Many real-world interconnected systems are governed by non-pairwise interactions between agents frequently referred to as higher order interactions. The resulting higher order interaction structure can be encoded by means of a hypergraph or hypernetwork. This talk will focus on dynamics of such hypernetworks. We define a class of maps that respects the higher order interaction structure, so-called admissible maps, and investigate how robust patterns of synchrony can be classified. Interestingly, these are only defined by higher degree polynomial admissible maps. This means that, unlike in classical networks, cluster synchronization on hypernetworks is a higher order, i.e., nonlinear effect. This feature has further implications for the dynamics. In particular, it causes the phenomenon of ``reluctant synchrony breaking'' on hypernetworks, which occurs when bifurcating solutions lie close to a non-robust synchrony space.

This talk is dedicated to a common descent method designed for nonsmooth multiobjective optimization problems (MOPs) with objective functions defined on a general Hilbert space that are only locally Lipschitz continuous. The only strategy to handle nonsmooth MOPs in infinite dimensions besides the presented method relies on scalarization techniques, which are not suitable for MOPs with nonconvex objective functions or for MOPs with more than two objective functions. The class of nonsmooth MOPs on infinite dimensional Hilbert spaces is particularly important since it allows the formulation of PDE-constrained MOPs.

For the analysis of the presented method, we first introduce optimality conditions suitable for nonsmooth MOPs. We generalize the so-called Goldstein epsilon-subdifferential to the multiobjective setting in Hilbert spaces and describe its main properties.

Then, we introduce the mentioned descent method. The method uses an approximation of the epsilon-Goldstein subdifferential to compute a common descent direction that provides sufficient descent for all objective functions. In the main result, we show that, under reasonable assumptions, the method produces sequences that possess Pareto critical accumulation points.

Finally, we present the behaviour of the common descent method for a (PDE-constrained) multiobjective obstacle problem in one and two spatial dimensions. We show that the method is capable of producing several different optimal solutions and discuss the behaviour of the approximated subdifferential.

Online social media are nowadays an integral part of people's everyday life that can influence our behaviour and opinions. Despite recent advances, the changing role of traditional media and the emerging role of "influencers" are still not well understood, and the implications of their strategies in the attention economy even less so. In this talk, we will propose a novel agent-based model (ABM) that aims to model how individuals (agents) change their opinions (states) under the impact of media and influencers. We will show the rich behavior of this ABM in different regimes and how different opinion formations can emerge, e.g. fragmentation. In the limit of infinite number of agents, we will derive a corresponding mean-field model given by a PDE. Based on the mean-field model, we will show how strategies of influencers can impact the overall opinion distribution and that optimal control strategies allow other influencers (or media) to counteract such attempts and prevent further fragmentation of the opinion landscape.

Dieser Vortrag ist gleichzeitig auch im Kolloquium Angewandte Mathematik angekündigt.

At first glance, smooth multiobjective optimization (MOO) and nonsmooth single-objective optimization (NSO) are two distinct subclasses of general optimization. But upon closer analysis, it turns out that there are several parallels. When considering first-order information then, in both areas, there is not only one but multiple gradients that have to be considered simultaneously: In MOO, these are the gradients of the different objective functions, and in NSO, these are all the subgradients from the Clarke subdifferential. As such, there are strong structural similarities when considering results like optimality conditions and descent directions.
In addition to theoretical results, there are also applications where MOO and NSO naturally meet: In many practical problems, a weighted, nonsmooth regularization term is added to a smooth objective function to enforce additional properties of the solution. For example, a sparse minimizer of a function can be found by adding a weighted L1-norm to the function. By varying the weight (also known as the regularization parameter), minimizers with varying degrees of regularity can be computed. Traditionally, these problems are treated via NSO. But since a regularized objective function can be interpreted as a simple weighted sum scalarization, regularization problems may also be treated via MOO.
In this talk, I will present several of these similarities and show how they can be used to obtain new insights and results.

Population protocols and related models allow to study the dynamics of distributed systems consisting of a vast number of simple and identical agents. The standard model assumes a complete network of $n$ agents modeled as simple finite state machines. Pairwise interactions between agents happen either adversarially or in a randomized way and cause the agents to update their respective state, depending on their own state and that of their interaction partner.

Despite their simplicity, population protocols can solve fundamental distributed problems like leader election, majority, or consensus problems. My talk will start with a general introduction into the topic and then dive into some of our recent and ongoing work on how to efficiently solve consensus-related problems in population protocols (and variants).
 

Dieser Vortrag ist gleichzeitig auch im Kolloquium Angewandte Mathematik angekündigt.