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: We will discuss two recent results on positive scalar curvature on circle bundles. In the context of trivial circle bundles over 4-manifolds, we consider two problems: Gromov's width inequality conjecture, and Rosenberg's S^1-stability conjecture. Both conjectures have counterexamples in dimension 4 based on Seiberg-Witten invariants. Nevertheless, we show that in the simply connected case both of these results are true upon considering 4-manifolds up to homeomorphism. We also obtain a result up to stabilization in the non-simply connected case. In the context of nontrivial bundles, we construct infinitely many examples of PSC circle bundles over enlargeable manifolds in all dimensions greater than 3. This answers a question of Gromov. We shall emphasize some analogies between symplectic geometry and positive scalar curvature that we encountered in the process of finding these results. Based on joint work with B. Sen.
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, I will describe joint work with Langte Ma studying dimension reduction phenomena for absolute minimizers of the Yang-Mills functional. This work is motivated by special holonomy geometry and I will emphasize applications to gauge theory on special holonomy manifolds. I will explain the general approach to such phenomena that we develop, characterizing the moduli space of generalized anti-self-dual instantons on certain bundles over product Riemannian manifolds equipped with a parallel codimension-4 differential form. One outcome of this is an explicit description of instanton moduli spaces over certain product special holonomy manifolds.
Abstract: We study the model theory of the Farey graph $F$ by realizing it as the generic of a smooth class $(K,\leq)$. By varying the relation $\leq$ we may obtain distinct generics that are either atomic or saturated. This will allow us to demonstrate a quantifier elimination for $Th(F)$. The Farey graph is also the simplest nontrivial curve complex of a surface, where $F=C(\Sigma_{1,1})$. Modifications of this technique to obtain results for the general model theory of the curve complex $C(\Sigma_{g,n})$ will be discussed.
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: One of the oldest problems in dynamical systems is the stability of the Solar System. That is, consider N bodies moving following Newton's law of gravitation, one of them with large mass (the Sun) and the others with small masses (the planets). If one neglects, the gravitational interaction between planets, the classical Kepler's laws assert that the planets move on ellipses. Then, one wants to understand whether the effect of the planet's mutual attraction causes long term changes on the shape and relative position of the Keplerian ellipses. In this talk I will explain how to construct unstable motions in a planetary 4 body problem, which lead to drastic changes in the semimajor axes, eccentricity and inclination of these ellipses. The results are based on joint work with Andrew Clarke and Jacques Fejoz.
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.