Events

Quantum walks (QWs) can be viewed as quantum analogs of classical random walks. Mathematically, a QW is described as a unitary, local operator acting on a grid and can be written as a product of shift and coin operators. We highlight differences to classical random walks and stress their connection to quantum algorithms (see Grover’s algorithm). If the QW is assumed to be translation invariant, applying the Fourier transform yields a multiplication operator, whose bandstructure we briefly study. After equipping the underlying lattice with random phases, we turn to dynamical localization. This means that the probability to move from one lattice site to another decreases on average exponentially in the distance, independently of how many steps the QW may take. We sketch the proof of dynamical localization on the hexagonal lattice in the regime of strong disorder, which uses a finite volume method.

Decision makers desire to implement decision rules that, when applied to individuals in the population of interest, yield the best possible outcomes. For example, the current focus on precision medicine reflects the search for individualized treatment decisions, adapted to a patient's characteristics. In this presentation, I will consider how to formulate, choose and estimate effects that guide individualized treatment decisions. In particular, I will introduce a class of regimes that are guaranteed to outperform conventional optimal regimes in settings with unmeasured confounding. I will further consider how to identify or bound these "superoptimal" regimes and their values. The performance of the superoptimal regimes will be illustrated in two examples from medicine and economics.

A fundamental challenge in causal induction is inferring the underlying graph structure from observational and interventional data. While traditional algorithms rely on candidate graph generation and score-based evaluation, my research takes a different approach. I developed neural causal models that leverage the flexibility and scalability of neural networks to infer causal relationships, enabling more expressive mechanisms and facilitating analysis of larger systems. Building on this foundation, I further explored causal foundation models, inspired by the success of large language models. These models are trained on massive datasets of diverse causal graphs, learning to predict causal structures from both observational and interventional data. This "black box" approach achieves remarkable performance and scalability, significantly surpassing traditional methods. This model exhibits robust generalization to new synthetic graphs, resilience under train-test distribution shifts, and achieves state-of-the-art performance on naturalistic graphs with low sample complexity. We then leverage LLMs and their metacognitive processes to causally organize skills. By labeling problems and clustering them into interpretable categories, LLMs gain the ability to categorize skills, which acts as a causal variable enabling skill-based prompting and enhancing mathematical reasoning. This integrated perspective, combining causal induction in graph structures with emergent skills in LLMs, advances our understanding of how skills function as causal variables. It offers a structured pathway to unlock complex reasoning capabilities in AI, paving the way from simple word prediction to sophisticated causal reasoning in LLMs.

Currently, there is no universal definition of a random bifurcation. Many of the existing definitions either fail or are often too complicated to validate in applications. To address this, new definitions of a random bifurcation were proposed and subsequently validated for the supercritical pitchfork bifurcation with additive Brownian noise ([1]). However, the in [1] used stochastic integration suffers from multiple drawbacks, which can be circumvented by using rough integrals instead. In my master thesis, we translated their work to the rough path formalism (made precise in [2], [3]). We were able to reproduce the results in [1] using rough instead of stochastic integration. Further, taking advantage of the rough path formalism, we could additionally investigate the case, where the additive noise is given by fractional Brownian motion with Hurst parameter H ∈ ( 1 3 , 1 2 ). In this talk, we discuss previously existing definitions of a random bifurcation and compare them to the in [1] newly proposed concepts. In particular, we point out where existing definitions fall short. Further, we introduce the rough path formalism and argue why it is such a fitting choice when investigating random bifurcations. Moreover, we compare rough to stochastic integrals and highlight the major advantages of the former. Finally, we present our bifurcation results for the rough fractional pitchfork equation with additive (fractional) Brownian noise. References [1] M. Callaway, T. S. Doanb, J. S. W. Lamb and Martin Rasmussen. “The dichotomy spectrum for random dynamical systems and pitchfork bifurcations with additive noise”. In: Annales de l’Institut Henri Poincar´e - Probabilit´es et Statistiques 53.4 (2017). doi: 10.1214/16-AIHP763. [2] P.K. Friz and M. Hairer. A Course on Rough Paths. Berlin: Springer, 2014. [3] P.K. Friz and N.B. Victor. Multidimensional Stochastic Processes as Rough Paths: Theory and Ap- plications. Cambridge: Cambridge University Press, 2009.