics etc. with beautiful interconnections. The purpose of this talk is to present a part of the Markov theory that is built around an intriguing conjecture and Markov's theorem which combines approximation of irrationals and Diophantine equations in a totally unexpected way. In the end, we will introduce an involution of the real line called Jimm induced by the outer automorphism of the extended modular group $ \mathrm{PGL(2,\mathbb{Z})} $ that may be relevant to the subject.

Abstract: Population size estimation by means of capture-recapture methods using zero-truncated count distributions has become popular. The estimator of Chao is likewise frequently used as it is asymptotically unbiased if the model holds and provides a lower bound in the case of population heterogeneity. However, if one-inflation occurs Chao’s estimator can seriously overestimate as it builds largely on the count of ones, the singletons, in the sample. The talk highlights how one-inflation can be detected and how it can be dealt with, and ultimately provides a more reasonable population size estimator. Two examples will motivate and illustrate one-inflated modelling: the size of a dice-snake population in Graz (Austria) as well as the size of the flare star cluster in the Pleiades.

{\bf Maria Charina} (Univ. Wien): Reelle Nullstellen von Polynomen und Origami

{\bf Dennis Elbrächter} (Univ. Wien): Universal sparsity of deep neural networks

{\bf Svenja Hüning} (TU Graz): Convergence of subdivision processes in nonlinear geometries

{\bf Thomas Lang} (Univ. Passau): Segmentation of CT Scans using Support Vector Machines

Seit Inkrafttreten von Solvency II per 1.1.2016 sind Versicherungsunternehmen verpflichtet, den Wert der eingegangenen Verpflichtungen marktkonsistent und unter Berücksichtigung realistischer Annahmen zu bestimmen ("Best Estimate").
Speziell für die klassische Lebensversicherung führen diese Bedingungen zu besonderen Herausforderungen, da es bei diesen Produkten eine enge Verflechtung von Aktiv-, Passivseite und Managementregeln gibt.
Im Rahmen des Vortrags werden wir auf einige Probleme im Zusammenhang mit der Best Estimate Berechnung eingehen und die wichtigsten Validierungsschritte vorstellen.

The talk will provide an overview of some of my recent research topics, with a common theme of generalising the standard graph notions of connectedness and components to higher-dimensional structures.

These include the $k$-core of a graph, i.e. the unique largest subgraph of minimum degree at least $k$, which we analyse by means of a message-passing algorithm introduced in physics literature. We show how an understanding of this local algorithm helps us to determine the global structure of the $k$-core and its interaction with other vertices.

We also consider $j$-tuple-connected components in $k$-uniform hypergraphs, a notion of connectedness related to $j$-tight paths. We observe some phase transition phenomena analogous to famous and classical graph results, but also discuss why the hypergraph case is richer and more complex.

We will consider Maxwell's equations and recall the construction of the related Picard's extended Maxwell system, which proves to be useful in spectral theory and homogenisation. We will provide some information of the extended Maxwell system on compact embeddings and an analysis of the kernel and its relation to the geometry of the underlying domain. Furthermore, we shall show that the extended Maxwell operator is strongly related to the Dirac operator.

The talk is based on joint work with Rainer Picard and Sascha Trostorff; see also [Picard, R.; Trostorff, S.; Waurick, M. On a Connection between the Maxwell System, the Extended Maxwell System, the Dirac Operator and Gravito-Electromagnetism. Math. Meth. Appl. Sci., 40(2): 415-434, 2017].

Random simplicial complexes have received considerable attention in the last years as a higher-dimensional analogue of random graphs. Two models of ``binomial'' random simplicial complexes of dimension d have been studied. In both models, the vertex set is $\{1,...,n\}$ and each $d$-simplex is present with some global probability $p=p(n)$ independently. The first model, due to Linial, Meshulam, and Wallach, furthermore contains all simplices of dimension smaller than $d$. By contrast, the other model, recently introduced by Cooley, Del Giudice, Kang, and Sprüssel, only contains those simplices of dimension $1$ up to $d-1$ that are contained in some d-simplex. For both models, higher-order connectivity of the complex can be defined via the vanishing of cohomology groups, and sharp thresholds for these properties have been determined for various choices of coefficients for cohomology.

Both models mentioned above are ``uniform'' in the sense that the randomness lies only in the choice of the d-simplices. In this talk, we present a ``non-uniform'' model in which the simplices of all dimensions from $1$ up to $d$ are chosen randomly. In particular, both uniform models are special cases of the non-uniform model. We determine a sharp threshold for the aforementioned notion of higher-order connectedness in the non-uniform model, where the coefficients of the cohomology groups are chosen from any abelian group. This result implies the corresponding results for the uniform models.

This talk is based on joint work with Oliver Cooley, Nicola Del Giudice, and Mihyun Kang.

New results towards the Duffin-Schaeffer conjecture, which is a fundamental unsolved problem in metric number theory, have been established recently assuming extra divergence. Given a non-negative function $\psi: \mathbb{N}\to\mathbb{R}$ we denote by $W(\psi)$ the set of all $x\in\mathbb{R}$ such that $|nx-a|<\psi(n)$ for infinitely many $a,n$. Analogously, we write $W'(\psi)$ if we additionally require $a,n$ to be coprime.

Aistleitner et al. proved that $W'(\psi)$ is of full Lebesgue measure if there exists an $\varepsilon>0$ such that $\sum_{n=2}^\infty\psi(n)\varphi(n)/(n(\log n)^\varepsilon)=\infty$. This result seems to be the best one can expect from the method used. Assuming the extra divergence $\sum_{n=2}^\infty\psi(n)/(\log n)^\varepsilon=\infty$ we prove that $W(\psi)$ is of full measure. This could also be deduced from the results in Aistleitner et al., but we believe that our proof is of independent interest, since its method is totally different from theirs. As a further application of our method, we prove that a variant of Khintchine's theorem is true without monotonicity, if the support of $\psi$ can be restricted subject to a condition on the set of divisors.

Capital requirements in Solvency 2 are assembled from many components. In the
talk we will focus on counter-party default risk. We will start from the mysterious
looking instructions how to compute the contribution to capital requirements and
try to explain the underlying statistical model and the meaning of parameters. The
results will then be compared to simulated results in the real case of a reinsurance
company.

Abstract:
It is an elementary exercise to show that the sequence of the
square roots of the positive integers is equidistributed modulo 1. I
will discuss results concerning the fine-scale statistics of this
sequence, such as the determination of its gap distribution by Elkies
and McMullen (using homogeneous dynamics) and its pair correlation in
joint work with Jens Marklof and Ilya Vinogradov (based on the
Elkies-McMullen approach along with some analytic number theoretic
estimates). I will also mention an ongoing project with Carlo Pagano
whose goal is to understand such statistics for the square roots of
subsets of the integers (such as the square-free integers).

Detailliertes Programm siehe
www.applied.math.tugraz.at/tagungen/anaday19/

We first report on a real world case study of vehicle routing. Patient transits in the Auckland City Hospital are carried out by so called orderlies that transfer patients from and to appointments within the hospital complex. For some transits the assistance of a nurse is required. Ad-hoc dispatching of staff members, nurses and orderlies, to transits has been identified as one major source for delays. We present automated, optimized dispatching algorithms which rely
on a network formulation which is strongly related to an established approach for the VRP with soft time windows. However, the need to synchronize the routes of staff members of different types (nurses and orderlies) adds a whole new layer of complexity to the problem, as routes cannot be assessed independently. We present a set of algorithms with varying complexity, ranging from simple heuristics to the use of critical path methods to combine mixed integer formulations for the separated orderly and nurse problems. To address a transit service's stochasticity, volatility and the resulting need for constant re-optimization, we embed the optimization algorithms in a discrete event simulation to evaluate their performance under realistic circumstances.
Some elements of the underlying structure of the above outlined problem have not been explicitly addressed by the literature on vehicle routing. We present machine learning models and some preliminary results on the predictability of optimal solution structures for a sampled version of the VRP with time windows that can be found in numerous applications. Further, we outline some possibilities to make use of such predicted solution structures within heuristics methods, as well as exact algorithms for vehicle routing.

For a finitely generated group, the Cayley graph is a metric space encoding the structure of the group. Gromov introduced the notion of a $\delta$-hyperbolic group, a finitely generated group with a negatively curved Cayley graph, that is, for any triangle in the graph with geodesic sides, each side is contained in the $\delta$-neighborhood of the union of the two other sides. Hyperbolic groups are ``prevalent'' among finitely generated groups.

Grigorchuk, Nekrashevych and Sushchanskii defined the rational group as the full group of homeomorphisms of a Cantor space and which admit precisely finitely many types of ``local actions'' described by finite state transducers (one of many models of computing machines). This is a rather large group and, by construction, it contains all groups generated by finite state automata (for example, the Grigorchuk group of intermediate word growth).

In this talk I will introduce these groups and some of their properties and explain how to embed a class of hyperbolic groups in the rational group.

Parts of this talk are joint with James Belk, Collin Bleak and James Hyde.

Abstract: In this talk I will describe some recent work on counting
monic cubic polynomials with respect to various heights. In particular,
we prove that the number of irreducible, monic, Galois cubic polynomials
$x^3 + a_2 x^2 + a_1 x + a_0$ with $\max\{|a_2|, |a_1|, |a_0|\} \leq X$
and whose $I,J$-invariants are co-prime is $O(X \log X)$.

In this talk, we will establish a central limit theorem for the capacity of
the range process for a class of d-dimensional symmetric alpha-stable random
walks with the index satisfying $d \geq 3\alpha$. Our approach is based on
controlling the limit behavior of the variance of the capacity of the range
process which then allows us to apply the Lindeberg-Feller theorem.

We investigate the connectivity of the flip-graph of all (full) triangulations of a given
finite point set $P$ in general position in the plane and prove that, for $n:=|P|$ large enough,
both edge- and vertex-connectivity are determined by the minimum degree occurring in the
flip-graph, i.e.\ the minimum number of flippable edges in any triangulation of $P$. It is known
that every triangulation allows at least $(n-4)/2$ edge-flips.

This result is extended to so-called subtriangulations, i.e. the set of all triangulations of
subsets of P which contain all extreme points of $P$, where the flip operation is extended to
bistellar flips (edge-flips, and insertion and removal of an inner vertex of degree three).
Here we prove $(n-3)$-edge-connectedness (for all $P$) and $(n-3)$-vertex-connectedness of $n$ large
enough ($(n-3)$ is tight, since there is always a subtriangulation which allows exactly n-3
bistellar flips). This matches the situation known (through the secondary polytope) for
so-called regular triangulations (i.e. subtriangulations obtained by liftings).

(joint work with Uli Wagner, IST Austria)

Let $A$ be the set of all integers of the form $\gcd(n, F_n)$, where $n$ is a positive integer and $F_n$ denotes the $n$-th Fibonacci number. We show that
$\# A(x) \gg x / \log x$ and $\# A(x) =o(x)$ as $x\to \infty$. As a consequence, we obtain that the set of all integers $n$ such that $n$ divides $F_n$ has zero asymptotic density relative to $A$.

Remark: Paolo Leonetti is a new PostDoc researcher at the Institute of Analysis and Number Theory since February 2019.

The talk is focused on Bernstein-type estimates for Jacobi polynomials and their applications to various branches in mathematics. This is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive estimates for a certain class of Schrödinger equations whose Hamiltonian is given by the generalized Laguerre operator, i.e., the Jacobi matrix associated with generalized Laguerre polynomials. These operators feature prominently in the recent study of nonlinear waves in (2+1)-dimensional noncommutative scalar field theory since they appear as the linear part in the nonlinear Klein--Gordon and the nonlinear Schrödinger equations investigated in the recent of Chen, Fröhlich and Walcher (2003) and Krueger and Soffer (2015), respectively.

We show that dispersive estimates for the evolution group are connected with Bernstein-type inequalities for Jacobi polynomials. We use known uniform estimates for Jacobi polynomials to establish some new dispersive estimates. In turn, the optimal dispersive decay estimates lead to new Bernstein-type inequalities.

The talk is based on joint work with T. H. Koornwinder (Amsterdam) and G. Teschl (Vienna).

Abstract: Mathematical models often contain uncertainty in parameters and measurements. In this talk we focus on partial differential equations where some parameters are modeled by random variables. The main example comes for the diffusion equation where the diffusion parameters is modeled as a random field which randomly fluctuates around a given mean. We discuss recent progress on numerical methods in quantifying this uncertainty.

Real algebra and geometry studies semialgebraic sets, i.e. solution sets to systems of polynomial inequalities. The classical theory provides Positivstellensätze, which are of the same spirit as Nullstellensätze in the case of varieties, i.e. solution sets to systems of polynomial equations. Many interesting (and hard) such results have been developed in the last decades, and surprising applications in optimization and convexity have arisen. A much more recent development is the theory of non-commutative semialgebraic sets. Triggered by questions in electrical engineering, control theory and quantum physics, several exciting results have been proven. But beyond the mentioned applications, the non-commutative theory also sheds light on classical questions.

In this talk, I will give an introduction to both the classical theory and their non-commutative extension, as well as some interesting applications.

Let $s_n$ be the sequence of all positive integers that are sums of two squares, arranged in increasing order. Finding nice estimates for the gap sequence $g_n=s_{n+1}-s_n$ is a classical problem in analytic number theory. In my talk, I will construct certain series involving values of Bessel function that will allow us to prove new results about the power moments of the sequence $g_n$. We will also discuss possible generalizations of these results.

Remark: This talk is part of a joint Austrian-Russian FWF-RSF research project on Diophantine approximation and geometry of numbers.

The aim of the conference is to present and discuss recent results on differential operators on metric graphs and domains with waveguide-like geometry. Some of the main topics are spectral and scattering theory, asymptotic analysis and homogenization, point interactions, and random models. For a detailed conference program see

www.math.tugraz.at/diffop2019/program.html

A significant part of my research deals with understanding the behavior of the following cluster growth models: internal diffusion limited aggregation, the rotor model, and the divisible sandpile model.
These models can be run on any infinite state space, and they are based on particles moving around according to some rule (that can be either random or deterministic) and aggregating. Describing the limit shape of the cluster whjich these particles produce is one of the main questions one would like to answer. For some of the models, the limit shape is hard to understand, and according to simulations, the fractal nature of the sets they produce is, from the mathematical point of view,  far away from being understood. I will present several results concerning the limit shape of the clusters.  In particular, I will present a limit shape universality result on the Sierpinski gasket graph, and conclude with some future research directions one can pursue within this topic.

Automata and graphs are associated in many ways. For invertible automata one can
define the automaton group and observe its action on the set of finite words over the
input alphabet. This leads to the construction of Schreier graphs.
The sum of all distances in a graph, called Wiener index, is a graph property of wide interest. Harry Wiener showed that the properties of molecules are related to the Wiener index of chemical structural formulas. In my presentation I am going to introduce all necessary tools and prove an upper bound for the Wiener index of Schreier graphs of the Basilica automaton.

Schedule:

9:30-10:00 Coffee

10:00-10:50: Michael Kerber: Algorithmic advances for multi-parameter persistence

10:50-11:10: Coffee

11:10-12:00: Mickael Buchet: On the structure of indecomposable modules of multi-dimensional commutative grids

12:00-12:30: Arnur Nigmetov: Metric spaces with expensive distances

12:30-14:00: Lunch break

14:00-14:50: Hubert Wagner: Bregman geometry and Information Topology

14:50-15:20: Hannah Schreiber: Discrete Morse Theory for Computing Zigzag Persistence

15:20-15:50: Coffee

15:50-16:40: Herbert Edelsbrunner: Tri-partitions of complexes

16:40-17:30: Georg Osang: Persistence of Multi-Covers

17:30- Discussions

More information concerning the talks and the speakers can be found on the webpage: https://www.math.tugraz.at/GAG/
If you want to attend the workshop, please let us know in order to organize the coffee breaks.

The so-called Japanese employment system consists of three unique components: long-term employment, seniority-based wage system and enterprise unions. Partly because the system worked properly, Japan successfully achieved the rapid growth in 1960s.
In the talk, first, we give a brief description of Japanese employment system. Then, we introduce two leading theories explaining the economic rationality of seniority-based wage system. Finally, we show an application of such theories as these to empirical analysis taking one of my studies as an example.The details are as follows.

1. An overview of Japanese employment system
 Japanese employment system
 has three unique components such as
 Long-term employment
 Seniority-based wages system
 Enterprise unions
 worked well especially in 1960s, but has been working poorly since early 1990s ,when the bubble economy collapsed
 is undergoing the drastic reform affected by the progress in the reform in corporate governance
2. Focus on seniority-based wage system
 There exist two leading theories explaining the rationality of seniority-based wage system as below
 Human capital theory ( by G. Becker )
 Delayed compensation theory ( by P. Lazear )
 The former sheds light on the role of the system in employees’ skill formation, and the latter regards it as an incentive device
 In Japan, delayed compensation theory seems more plausible
3. An application of economic theories to empirical analysis
 We show how economic theories are applied to empirical analysis taking one of my studies as an example which employed the theories explained above for examining “Employment Ice Age” (the period just after the bubble economy collapsed, when Japanese firms seriously cut back on hiring of new university graduates influenced by negative macroeconomic shocks)

A classical variation of Dirichlet's theorem on Diophantine approximation asks for how well a given number $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ can be approximated by fractions with prime denominators. This problem has attracted the attention of a number of researchers, amongst these, Vinogradov, Vaughan, Harman, Jia, and Heath-Brown. Ten years ago, Matomäki achieved unconditionally the result that, for any $\epsilon>0$, there are infinitely many primes $p$ such that $\min_{a\in\mathbb{Z}} \lvert p\alpha-a \rvert < p^{-1/3+\epsilon}$.---A result previously known only on assuming the Generalised Riemann Hypothesis.

In this talk we consider a natural two-dimensional variation of the aforementioned problem. Namely, for an imaginary quadratic number field $\mathbb{K}\subset\mathbb{C}$ and $\alpha\in\mathbb{C}\setminus\mathbb{K}$, we seek to establish the existence of infinitely many prime elements $p$ in the ring of algebraic integers $\mathcal{O}$ of $\mathbb{K}$ such that $p\alpha$ is `close' to some element of $\mathcal{O}$ in a manner to be made precise in the talk.
The main ingredients of our attack on the problem are a smoothed version of a sieve method due to Harman, Poisson summation, and some point distribution results in the setting of $\mathcal{O}$ in the spirit of Vinogradov. In particular, the introduction of smoothing allows for stronger results than those based on truncating certain Fourier series. For technical reasons we only obtain results for those $\mathbb{K}$ with class number one.

The talk is based on joint work in progress with Stephan Baier.}

We consider parallel machine scheduling with job assignment restrictions,
i.e., each job can only be processed on a certain subset of the machines.
Moreover, each job requires a set of renewable resources. Any resource can
be used by only one job at any time.  The objective is to minimize the
makespan. We present approximation algorithms with constant worst-case bound
in the case that each job requires only a fixed number of resources. For
some special cases optimal algorithms with polynomial running time are
given. If any job requires at most one resource and the number of machines
is fixed, we give a PTAS. On the other hand we prove that the problem is
APX-hard, even when there are just three machines and the input is
restricted to unit-time jobs.