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Historically, the synergy between physics and geometry, from the times of Archimedes and Newton to the era of Einstein, has repeatedly been the catalyst for pivotal breakthroughs in physics and mathematics. In this talk, I will introduce a new narrative demonstrating how physics and geometry intertwine, leading to unexpected and significant results in critical phenomena in physics. Specifically, I will elucidate how non-commutative geometry—a mathematical framework born from the insights of physicists—offers fresh perspectives on conformal field theory, a subject with profound applications across various physics domains, from condensed matter to quantum gravity, and string theory.

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Current and future weak lensing surveys contain significant information about our universe. However, their optimal cosmological analysis is challenging, with traditional analyses often resulting in information loss due to reliance on summary statistics like two-point correlation functions. While deep learning methods offer promise in capturing the complex non-linear features of these cosmological fields, they often suffer from issues such as inadequate uncertainty quantification, susceptibility to distribution shifts, and interpretability limitations, which hinder their scientific applicability. In this talk, I propose a novel approach leveraging generative probabilistic modeling with Normalizing Flows to learn the data likelihood function at the field level, facilitating more effective cosmological information extraction. This framework not only enables anomaly detection of distribution shifts to improve the robustness of the analysis, but also fostering interpretability via generated samples. I will also discuss incorporating physical prior knowledge, such as symmetries and multiscale structure, into the model architectures to improve their generalization capabilities. Finally, I will explore the broader implications of deep probabilistic models in physics, highlighting their potential applications in diverse areas ranging from astronomical observations to high-energy physics and lattice field theory.

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