We present a rigorous framework for determining the equilibrium configurations of uniformly rotating, self-gravitating fluid bodies. This work addresses the classical challenge of modeling rotational deformation in celestial objects such as stars and planets. By integrating foundational theory with modern mathematical tools, we develop a unified formalism that enhances the precision and generality of shape modeling in astrophysical contexts. Our method applies Lie group theory to vector flows and solves functional equations using the Neumann series. We extend Clairaut's classical linear perturbation theory into the nonlinear regime via Lie exponential mapping, yielding a system of nonlinear functional equations for gravitational potential and fluid density. These are analytically tractable using shift operators and Neumann series summation, enabling explicit characterization of density and gravitational perturbations. This leads to an exact nonlinear differential equation for the shape function, describing equilibrium deformation without assuming slow rotation. We validate the framework through exact solutions, including the Maclaurin spheroid, Jacobi ellipsoid, and unit-index polytrope. We also introduce spectral decomposition techniques for analyzing radial harmonics and gravitational perturbations. Using Wigner's formalism for angular momentum addition, we compute higher-order nonlinear corrections efficiently. The framework includes boundary conditions for Legendre harmonics, supporting the derivation of nonlinear Love numbers and gravitational multipole moments. This work offers a comprehensive, non-perturbative approach to modeling rotational and tidal deformations in astrophysical and planetary systems.

This talk is the continuation of the first lecture, given in preceding week in the Differential Equations Seminar.

We present a rigorous framework for determining the equilibrium configurations of uniformly rotating, self-gravitating fluid bodies. This work addresses the classical challenge of modeling rotational deformation in celestial objects such as stars and planets. By integrating foundational theory with modern mathematical tools, we develop a unified formalism that enhances the precision and generality of shape modeling in astrophysical contexts. Our method applies Lie group theory to vector flows and solves functional equations using the Neumann series. We extend Clairaut's classical linear perturbation theory into the nonlinear regime via Lie exponential mapping, yielding a system of nonlinear functional equations for gravitational potential and fluid density. These are analytically tractable using shift operators and Neumann series summation, enabling explicit characterization of density and gravitational perturbations. This leads to an exact nonlinear differential equation for the shape function, describing equilibrium deformation without assuming slow rotation. We validate the framework through exact solutions, including the Maclaurin spheroid, Jacobi ellipsoid, and unit-index polytrope. We also introduce spectral decomposition techniques for analyzing radial harmonics and gravitational perturbations. Using Wigner's formalism for angular momentum addition, we compute higher-order nonlinear corrections efficiently. The framework includes boundary conditions for Legendre harmonics, supporting the derivation of nonlinear Love numbers and gravitational multipole moments. This work offers a comprehensive, non-perturbative approach to modeling rotational and tidal deformations in astrophysical and planetary systems.

This is the first of two lectures. The second part will be held in seminar on the following week.

Diffusion-based generative models have become the standard for image generation. ODE-based samplers and flow matching models improve efficiency, in comparison to diffusion models, by reducing sampling steps through learned vector fields. However, the theoretical foundations of flow matching models remain limited, particularly regarding the convergence of individual sample trajectories at terminal time - a critical property that impacts sample quality and being critical assumption for models like the consistency model. In this paper, we advance the theory of flow matching models through a comprehensive analysis of sample trajectories, centered on the denoiser that drives ODE dynamics. We establish the existence, uniqueness and convergence of ODE trajectories at terminal time, ensuring stable sampling outcomes under minimal assumptions. Our analysis reveals how trajectories evolve from capturing global data features to local structures, providing the geometric characterization of per-sample behavior in flow matching models. We also explain the memorization phenomenon in diffusion-based training through our terminal time analysis. These findings bridge critical gaps in understanding flow matching models, with practical implications for sampling stability and model design.

Davies's efficient covering theorem states that we can cover any measurable set in the plane by lines without increasing the total measure. This result has a dual formulation, known as Falconer's digital sundial theorem, which states that we can construct a set in the plane to have any desired projections, up to null sets. The argument relies on a Venetian blind construction, a classical method in geometric measure theory. In joint work with Alex McDonald and Krystal Taylor, we study a variant of Davies's efficient covering theorem in which we replace lines with curves. This has a dual formulation in terms of nonlinear projections.

In this talk we will introduce the notion of a Kaplansky devissage and show its key properties. We will use these properties to prove that any projective module is some direct sum of countably generated projective submodules. We then define the direct sum property for modules and show that any countably generated module with the direct sum property is free. We finally show that any projective module over a local ring has the direct sum property letting us conclude that all projective modules over local rings are free.

Diffusion model is an emerging generative modeling technique, achieving the state-of-the-art performances in image and video synthesis, scientific simulation, inverse problems, and offline reinforcement learning. Yet, existing statistical analysis of diffusion models often requires restrictive theoretical assumptions or is suboptimal. In this talk, we present two recent works from our group toward closing these gaps between diffusion models and the theoretical limits in standard nonparametric and high-dimensional statistical settings, and discuss some future directions.

1) For subGaussian distributions on $\mathbb{R}^d$ with $\beta$-Hölder smooth densities ($\beta\le 2$), we show that the sampling distribution of diffusion models can be minimax optimal under the total variation distance, even if the score is learned from noise perturbed training samples with noise multiplicity equal to one and without density lower bound assumptions.

2) For conditional sampling under i.i.d. priors and noisy linear observations, we show that diffusion models (also known as stochastic localization) can successfully sample from the posterior distribution, provided the signal-to-noise ratio exceeds a computational threshold predicted by prior work on approximate message passing (Barbier et al., 2020). This improves previous thresholds established in the stochastic localization literature, and enhances the sampling accuracy of dominant noisy inverse sampling techniques used in machine learning - albeit in a stylized theoretical model.

(Based on works with Kaihong Zhang, Heqi Yin, Feng Liang arXiv: 2402.15602, and with Han Cui, and Zhiyuan Yu arXiv: 2407.10763)

Group synchronization is a mathematical framework used in a variety of applications, such as computer vision, to situate a set of objects given their pairwise relative positions and orientations subjected to noise. More formally, synchronization estimates a set of group elements given some of their noisy pairwise ratios. In this talk I will present an entirely new view on the task of group synchronization by considering the natural higher-order structures that relate the relative orientations of triples or n-wise sets of objects. Examples of these structures include triples of so-called ‘common lines’ in cryo-EM and trifocal tensors in multi-view geometry. Thus far very little mathematical or computational work has explored synchronizing these higher-order measurements. I will introduce the problem of higher-order group synchronization and discuss the formal foundations of synchronizability in this setting. Then I will present a message passing algorithm to solve the problem and compare its performance to classical pairwise synchronization algorithms.