Im Mai 2013 ging durch die Medien (u.a. die New York Times), dass ein Durchbruch in Richtung Primzahlzwillingsvermutung erzielt wurde: Der zuvor unbekannte Mathematiker Y. Zhang konnte zeigen, dass die Folge der Primzahllücken p_n+1-p_n einen bestimmten Wert unendlich oft annimmt. Als obere Schranke für diesen Wert bewies er H=70.000.000. Dies konnte mittlerweise durch das Polymath8-Projekt auf H=246 gedrückt werden. In dem Vortrag werde ich zunächst eine kurze Einführung in die Verteilung der Primzahlen und Primzahlmuster geben und dann über die zugrundeliegende GPY-Methode und die Ideen, die zur Verbesserung der Lückenschranke H geführt haben, berichten. Dabei handelt es sich insbesondere um neue Erkenntnisse zur Verteilung der Primzahlen in Restklassen - und zwar in Gestalt neuer Varianten des Satzes von Bombieri-Vinogradov, den ich vorstellen werde und dessen zentrale Bedeutung ich in verschiedenen Anwendungen in der Mathematik beleuchten möchte.

In May 2013, the international media (e.g. the New York Times) reported that there had been a breakthrough towards the twin prime conjecture: The mathematician Y. Zhang, previously unknown, had shown that the sequence p_n+1-p_n of prime gaps takes a certain value infinitely often. As an upper bound for this value, he proved that H=70.000.000 is possible. By now, the Polymath8-project was able to improve this bound to H=246. In the talk I will give a short introduction to the distribution of primes and prime patterns and then speak about the GPY-method underlying this breakthrough and about the ideas that have led to the latest improvements of the gap bound H. In particular, these ideas yield new insights to the theory of the distribution of primes in the form of new variants of Bombieri-Vinogradov's theorem. I will introduce this theorem and show its importance by giving some examples of applications in mathematics.

Mapping class groups of surfaces are a classical object of study both in geometry and in topology. Fixing a genus g, the mapping class group is the group of (orientation-preserving) self-homeomorphisms of a genus g surface, where two homeomorphisms are considered the same if one can be deformed into the other. In genus one, the surface is a torus and the mapping class group is isomorphic to the matrix group SL(2,Z). In particular, it is an arithmetic group. But in genus at least three, mapping class groups are no longer arithmetic and it is not even known whether they are linear. In this talk, we shall discuss a new way to get information about mapping class groups by exploiting their appearance in the study of certain topological invariants such as the famous Jones polynomial of classical knots, or Witten-Reshetikhin-Turaev invariants of 3-dimensional manifolds. These invariants are often called quantum invariants because they are related to theoretical quantum physics. Mapping class groups appear when one extends these invariants to what Atiyah and Segal called a Topological Quantum Field Theory (TQFT). More precisely, one gets finite-dimensional unitary representations of mapping class groups in this way. These representations are often called quantum representations. The main point of the talk will be to explain that TQFTs, at least in good cases, have an integral structure in the sense that the numbers that appear as matrix coefficients of the quantum representations are not arbitrary complex numbers but algebraic integers. In this way, quantum representations provide a new way to get homomorphisms from mapping class groups to arithmetic groups. We shall see that these homomorphisms to arithmetic groups allow to answer some previously open questions about mapping class groups.

For a transmission problem between a (thermo-)viscoelastic system with Kelvin-Voigt damping and a purely elastic system it is shown that neither the elastic damping by Kelvin-Voigt mechanisms nor the dissipative effect of the temperature in one material can assure the exponential stability of the system. The approach shows the lack of exponential stability using Weyl's theorem on perturbations of the essential spectrum. To prove polynomial stability we provide an extended version of the semigroup characterizations by Borichev and Tomilov. Observations on the lack of compacity of the inverse of the arising semigroup generators are included too.

The broad research thematic of flows on networks was addressed in recent years by many researchers, in the area of applied mathematics, with new models based on partial differential equations. The latter brought a significant innovation in a field previously dominated by more classical techniques from discrete mathematics or methods based on ordinary differential equations. We are in particular interested in flows described by transport processes on networks including a wide and increasing range of applications: from blood flow to air traffic management. The aim of the present talk is to present a view on large number of themes, results and applications related to this broad research direction. The authors cover different expertise (modeling, analysis, numeric, optimization and other).

A group G is called residually finite if for any non-trivial element g in G there is a homomorphism φ from G to a finite group K such that φ(g) is non-trivial in K. A classical theorem of Baumslag states that if G is a finitely generated residually finite group then the automorphism group Aut(G) is also residually finite. Unfortunately this does not always extend to the group of outer automorphisms Out(G)=Aut(G)/Inn(G). During my talk I will discuss Grossman's classical approach to proving that Out(G) is residually finite together with various new applications of it.

I will begin with a historical introduction to the representation theory of finite groups. Over the last forty years there have been fascinating developments in the theory of modular (i.e. characteristic p) representations, and many basic questions are still open. I will try to emphasise connections to other fields (number theory, algebraic topology, algebraic geometry).

The monodromy conjecture is a mysterious open problem in singularity theory. It relates arithmetic and topological/geometric properties of a multivariate polynomial over the integers. The case of interest is when the zero set of the polynomial has singular points. We will present some history, motivation, and partial results.