Abstract: The main goal of this workshop is to bring together researchers from homogeneous dynamics, Teichmüller dynamics and closely related fields to discuss recent developments in these areas. Of particular interest are the subjects of mixing, equidistribution, rigidity, flexibility, and applications, both from the qualitative and quantitative points of view.
Abstract: Hassett showed that there are natural reduction morphisms between moduli spaces of weighted pointed stable curves when we reduce weights. I will discuss some joint work with Ziquan Zhuang which constructs similar morphisms between moduli of stable pairs in higher dimension.
Abstract: We formulate and analyze ODE and PDE models for epidemiology which incorporate human behavioral concerns. Specifically, we assume that as a disease spreads and a governing body implements non-pharmaceutical intervention methods, there is a portion of the population that does not comply with these mandates and that this noncompliance has a nontrivial effect on the spread of the disease. Borrowing from social contagion theory, we then allow this noncompliance to spread parallel to the disease. We derive reproductive ratios and large time asymptotics for our models and demonstrate their behavior with simulations.
Abstract: Score-based diffusion models (SBDM) have recently emerged as state-of-the-art approaches for image generation. We develop SBDMs in the infinite-dimensional setting, that is, we model the training data as functions supported on a rectangular domain. Besides the quest for generating images at ever higher resolution, our primary motivation is to create a well-posed infinite-dimensional learning problem so that we can discretize it consistently at multiple resolution levels. We demonstrate how to overcome shortcomings of current SBDM approaches in the infinite-dimensional setting by ensuring the well-posedness of forward and reverse processes and derive the convergence of the approximation of multilevel training. We implement an infinite-dimensional SBDM approach and illustrate that approximating the score function with an operator network is beneficial for multilevel training.
Abstract: The Fourier coefficients of theta functions have featured prominently in numerous number theory applications and constructions in the Langlands program. For example, they play an important role in the recent work of Friedberg-Ginzburg generalizing the theta correspondence to higher covering groups. For their construction one wants to know the wavefront set of the theta representations, i.e. the largest nilpotent orbit with nonvanishing Fourier coefficient.
To investigate these Fourier coefficients, it can be valuable to study the analogous local question. In this talk we consider local depth 0 theta representations and describe how to compute their stable wavefront set. This is joint work with Emile Okada and Runze Wang.
Abstract: We find a natural four-dimensional analog of the moderate deviation results for the capacity of the random walk, which corresponds to Bass, Chen, and Rosen concerning the volume of the random walk range for $d=2$. We find that the deviation statistics of the capacity of the random walk can be related to the optimal constant of generalized Gagliardo-Nirenberg inequalities.
Abstract: Approximate message passing (AMP) emerges as an effective iterative algorithm for solving high-dimensional statistical problems. However, prior AMP theory, which focused mostly on high-dimensional asymptotics, fell short of predicting the AMP dynamics when the number of iterations surpasses o(log n / log log n) (with n the problem dimension). To address this inadequacy, this talk introduces a non-asymptotic framework towards understanding AMP. Built upon a new decomposition of AMP updates in conjunction with well-controlled residual terms, we lay out an analysis recipe to characterize the finite-sample convergence of AMP up to O(n / polylog(n)) iterations. We will discuss concrete consequences of the proposed analysis recipe in the Z2 synchronization problem; more specifically, we predict the behavior of randomly initialized AMP for up to O(n/poly(\log n)) iterations, showing that the algorithm succeeds without the need of a careful spectral initialization and also a subsequent refinement stage (as conjectured recently by Celentano et al.)