Day 3, May 31 Conference Program

Session Chair: Alejandro Ramírez, NYU Shanghai

9:00-10:00 AM

Polynomial lower bound on the effective resistance for the one-dimensional critical long-range percolation

By Jian Ding, Peking University

In this talk, I will discuss one-dimensional critical long-range percolation and show that the effective resistance has a polynomial lower bound. This is based on joint work with Zherui Fan and Lujing Huang.

10:00-10:30 AM

Morning Tea Break

10:30-11:30 AM

Generative modeling with stochastic interpolants and Follmer processes

By Eric Vanden-Eijnden, New York University

Generative models based on dynamical transport have recently led to significant advances in unsupervised learning. At mathematical level, these models are primarily designed around the construction of a map between two probability distributions that transform samples from the first into samples from the second.  While these methods were first introduced in the context of image generation, they have found a wide range of applications, including in scientific computing where they offer interesting ways to reconsider complex problems once thought intractable because of the curse of dimensionality. In this talk, I will discuss the mathematical underpinning of generative models based on flows and diffusions, and show how a better understanding of their inner workings can help improve their design. I will also discussion connections with optimal transport, the Schrödinger bridge problem, and Föllmer processes. These results indicate how to structure the transport to best reach complex target distributions while maintaining computational efficiency, both at learning and sampling stages.  I will also discuss applications of generative AI in scientific computing, in particular in the context of Monte Carlo sampling, with applications to the statistical mechanics and Bayesian inference, as well as probabilistic forecasting, with application to fluid dynamics and atmosphere/ocean science.

11:30 AM-12:10 PM

Random walk on the simple symmetric exclusion process

By Daniel Kious, University of Bath

In this talk, I will overview works on random walks in dynamical random environments. I will recall a result obtained in collaboration with Hilário and Teixeira and then I will focus on a work with Conchon-Kerjan and Rodriguez.

Our main interest is to investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, with density ρ ∈ [0, 1] and rate γ > 0. At each jump, the random walker is subject to a drift that depends on whether it is sitting on top of a particle or a hole.

We prove that the speed of the walk, seen as a function of the density, exists for all density but at most one, and that it is strictly monotonic. We will explain how this helps understand the non-existence of transient regimes with zero speed. We will provide an outline of the proof, whose general strategy is inspired by techniques developed for studying the sharpness of strongly-correlated percolation models.

12:10 PM-1:30 PM

Lunch Break

1:30-2:30 PM

Discussion between alumni and current and prospective students (Hong-Bin Chen, Qiheng Fang, Jiaming Xia)

Afternoon Tea Break

Session Chair: Feng Zhou, East China Normal University

2:45-3:25 PM

Causal vs Structural Connectivity in Pulse-Coupled Neuronal Networks

By David McLaughlin, New York University / NYU Shanghai

Causal connectivity, as measured physiologically, is often used to infer network function, as well as to estimate the neuronal network’s underlying structural connectivity. Unfortunately, the inferred causal connectivity can depend upon the causality measure used. Moreover, this network’s structural connectivity may differ significantly from that estimated from causal connectivity. Here, we focus on nonlinear networks with pulse signals as measured output, e.g., spiking neuronal networks with spike output, and address the above issues based on four commonly utilized causality measures, i.e., time-delayed correlation coefficient, time-delayed mutual information, Granger causality, and transfer entropy. First, we show, for spike-output networks, precise relationships between the four causality measures to leading order in a perturbation expansion. Taking simulated Hodgkin–Huxley networks and a real mouse brain network as two illustrative examples, we verify the quantitative relations among the four causality measures and demonstrate that the causal connectivity inferred by any of the four well coincides with the underlying network’s structural connectivity, therefore illustrating a direct link between the causal and structural connectivity. We emphasize that the structural connectivity of spiking networks can be reconstructed pairwise without conditioning on the global information of all other nodes in a network, thus circumventing the curse of dimensionality. Our framework provides a practical and effective approach for the structural reconstruction of spiking networks.

3:25-3:45 PM

Afternoon Tea Break

3:45-4:25 PM

Symmetry Breaking Bifurcations in Fluid-Structure Interaction

By Jun Zhang, NYU Shanghai / New York University

In this talk, I will introduce two recent experiments on fluid-structure interactions. In both cases, symmetric solid boundaries are set to interact freely with their surrounding fluids. The experiments and the emergent dynamics are inspired by natural phenomena. In the first case, the free falling of snowflakes is investigated using artificial solid plates that descend in water, and their collective optical effects are discussed. The second experiment is inspired by a recent discovery on the ‘super-rotation’ of Earth’s solid core. In our table-top experiment, a free body takes the central axis of a symmetric cavity containing fluid undergoing thermal convection. A spontaneous, smooth, and persistent co-rotation of the free body and large-scale flow is observed. The mechanism that powers the co-rotation and its stochastic reversals are explained in some detail.

4:25-5:05 PM

Conditional optimal control

By Mathieu Laurière, NYU Shanghai

In this talk, we present a class of conditional optimal control problems introduced by Pierre-Louis Lions in his lectures at College de France. In these problems, the cost is not an expectation but a conditional expectation, conditioned on the fact that the process does not exit a given domain. We first study closed-loop Markovian controls and show that the solution can be described using a forward-backward system of partial differential equations reminiscent of the one arising in mean field control problems. Then, we consider open-loop controls and address a conjecture proposed by Pierre-Louis Lions: For a relaxed model with soft killing of the process, we show that the two classes of controls achieve the same optimal value. Numerical results will also be presented. This talk is based on joint works with Yves Achdou, Rene Carmona and Pierre-Louis Lions.