## Geometry of algebraic fibred surfaces with vector bundles techniques

*Valentina Beorchia, Universita' **degli** Studi di Trieste*

In the talk, I shall consider algebraic fibred surfaces and their invariants, such as the relative canonical divisor and the relative Euler-Poincare' characteristic. We shall see the relation of such invariants with the gonality of the general fiber, and we will associate a suitable generically finite map with such fibrations. Moreover, some bounds for the invariants will be established in terms of the Chern classes of a vector bundle arising in our construction. This will follow from some Bogomolov - type inequalities, which hold for weakly positive vector bundles on ruled surfaces.

All the results have been obtained in collaboration with F. Zucconi from the University of Udine.

## On isometries and projections on some Banach spaces

*Dijana Ilišević, University of Zagreb*

Isometries are maps between metric spaces which preserve distance between elements. Although the beginnings of the study of isometries between Banach spaces coincide to the beginnings of the Banach space theory, it is still a very active area of research. Projections on Banach spaces are operators with a simple spectral structure, and they are often used as building blocks for many other operators, which is well known from the spectral theory of operators. In this talk isometries on various Banach spaces will be described, with a special emphasis to their spectral representation.

Some parts of this talk are based on joint work with several collaborators. The work of Dijana Ilišević has been fully supported by the Croatian Science Foundation under the project IP-2016-06-1046.

## The theory of knotoids and braidoids and applications in proteins

*Sofia Lambropoulou, National Technical University of Athens*

We will present the diagrammatic theory of knotoids, introduced by Turaev. We will review the lifting of planar knotoids in 3-space, by Gugumcu and Kauffman, its importance in the classification of proteins and its connection to the diagrammatic theory of rail knotoids, which is in turn related to the knot theory of the handlebody of genus 2. Finally, we will present the theory of braidoids, braidoiding algorithms as well as an equivalence relation on the set of braidoids corresponding to equivalence classes of planar knotoids.

## Topology and data

*Neža Mramor-Kosta, University of Ljubljana*

Topological data analysis (TDA) has recently become a hot topic in data analysis, and the goal of this talk is to discuss why. A dataset is typically presented as a point cloud, that is, a finite set of points in some metric space of possibly high dimension. The underlying idea of TDA is that the shape of the point cloud provides useful insight into properties of the data. A number of topological concepts, models and techniques are available for the study of shapes and their evolution. We will describe some of these, and indicate how they are used in data analysis tasks and machine learning algorithms.

## Are locality and renormalisation reconcilable?

*Sylvie Paycha, University of Potsdam, on leave from the University Clermont-Auvergne*

According to the principle of locality in physics, events taking place at different locations should behave independently, a feature expected to be reflected in the measurements. The latter are confronted with theoretic predictions which use renormalisation techniques in order to deal with divergences from which one wants to derive finite quantities. The purpose of this talk is to confront locality and renormalisation. Sophisticated (co)algebraic methods developed by physicists enable to keep track of locality while renormalising. They mostly use a univariate regularisation scheme such as dimensional regularisation. We shall present an alternative multivariate approach to renormalisation which encodes locality as an underlying algebraic principle. We shall apply it to various situations involving renormalisation, such as divergent multizeta functions and their generalisations, namely discrete sums on cones and discrete sums associated with trees.

This is based on joint work with Pierre Clavier, Li Guo, and Bin Zhang

## Introduction to a semigroup approach for dynamical networks

*Eszter Sikolya, Eötvös Loánd University, Budapest*

In this talk, we first give a short introduction to metric graphs. Then we define dynamical processes (e.g. flows) taking place on the edges of such graphs, satisfying certain conditions (Kirchhoff law) in the vertices. Using tools from the theory of operator semigroups we are able to prove well-posedness of such problems on L^{p} spaces. This approach makes also possible to describe asymptotic properties of the dynamical network: it turns out that the long-time behavior of the process depends on the eigenvalues of the adjacency matrix of the underlying graph.

The talk is based on results coming from joint works with Marjeta Kramar Fijavž (Ljubljana), Britta Dorn and Vera Keicher (Tübingen), Delio Mugnolo (Hagen).

## Cellular automata based algorithms

*Biljana Stamatovic, University of Donja Gorica *

Cellular automaton (CA) is a discrete structure which is used for creating different models (in computer science, physics, mathematics, social sciences ...) for dynamic agent-based systems. From the theoretical part of computational theory, just one example of CA - Game of life, is universal in the sense that it is Turing complete (it can compute everything that can be computed). CA can be considered as an alternative way of computation based on local data flow principles.

Three CA-based algorithms will be presented. One is for the identification of backbones in infinite clusters on 2D percolation site lattices. The second is for labeling of connected components in the n-dim binary lattice. The third is for identification n-connectivity, n > 2.

## How do Pure States extend?

## A story about the Kadison-Singer problem.

*Betül Tanbay, Boğaziçi University Istanbul*

The Kadison-Singer problem was born in 1959 and raised in different parts of the world from the USA to Scotland, from Turkey to India. By the time it was solved in 2013, many equivalent versions were expressed in different branches of mathematics, from set theory to operator algebras, from linear algebra to computer sciences. I was born in 1960 and have got stuck with this problem since 1989. Now that we know the answer to the original problem is positive, there are still quite intriguing questions remaining...for us all.

## Bayesian treatment effects modeling of earnings effects of maternity leave

*Helga Wagner, Johannes Kepler University Linz*

Analysis of treatment effects is an important issue in many application fields, e.g. to evaluate of the effectiveness of social programs, government policies or medical interventions. As each person is observed either under treatment or under control condition identification and estimation of treatment effects is not straightforward.

In this talk, I will give an introduction to the problem of identification and estimation of treatment effects. I will discuss implications for studies where participants are randomly assigned to the treatment or control group as well as for observational studies where participants can choose treatment or control based on their preferences and expectations of the outcome under both conditions.

There are many different statistical approaches to treatment effects analysis. I will focus on the Bayesian approach where inference on treatment effects is accomplished by specifying a joint model of treatment selection and the potential outcomes.

I employ this model to analyse the effects of long maternity leave on earnings of Austrian mothers. The analysis is based on data from the Austrian Social Security Register which contains individual employment histories since 1972 and also reports the number of births and maternity and parental leave spells for all Austrian employees and exploits a change in the parental leave policy in Austria that extended maternal benefits from 18 months since the birth of the child to 30 months.