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Quantum Geometry Speaker: Murad Alim (TUM) Abstract: Quantum theory has not only reshaped our understanding of the physical world; it has also become a powerful source of ideas for modern mathematics. In this talk, I will introduce aspects of the emerging field of quantum geometry, where insights from quantum field theory and string theory interact with symplectic, complex, and algebraic geometry. I will explain how dualities in physical theories often reveal that seemingly different mathematical structures share common underlying principles, leading to deep new results and unexpected bridges between diverse areas. A central example is mirror symmetry, a duality relating symplectic and complex geometry with far-reaching consequences for enumerative geometry, representation theory and number theory.

We consider a class of Unitary Quantum Walks on arbitrary graphs, parameterized by a family of scattering matrices. After explaining that these Scattering Quantum Walks encompass several known Quantum Walks, we further introduce two classes of Scattering Open Quantum Walks on arbitrary graphs based on that construction, whose asymptotic states we discuss.

Real-life financial time series exhibit heavy tails and clusters of extreme values. In this talk we will address models that exhibit these stylized facts. This is the class of regularly varying time series, introduced by Davis and Hsing (1995, AoP) and further developed by Basrak and Segers (2009, SPA). The marginal distribution of a regularly varying time series has tails of power-law type, and the dynamics caused by an extreme event in this time series is described by the spectral tail process. The perhaps best known financial time series models of this kind are Engle’s (1982) ARCH process, Bollerslev’s (1986) GARCH process and Engle’s and Russell’s (1998) Autoregressive Conditional Duration (ACD) model. The length and magnitude of extremal clusters in such a series can be described by an analog of the autocorrelation function for extreme events: the extremogram. The extremal index is another useful tool for describing expected extremal cluster sizes. Both objects can be expressed in terms of the spectral tail process and allow for statistical estimation. The probabilistic and statistical aspects of regularly varying time series are summarized in the recent monograph by Mikosch and Wintenberger (2024) “Extreme Value Theory for Time Series. Models with Power-Law Tails”. The talk is based on joint work with Olivier Wintenberger (Sorbonne).

The rapid advances in artificial intelligence (AI) have largely been driven by scaling deep neural networks (DNNs) - increasing model size, data, and computational resources. However, performance is ultimately governed by network dynamics. The lack of a principled understanding of DNN dynamics beyond heuristic-based design has contributed to challenges with their robustness, suboptimal performance, high energy consumption and pathologies in continual and AI-generated content learning. In contrast, the human brain does not seem to suffer these problems, and converging evidence suggest that these benefits are achieved by dynamics being poised at a critical phase transition. Inspired by this principle, we propose that criticality provides a unifying framework linking structure, dynamics, and function also in DNNs. First, by analyzing more than 80 state-of-the-art models, we report that a decade of AI progress has implicitly driven successful networks towards criticality – explaining why certain architectures succeeded while others failed. Second, we demonstrate that incorporating criticality explicitly into training improves robustness and accuracy preventing key limitations of current models. Third, we show that catastrophic AI pathologies, including the performance degradation in continual learning and in model collapse - where performance degrades when training on AI-generated data - constitute a loss of critical dynamics. By maintaining networks at criticality, we provide a principled solution to this fundamental AI problem, demonstrating how criticality-based optimization mitigates performance degradation. This work highlights criticality as substrate-independent principle of intelligence, connecting AI advancement with core principles of brain function. It provides theoretical insights along with immediate practical value solving major AI challenges to ensure long-term DNN performance and resilience as models grow in scale and complexity.