Originated from classical questions in analytic number theory, the Vinogradov Mean Value Method introduced in 1935 has been successful in many important number-theoretic applications. Much attention has been drawn on the main conjecture, which has fully been confirmed in 2015. We present the main two approaches towards the proof and discuss some consequences, especially for sieve theory and the distribution of prime numbers.

I will present several results stating that some sets, such as graph of certain analytic functions, contain few rational points. I also will explain how to prove these statements. The techniques involved mix elementary real analysis and geometry of tame sets.

The aim of the Minimal Model Program is to generalize the classification of complex projective surfaces, known in the early 20th century, to higher dimensional varieties. Besides providing a historical introduction, we will discuss some recent results and new aspects of this Program.

First, I will review the connection between restrictions of the Fourier transform to curved hypersurfaces and the analysis of wave equations. Then, I will link this to the mathematics of the Dirac equation, which is a fundamental equation in relativistic quantum mechanics. Finally, I will describe recent progress on nonlinear Dirac equations.

In this talk I will summarise recent developments in the structure theory of infinite permutation groups. An important class of permutation groups are those that are primitive. These groups are indecomposable in some sense, and are often thought of as being the "atoms" of permutation group theory.
I will present a recent result which describes the structure of all subdegree-finite primitive permutation groups. This class of groups includes, for example, all automorphism groups of locally finite primitive graphs. These groups can be used to examine permutation representations of certain topological groups.

The quadratic shortest path problem (QSPP) is the problem of finding a path in a digraph such that the sum of weights of arcs and the sum of interaction costs over all pairs of arcs on the path is minimized. This problem has recently been proven NP-hard.
In this talk we first consider linearizable QSPP instances whose optimal solution can be found by solving the corresponding instance of the shortest path problem. We also present a polynomial-time algorithm that verifies whether a QSPP instance on a digraph is linearizable. If the instance is linearizable, then our algorithm returns the corresponding linearization vector. For non-linearizable instances we exploit our algorithm to compute lower bounds for the original problem. To the best of our knowledge, this is the first study in which the linearization problem is exploited to compute bounds for the corresponding combinatorial optimization problems.
Further, we present several semidefinite programming relaxations for the QSPP. Semidefinite programming is a special case of cone programming. We use the alternating direction method of multipliers (ADMM) to solve our semidefinite programming relaxations. Finally, we show how to incorporate the ADMM within a branch and bound framework for solving large scale instance of the QSPP.

One central task in comparative politics is to locate party positions in a certain political space. For this purpose, several empirical methods have been proposed using text as data sources. In general, the analysis of texts to extract information is a difficult task. Its data structure is very complex and political texts usually contain a large number of words such that a simultaneous analysis of word counts becomes challenging. In this paper, we consider Poisson models for each word count simultaneously and provide a statistical analysis suitable for political text data. In particular, we allow for multi-dimensional party positions and develop a data-driven way of determining the dimension of positions. Allowing for multi-dimensional political positions gives new insights in the evolution of party positions and helps our understanding of a political system. Additionally, we consider a novel model which allows the political lexicon to change over time and develop an estimation procedure based on LASSO and fused LASSO penalization techniques to address high-dimensionality via significant dimension reduction. The latter model extension gives more insights into the potentially changing use of words by left and right-wing parties over time. Furthermore, the procedure is capable to identify automatically words having a discriminating effect between party positions. To address the potential dependence structure of the word counts over time, we included integer-valued time series processes into our modeling approach and we implemented a suitable bootstrap method to construct confidence intervals for the model parameters. We apply our approach to German party manifesto data of the five main parties over all seven federal elections after German reunification. The approach is simply implemented as it does not require any a priori information (from external source) nor expert knowledge to process the data. The data studies confirm that our procedure is robust, runs stable and leads to meaningful and interpretable results. The talk reports on joint work with Carsten Jentsch, Mannheim and Eun Ryung Lee, Seoul.

The talk will discuss the notion of support for modular representations of finite groups. The classification of the indecomposable representations is a classical challenge in representation theory, and the (cohomological) support provides an elegant method to achieve this. I will also discuss the more general context of tensor triangular geometry, because there are similar notions of support in other subjects, for instance in algebraic geometry and stable homotopy theory.