News & Events

Please register for the workshop. More information can be found at s. https://www.fm.math.lmu.de/en/news/events-overview/event/frontiers-in-mathematical-finance-between-theory-and-applications.html

Interconnected real-world systems oftentimes contain non-pairwise interactions between agents. These groupwise interactions are referred to as higher order interactions and can be encoded by means of hypergraphs or hypernetworks. Countless works in recent years have pointed out how this structural feature crucially shapes the collective behavior. This talk will, in particular, focus on dynamics of systems with higher order interactions. We observe that the restriction to undirected higher order interactions obstructs the emergence of certain heteroclinic structures in phase space. The directed counterparts, on the other hand, do not. Motivated by this, we define a general class of directed hypernetworks and corresponding maps that respect a given interaction structure, so-called admissible maps. For this class, all robust patterns of (cluster) synchrony that a given hypernetwork supports can be classified. Interestingly, these are only determined by higher degree polynomial admissible maps. In particular, unlike in classical networks, cluster synchronization is a higher order, that is, nonlinear effect. This feature induces a novel type of “reluctant” synchrony breaking bifurcation when a high order tangency of the solution branch to a non-robust synchrony space causes formerly synchronous nodes to separate unusually slowly.

After motivating the Stefan problem from the random growth model perspective, I will discuss its discontinuities in time. These turn out to be characterized by the cascade equation, a second-order hyperbolic PDE. Questions of existence and regularity for the latter can be answered by expressing its solution as the value function of a player in an equilibrium of a suitable mean field game. Based on joint work with Yucheng Guo and Sergey Nadtochiy.

While the theory of causality is widely viewed as an extension of probability theory, a view which we share, there was no universally accepted, axiomatic framework for causality analogous to Kolmogorov's measure-theoretic axiomatization for the theory of probabilities. Instead, many competing frameworks exist, such as the structural causal models or the potential outcomes framework, that mostly have the flavor of statistical models. To fill this gap, we propose the notion of causal spaces, consisting of a probability space along with a collection of transition probability kernels, called causal kernels, which satisfy two simple axioms and which encode causal information that probability spaces cannot encode. The proposed framework is not only rigorously grounded in measure theory, but it also sheds light on long-standing limitations of existing frameworks, including, for example, cycles, latent variables, and stochastic processes. Our hope is that causal spaces will play the same role for the theory of causality that probability spaces play for the theory of probabilities.