TBA

Internal waves are traveling waves that propagate along the interface dividing two immiscible fluids. In this talk, we discuss recent progress on rigorously constructing two related species of extreme internal waves: overturning bores and gravity currents. Extreme refers to the fact that there is a stagnation point on the interface, which allows for the boundary to be non-smooth. 

Hydrodynamic bores are front-type traveling wave solutions to the two-layer free boundary Euler equations in two dimensions. We  prove that there exists a family of solutions of this form that starts at  trivial solution where the interface is flat and continues until the interface develops a vertical tangent. This type of behavior was first observed over 40 years ago in computations of internal gravity waves and gravity water waves with vorticity via numerical continuation. Despite considerable progress over the past decade in constructing global families of water waves that potentially overturn, a rigorous proof that overturning definitively occurs has been stubbornly elusive.  

Gravity currents arise when a heavier fluid intrudes into a region of lighter fluid. Typical examples are  muddy water flowing into a cleaner body of water and haboobs (dust storms). We give the first rigorous proof of a conjecture of von Kármán on the structure of gravity currents near the rigid boundary. 

The Israel-Stewart theory models relativistic viscous fluids, with important applications in astrophysics and cosmology. In this talk, I will present recent progress on an Israel-Stewart type system with bulk viscosity in the presence of vacuum. By allowing vacuum, we introduce degeneracy near the boundary. In this case, the decay rates of fluid variables play a crucial role in solving the problem.

We focus on decay rates that ensure the boundary maintains a finite, nonzero acceleration-a condition we refer to as the physical vacuum boundary condition. This allows us to model physical phenomena such as star rotation. The core of the talk is on establishing the local well-posedness of the linearized system under these vacuum conditions. Classical hyperbolic theory fails due to the degeneracy near the free boundary, so we incorporate weights in our functional framework and derive weighted energy estimates to construct solutions.

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.

The fundamental matrix of a pair of pinhole cameras lies at the core of many systems that reconstruct 3D scenes from 2D images. However, for more than two cameras, the relations between the various fundamental matrices of camera pairs are not yet completely understood. In joint work with Viktor Korotynskiy, Anton Leykin, and Tomas Pajdla, we characterize all polynomial constraints that hold vanishing for an arbitrary triple of fundamental matrices. Unlike most constraints in previous works, our constraints hold independently of the relative scaling of of the fundamental matrices, which is unknown in practice. We also provide a partial characterization for essential matrix triples arising from calibrated cameras.

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.

The generalized phase retrieval problem over compact groups aims to recover a set of matrices, representing an unknown signal, from their associated Gram matrices, leveraging prior structural knowledge about the signal. This framework generalizes the classical phase retrieval problem, which reconstructs a signal from the magnitudes of its Fourier transform, to a richer setting involving non-abelian compact group. Our main result shows that for a suitable class of semi-algebraic priors, the generalized phase retrieval problem not only admits a unique solution (up to inherent group symmetries), but also satisfies a bi-Lipschitz property.  This implies robustness to both noise and model mismatch, an essential requirement for practical use, especially when measurements are severely corrupted by noise. 

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.