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Representing and processing data in spherical domains presents unique challenges, primarily due to the curvature of the domain, which complicates the application of classical Euclidean techniques. Implicit neural representations (INRs) have emerged as a promising alternative for high-fidelity data representation; however, to effectively handle spherical domains, these methods must be adapted to the inherent geometry of the sphere to maintain both accuracy and stability. In this context, we propose Herglotz-NET (HNET), a novel INR architecture that employs a harmonic positional encoding based on complex Herglotz mappings. This encoding yields a well-posed representation on the sphere with interpretable and robust spectral properties. Moreover, we present a unified expressivity analysis showing that any spherical-based INR satisfying a mild condition exhibits a predictable spectral expansion that scales with network depth. Our results establish HNET as a scalable and flexible framework for accurate modeling of spherical data. Finally, inspired by the HNET analysis, we propose a common framework to understand the expressivity of the positional encodings used in Euclidean and spherical INRs. ____________________ Invited by Prof. Johannes Maly

Diffusion models are one of the key architectures of generative AI. Their main drawback, however, is the high computational cost. The first part of the talk introduces diffusion models, outlining their main ideas and their role in modern generative AI. The second part of the talk will focus on our recent research showing how the concept of sparsity, well known especially in statistics, can provide a pathway to more efficient diffusion pipelines. Our mathematical guarantees prove that sparsity can reduce the influence of the input dimension on computational complexity to that of a much smaller intrinsic dimension of the data. Our empirical findings further confirm that inducing sparsity can indeed lead to better samples at a lower cost.

We present the relativistic quantum physics model hierarchy from Dirac-Maxwell to Vlasov/Euler-Poisson that models fast moving charges and their self-consistent electro-magnetic field. Our main interest is (asymptotic) analysis of these nonlinear time-dependent PDE, with focus on the Pauli-Poisswell/Darwin system which is the consistent model at first/second order in 1/c (c = speed of light) that keeps both relativistic effects "spin" and "magnetism". Emphasis is on the (semi)classicial limit for vanishing Planck constant. We use both WKB methods and Wigner functions where we extend the 1993 results of P. L. Lions & Paul and Markowich & Mauser on the limit from Schrödinger-Poisson to Vlasov-Poisson, with similar subtilities of pure quantum states vs mixed states. In the hope of taming the mathematical complications stemming from the magnetic field, we are interested also in developing "quantum/semiclassical velocity averaging lemmata" building on the 1988 ideas of Golse, Perthame, Sentis and P.L.Lions. This talk aims to explain the models, the new results & ideas of proofs/techniques, as worked out in joint works mainly with Jakob Möller (X) and also Pierre Germain (ICL), Changhe Yang (Caltech), François Golse (X).

Mean field limits are an important tool in the context of large-scale dynamical systems, in particular, when studying multiagent and interacting particle systems. While the continuous-time theory is well-developed, few works have considered mean field limits for deterministic discrete-time systems, which are relevant for the analysis and control of large-scale discrete-time multiagent system. We prove existence results for the mean field limit of very general discrete-time dynamical systems, which in particular encompass typical multiagent systems. As a technical novelty, we utilize kernel mean embeddings, which are an established tool in machine learning and statistics, but have been rarely used in the context of multiagent systems and kinetic theory. Our results can serve as a rigorous foundation for many applications of mean field approaches for discrete-time dynamical systems, from analysis, simulation and control to learning.