Date/Time : 26 May 2025 14:00
Venue : S17 #04-05 (SR2)
Speaker : Gus Lehrer, University of Sydney

I shall show how to use algebraic geometric methods to convert invariants of OSp into invariants of GL. This will be used to obtain precise versions of the first and second fundamental theorems for OSp.

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Date/Time : 30 Apr 2025 14:00
Venue : S17 #04-04 (SR3)
Speaker : Lin Shu, Peking University

There are a variety of interesting chaotic dynamical systems in negatively curved spaces. Entropies related to these systems are important quantities for understanding not only the complexities of their dynamics but also the complexities of the underlying spaces. This partially explains why the study of entropies in negatively curved spaces has always been a vibrant topic in dynamical systems. In this talk, we will give a biased introduction to this heavy topic by focusing on some natural diffusions (for instance, foliated Brownian motion) associated with the geodesic flow in a negatively curved space. We will mention some of our results (joint with F. Ledrappier) on stochastic entropies of these diffusions, in an attempt to use these quantities to characterize the geometry of the underlying space, to find connections between classical dynamical measures, and to understand the linear response theory for heat conduction, etc.

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Date/Time : 30 Apr 2025 15:00
Venue : S17 #04-05 (SR2)
Speaker : Liu Jinpeng, Tsinghua University

Nonlinear dynamics play a prominent role in scientific computation and artificial intelligence. Whereas previous quantum algorithms for general nonlinear equations have been severely limited due to the linearity of quantum mechanics, we gave the first efficient quantum algorithm for nonlinear differential equations with sufficiently strong dissipation. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in the evolution time. Furthermore, we design the first quantum algorithm for training classical sparse neural networks with end-to-end settings. We benchmark instances of ResNet with sparse pruning applied to Cifar-100 dataset and DPM-Solver for U-ViT applied to ImageNet-100 dataset, and we find that a quantum enhancement is possible at the early stage of learning. Our work shows that fault-tolerant quantum computing can contribute to the training and inference processes of most state-of-the-art large language models and diffusion models.

References:
[1]Efficient quantum algorithm for dissipative nonlinear differential equations. Proceedings of the National Academy of Science 118, 35 (2021).
[2]Towards provably efficient quantum algorithms for large-scale machine learning models. Nature Communications 15, 434 (2024)
[3]Towards efficient quantum algorithms for diffusion probability models. arXiv:2502.14252

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Date/Time : 30 Apr 2025 16:00
Venue : S17 #04-05 (SR2)
Speaker : Silu Yin, Hangzhou Normal University

In this talk, we investigate the ill-posedness of the non-strictly hyperbolic compressible ideal MHD (IMHD) system in two dimensions，which exhibits multiple propagation speeds. For scalar quasilinear wave equation, Smith-Tataru showed its local well-posedness in $H^{11/4+}(R^2)\times H^{7/4+}(R^2)$. Here, we construct a counterexample to the local existence of low-regularity solutions to the IMHD system in the borderline space $H^{7/4}(R^2)$. Our proof is based on a coalition of a carefully designed algebraic approach and Christodoulou’s geometric approach. We give a complete description of solutions’ dynamics up to the earliest singular event, when a shock forms. This talk is based on a joint work with Xinliang An and Haoyang Chen.

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Date/Time : 30 Apr 2025 15:15
Venue : S17 #04-04 (SR3)
Speaker : Fabrizio Bianchi, University of Pisa

For a rational map, the classical Mané-Manning formula relates the entropy, the Lyapunov exponent, and the Hausdorff dimension of any invariant measure. I will describe a generalisation of this formula valid for invariant measures for endomorphisms of P^k(C) in any dimension with strictly positive Lyapunov exponents. This is joint work with Y. M. He.

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Date/Time : 30 Apr 2025 14:00
Venue : S17 #04-05 (SR2)
Speaker : Zak Shumaylov,

In the last five years, neural solvers have emerged as powerful tools for learning solution operators of partial differential equations (PDEs). Paralleling the central role of symmetries in classical numerical analysis – where they guide the development of stable and structure-preserving algorithms, these principles are now proving equally vital in machine learning. Incorporation of symmetries often leads to faster learning, better generalization, with solutions respecting fundamental physical laws. This talk explores how similar principles can be be integrated into modern neural PDE solvers, focusing on Lie point symmetries, generalized symmetries, and Hamiltonian structures. These approaches demonstrate how revisiting classical symmetry ideas leads to more robust, data-efficient, and physically consistent learning frameworks

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Date/Time : 30 Apr 2025 16:30
Venue : S17 #04-06 (SR1)
Speaker : Grigorios Chrysos , University of Wisconsin-Madison

Activation functions play a pivotal role in deep neural networks, enabling them to tackle complex tasks like image recognition. However, activation functions also introduce significant challenges for deep learning theory, network dynamics analysis, and properties such as interpretability and privacy preservation. In this talk, we revisit the necessity of activation functions across various scenarios. Specifically, we explore expressing network outputs through high-order interactions among input elements using multilinear algebra. This approach allows us to attain the necessary expressivity via these high-order interactions. Yet, the question remains: Is this expressivity alone sufficient for effective learning? Our recent research, presented at ICLR’24, unveils meticulously designed networks, which achieve strong performance even in demanding tasks, such as ImageNet image recognition.

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Date/Time : 29 Apr 2025 15:00
Venue : S17 #04-05 (SR2)
Speaker : Federico Pasqualotto, University of California San Diego

In this talk, I will first review the singularity formation problem in incompressible fluid dynamics, describing how particle transport poses the main challenge in constructing blow-up solutions for incompressible fluids. I will then outline a new mechanism that allows us to overcome the effects of particle transport, leveraging the instability seen in the classical Taylor–Couette experiment. Using this mechanism, we construct the first swirl-driven singularity for the incompressible Euler equations in R^3. This is joint work with Tarek Elgindi (Duke University).

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Date/Time : 29 Apr 2025 16:00
Venue : S17 #04-05 (SR2)
Speaker : Liu Jinpeng, Tsinghua University

Fault-tolerant quantum computers are expected to excel in simulating unitary dynamics, such as the dynamics of a quantum state under a Hamiltonian. Most applications in scientific and engineering computations involve non-unitary dynamics. First, we propose a simple method for simulating a general class of non-unitary dynamics as a linear combination of Hamiltonian simulation (LCHS) problems. The LCHS method can achieve optimal cost in terms of state preparation. Second, we generalize the LCHS formula to simulate time-evolution operators in infinite-dimensional spaces, including scenarios involving unbounded operators. We demonstrate the applicability of LCHS to a wide range of non-Hermitian dynamics, including linear parabolic PDEs and open quantum system dynamics. Our analysis provides insights into simulating general linear dynamics using a finite number of quantum dynamics and includes cost estimates for the corresponding quantum algorithms.

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