Abstract: The Hilbert scheme of d points on a smooth variety X, denoted by Hilb^d(X), is an important moduli space with connections to various fields, including combinatorics, enumerative geometry, and complexity theory, to name a few. In this talk, I will focus on the case where X is a threefold, as there are several open questions regarding its singularities. I will describe the structure of the smooth points of this Hilbert scheme and, time permitting, discuss the structure of the mildly singular points. This is all joint (ongoing) work with Joachim Jelisiejew and Alessio Sammartano.
Abstract: This talk is concerned with how diffusion generative models leverage (unknown) low-dimensional structure to accelerate sampling. Focusing on two mainstream samplers --- the denoising diffusion implicit model (DDIM) and the denoising diffusion probabilistic model (DDPM) --- and assuming accurate score estimates, we prove that their iteration complexities scale linearly in some intrinsic dimension of the target distribution. Our results apply to a broad family of target distributions without requiring smoothness or log-concavity assumptions. Our findings provide the first rigorous evidence for the low-dimensional adaptation ability of the DDIM-type samplers, and significantly improves over the state-of-the-art DDPM theory regarding total variation convergence.
Abstract: During the COVID-19 pandemic, variants constantly emerged and interacted with existing ones. Our data-driven research showed that a variant with a basic reproduction number as high as 10 can defy conventional theory. Motivated by this, the talk will present two works on the dynamics of epidemic systems with two strains that provide partial cross-immunity to each other.
In the first part, we challenge the validity of the exclusion principle at a limit in which one strain has a vast competitive advantage over the other strains. We show that when one strain is significantly more transmissible than the other, an epidemic system with partial cross-immunity can reach a stable endemic equilibrium in which both strains coexist with comparable prevalence. Thus, the competitive exclusion principle does not always apply.
The second part explores conditions under which a two-strain epidemic model with partial cross-immunity can lead to self-sustained oscillations. Contrary to previous findings, our results indicate that oscillations can occur even with weak cross-immunity and weak asymmetry. Using asymptotic methods, we reveal that the steady state of coexistence becomes unstable near specific curves in the parameter space, leading to oscillatory solutions for any basic reproduction number greater than one. Numerical simulations support our theoretical findings, highlighting an unexpected oscillatory region.
Abstract: In this talk, we will present a highly parallel and derivative-free martingale neural network method, based on the probability theory of Varadhan’s martingale formulation of PDEs,
to solve Hamilton-Jacobi-Bellman (HJB) equations arising from stochastic optimal control problems (SOCPs), as well as general quasilinear parabolic partial differential equations (PDEs).
In both cases, the PDEs are reformulated into a martingale problem such that loss functions will not require the computation of the gradient or Hessian matrix of the PDE solution, and can be computed in parallel in both time and spatial domains. Moreover, the martingale conditions for the PDEs are enforced using a Galerkin method realized with adversarial learning techniques, eliminating the need for direct computation of the conditional expectations associated with the martingale property. For SOCPs, a derivative-free implementation of the maximum principle for optimal controls is also introduced. The numerical results demonstrate the effectiveness and efficiency of the proposed method, which is capable of solving HJB and quasilinear parabolic PDEs accurately and fast in dimensions as high as 10,000.
Abstract: We introduce a family of intrinsic Gaussian noises on compact manifolds that we call “colored noise” on manifolds. With this noise, we study the parabolic Anderson model (PAM) on compact manifolds. Under some curvature conditions, we show the well-posedness of the PAM and provide some preliminary (but sharp) bounds on the second moment of the solution. It is interesting to see that global geometry enters the play in the well-posedness of the equation.
Abstract: Recently, there has been a surge of interest in hypothesis testing methods for combining dependent studies without explicitly assessing their dependence. Among these, the Cauchy combination test (CCT) stands out for its approximate validity and power, leveraging a heavy-tail approximation insensitive to dependence. However, CCT is highly sensitive to large p-values and inverting it to construct confidence regions can result in regions lacking compactness, convexity, or connectivity. In this talk, we will propose a "heavily right" strategy by excluding the left half of the Cauchy distribution in the combination rule, retaining CCT's resilience to dependence while resolving its sensitivity to large p-values.
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