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 : 23 Apr 2025 15:00
Venue : S17 #04-05 (SR2)
Speaker : Jialun Li, Ecole Polytechnique de Paris

I will begin with the classical setting of the modular surface: the unit tangent bundle of the modular curve T^1(SL(2,Z)\H)=SL(2,Z)\SL(2,R), where periodic geodesics correspond to periodic orbits of the diagonal subgroup. Margulis and Bowen established dynamical frameworks for studying these orbits, while Selberg and Sarnak developed spectral and number-theoretic approaches.

We then turn to the higher-rank generalization, periodic diagonal orbits on SL(3,Z)\SL(3,R). In this case, each periodic diagonal orbit is a two-dimensional flat torus embedded in the space. Different counting problems emerge, depending on how one orders these tori–whether by dynamical ordering, geometric ordering or arithmetic ordering. The dynamical and geometric ordering are closely connected through the shapes of the flat tori. Finally, I will discuss my recent joint work with Thi Dang and Nihar Gargava on the density of shapes of periodic tori.

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Date/Time : 19 Mar 2025 14:00
Venue : S17 #04-06 (SR1)
Speaker : Jiajie Chen, New York University

Whether the 3D incompressible Euler equations can develop a finite-time singularity from smooth initial data is a long-standing open problem in mathematical fluid mechanics. In this talk, I will begin by providing an overview of this problem and then introduce a framework for stable, nearly self-similar blowup, developed in joint work with Tom Hou (Caltech). Using this framework, we establish singularity formation in the incompressible Euler equations with smooth initial data and boundary. Additionally, I will briefly present recent results on constructing singularities in fluids and related equations using the self-similar blowup method.

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Date/Time : 05 Mar 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : James Farre, Max Planck Institute for Mathematics in the Sciences

The geodesic flow on the unit tangent bundle of a hyperbolic surface is a prototypical example of an Anosov flow, exhibiting rich, structurally stable dynamics. While invariant measures and closed sets for the geodesic flow can be quite complicated, the closely related horocycle flow exhibits striking rigidity. For example, on a closed surface, every horocycle is dense and equidistributed in its unit tangent bundle; this is very far from being true for geodesics.

Despite this rigidity, the behavior of horocycles remains mysterious in the setting of hyperbolic surfaces with infinite topology. In this talk, I will explore the dynamics of horocyclic trajectories for a particular class of infinite-area regular covers of closed hyperbolic surfaces. Surprisingly, these trajectories are intimately connected to an exceptional set for the geodesic flow: the distance-minimizing geodesic lamination. Both objects arise naturally in a geometric optimization problem: what does an optimal Lipschitz map to the circle in a given homotopy class look like?

This is joint work with Or Landesberg and Yair Minsky.

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Date/Time : 19 Feb 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : He Weikun, AMSS, Chinese Academy of Sciences

Diophantine approximation studies how well a real number can be approximated by rational numbers. Following a question asked by Mahler in 1984, we are interested in Diophantine properties of points in the middle third Cantor set or more generally fractal sets. I will survey briefly this topic and present a recent joint work with Timothée Bénard and Han Zhang, where we obtained a Khintchine dichotomy for the middle third Cantor set.

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Date/Time : 22 Jan 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : Zhihan Wang, University of Chicago

A central question in geometric flow is to understand the change of geometry and topology after passing through singularities.

In this lecture, I will focus on mean curvature flow, the negative gradient flow of area functional, and explain how the local dynamics influence the shape of the flow near a singularity, as well as how the geometry and topology of the flow change after passing through a singularity with generic dynamics.

This talk is based on the joint work with Ao Sun and Jinxin Xue.

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Date/Time : 27 Nov 2024 14:00
Venue : S17 #04-06 (SR1)
Speaker : Patrick Lutz, University of California, Berkeley

The field of computability theory studies the complexity of uncomputable problems (where a “problem” is usually formally defined as either a set of natural numbers or a set of binary strings). In this study, a special role is played by the Halting Problem—i.e. the problem of determining whether a given program stops after a finite number of steps or runs forever. Not only is it the first problem proved to be uncomputable, it also seems to be the simplest “natural” uncomputable problem.

Martin’s Conjecture is a long-standing open question in computability theory which partially explains why the Halting Problem plays such a special role. A key idea behind Martin’s Conjecture is to view the Halting Problem not just as an individual problem, but as an operator on problems, which takes any problem to a strictly harder one. Martin’s Conjecture consists of a classification of such operators, which says, in part, that the Halting Problem is the minimal non-trivial operator. I will discuss the background and motivation for Martin’s Conjecture, as well as recent progress by Benjamin Siskind and myself which essentially completes a proof of the conjecture for a special class of operators called “order-preserving” (i.e. those which preserve the relative complexity of problems).

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Date/Time : 09 Apr 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Dougal Davis, University of Melbourne

One of the oldest open problems in representation theory is to classify the irreducible unitary representations of a real reductive Lie group G_R. Such representations play a fundamental role in harmonic analysis and the Langlands program and arise in physics as the Hilbert spaces of states of quantum systems with G_R-symmetry. The orbit method, which goes back to the work of Kirillov in the 1960s, takes the physical perspective seriously and proposes to construct all unitary representations as quantisations of orbits of the group on the dual of its Lie algebra. This has proved difficult to implement for reductive groups, however: the best general results use deformation quantisation to construct interesting examples of representations but provide no tools to prove that they are unitary. In this talk, I will explain joint work in preparation with Lucas Mason-Brown overcoming this difficulty: we give a uniform proof that, under suitable hypotheses, the orbit method always produces unitary representations. The main tool is recent joint work of myself and Kari Vilonen, which gives a new characterisation of unitary representations in terms of Hodge theory on complex flag varieties.

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Date/Time : 27 Mar 2024 16:00
Venue : S17 #04-05 (SR2)
Speaker : Xiaochuan Tian, University of California, San Diego

Meshfree and particle methods are widely used in computational studies of partial differential equations, offering many advantages compared to traditional mesh- or grid-based numerical methods. Nevertheless, many practical questions revolve around meshfree methods, encompassing concerns about stability and accuracy. We propose a new paradigm of designing meshfree methods for solving partial differential equations (PDEs) through nonlocal analysis, inspired by the recent development of nonlocal calculus and its applications. We propose that the development of stable, accurate, and efficient meshfree methods relies on two key factors: (1) the formulation of well-posed continuum nonlocal models to approximate the PDE models and (2) the development of asymptotically compatible schemes for robust discretization of nonlocal models that allow a flexible coupling of the modeling and discretization parameters. We will review several aspects of nonlocal calculus and asymptotically compatible schemes and demonstrate the idea with a monotone meshfree method for solving linear elliptic equations in non-divergence form.

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