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Abstract: Our program aims to gather several experts in the field, providing them a unique opportunity to interact, tackle longstanding questions and explore new emerging directions.

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Abstract: Deep learning has been wildly successful in practice and most state-of-the-art artificial intelligence systems are based on neural networks. Lacking, however, is a rigorous mathematical theory that adequately explains the amazing performance of deep neural networks. In this talk, I present a new mathematical framework that provides the beginning of a deeper understanding of deep learning. This framework precisely characterizes the functional properties of trained neural networks. The key mathematical tools which support this framework include transform-domain sparse regularization, the Radon transform of computed tomography, and approximation theory. This framework explains the effect of weight decay regularization in neural network training, the importance of skip connections and low-rank weight matrices in network architectures, the role of sparsity in neural networks, and explains why neural networks can perform well in high-dimensional problems. At the end of the talk we shall conclude with a number of open problems and interesting research directions.

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Abstract: Nonlinear oscillators have a broad range of applications in engineering, including rotors, energy harvesters, sensors, and precision timing devices. The dynamics of a single oscillator with cubic nonlinearity and external periodic forcing is surprisingly rich. Depending on parameters, it may admit multiple attractors that may be periodic or chaotic. Their basin boundaries may be fractal. Linking oscillators into arrays and adding noise further complicate their dynamics. I will discuss a method for finding the most probable escape paths from the basins of attractors of noisy oscillators, sensor design, and a few open mathematical problems related to nonlinear oscillators.

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Abstract: I will present a general theory for coexistence and extinction of ecological communities that are influenced by stochastic temporal environmental fluctuations. The results apply to discrete time stochastic difference equations that can include population structure, eco-environmental feedback or other internal or external factors. Using the general theory, I will showcase some interesting examples. I will end my talk by explaining how the population size at equilibrium is influenced by environmental fluctuations.

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Abstract: Alice and Bob are playing a guessing game. A room contains infinitely many boxes, labeled with the positive integers. Each box contains a real number. Alice will go into the room, open some but not all boxes, and then guess the contents of an unopened box. Then she leaves, the boxes are closed, and Bob enters, opens some but not all boxes, and guesses the contents of an unopened box. Alice and Bob can plan their strategies before the game, but once the game starts they cannot communicate. Paradoxical Theorem: There are strategies that Alice and Bob can follow that guarantee that at least one of them will guess correctly. The proof uses the axiom of choice, but a recent theorem of Elliot Glazer calls into question whether the axiom of choice can be blamed for the paradox.

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Abstract: Generative machine learning models, including variational auto-encoders (VAE), normalizing flows (NF), generative adversarial networks (GANs), diffusion models, have dramatically improved the quality and realism of generated content, whether it's images, text, or audio. In science and engineering, generative models can be used as powerful tools for probability density estimation or high-dimensional sampling that critical capabilities in uncertainty quantification (UQ), e.g., Bayesian inference for parameter estimation. Studies on generative models for image/audio synthesis focus on improving the quality of individual sample, which often make the generative models complicated and difficult to train. On the other hand, UQ tasks usually focus on accurate approximation of statistics of interest without worrying about the quality of any individual sample, so direct application of existing generative models to UQ tasks may lead to inaccurate approximation or unstable training process. To alleviate those challenges, we developed several new generative diffusion models for various UQ tasks, including a score-based nonlinear filter for recursive Bayesian inference, and a training-free ensemble score filter for tracking high dimensional stochastic dynamical systems. We will demonstrate the effectiveness of those methods in various UQ tasks including tracking high-dimensional Lorenz 96 systems and data assimilation for multiple geophysical models.

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Abstract: In 1994, the field of quantum computing had a significant breakthrough when Peter Shor introduced a quantum algorithm that factors integers in (probabilistic) polynomial time.
In these talks, I'll explain the mathematical aspects of Shor's algorithm.
Part II will follow on 3/5.

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Abstract: I will discuss recent work with Hoi Nguyen (https://arxiv.org/abs/2409.03099 ) on products of discrete random matrices with IID integer entries from any non-degenerate distribution. Roughly speaking, the analogues of singular values for these products converge to the single-time marginal distribution of an interacting particle system, the reflecting Poisson sea. I will also discuss the technique we developed to show this, which is a general-purpose 'rescaled moment method' for fluctuations of random abelian groups.

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Abstract: Two simply-connected smooth 4-manifolds are homeomorphic if and only if their product with S^2, are diffeomorphic. The Donaldson 4-6 question ask whether this fact can be lifted to the symplectic category. Explicitly, it conjectures that the underlying smooth manifolds of two (simply-connected) symplectic manifolds are diffeomorphic if and only if their product with S^2, equipped with the standard area form, are symplectic deformation equivalent. I will describe one example of a smooth 4-manifold admitting two symplectic forms which remain deformation inequivalent after taking the product with S^2, giving counterexamples to one implication of the conjecture. On the other hand, I will explain why two symplectic manifolds, whose stabilisations are deformation equivalent, have the same Gromov-Witten invariants. This is joint work with Luya Wang.

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Abstract: Networks arise naturally in many scientific fields as a representation of pairwise connections. Statistical network analysis has most often considered a single large network, but it is common in a number of applications, for example, neuroimaging, to observe multiple networks on a shared node set. When these networks are grouped by case-control status or another categorical covariate, the classical statistical question of two-sample comparison arises. In this work, we address the problem of testing for statistically significant differences in a given arbitrary subset of connections. This general framework allows an analyst to focus on a single node, a specific region of interest, or compare whole networks. Our ability to conduct mesoscale testing on a meaningful group of edges is particularly relevant for applications such as neuroimaging and distinguishes our approach from prior work, which tends to focus either on a single node or the whole network. Our approach can leverage all available network information, and learn informative projections which improve testing power when low-dimensional latent network structure is present.