Georgia Institute of Technology College of Sciences

Astronomy is arguably the oldest scientific discipline. Precise measurements of the motion of celestial bodies date back to the ancient Babylonians, Chinese, Greeks, and indigenous peoples outside Eurasia. Starting in the 19th century, systematic applications of physical principles to the formation and dynamics of stars marked the birth of astrophysics as a subfield of physics. Present-day astrophysics employs an array of theoretical and observational tools to construct sophisticated and predictive models of the origin, evolution, and death of stars.

While stars can be largely described within Newtonian physics, some of their most interesting properties, such as bounds on their mass-radius ratio, their potential collapse into a black hole, or effects of viscosity on gravitational waves emitted by mergers of neutron stars, can only be studied via applications of general relativity. Moreover, as a matter of principle, we ought to be able to fully understand stars as general-relativistic phenomena. The mathematical treatment of stars within general relativity, however, has lagged behind. Little progress has been made on this front since the discovery of the Tolman-Oppenheimer-Volkoff (TOV) equations and the Oppenheimer-Snyder solution in the late 1930s. The TOV equations describe a static (i.e., time independent), perfectly spherically symmetric star, whilst the latter describes the collapse of a perfectly spherically symmetric star with no pressure into a black hole. Despite being landmark results in general relativity, both situations are highly idealized. Inferences about generic properties of general-relativistic stars derived from such models are, therefore, a priori unjustified.

In this talk, I will discuss the problem of formulating a sound mathematical theory of general-relativistic star evolution based on the Einstein-Euler system. After setting up the problem, I will explain its main challenges, but also discuss the rich physics and mathematics involved in its study. A fundamental difficulty involves understanding the mathematics of the fluid-vacuum interface which separates the body of the star from vacuum. This interface displays singular behavior which is not amenable to current mathematical techniques. This difficulty, however, can be circumvented if we consider stars that are spherically symmetric but not static. The resulting evolution problem corresponds to a dynamic (i.e., time-dependent) generalization of the TOV equations.

This is joint work with Jared Speck.

We discuss a novel class of swarm-based gradient descent (SBGD) methods for nonconvex optimization. Each agent in the swarm is characterized by its position and mass. 

The dynamics combines two mechanisms: persistent transfer of mass from agents positioned on “higher ground” to those with lower objective values, and a mass-dependent time-stepping protocol. This coupling creates a dynamic distinction between “leaders” and “explorers.” Heavier agents act as leaders, use small time steps to refine promising regions near local minima, while light agents take larger steps, exploring the landscape for lower objective values. The swarm dynamics adaptively balances exploitation of local refinement with global exploration. 

We present convergence results and numerical experiments illustrating the effectiveness of SBGD for global optimization.

How close is a random matrix to being singular? This question measures the stability of random linear systems. Classical models treat every direction equally, but in many settings different directions have different scales or variances. We study what happens when a random matrix is deformed by a fixed invertible matrix. Even a diagonal deformation raises a basic question: is stability controlled by the weakest direction, or by the collective geometry of all directions? I will explain why standard estimates miss this geometry and how random subspaces lead to sharper bounds.

The abelian Yang-Mills-Higgs model on (1 + 2)-dimensional Minkowski space is a classical relativistic field theory, where a scalar complex field is coupled to an electromagnetic field. It admits topological soliton solutions called vortices. We present a proof of the asymptotic stability of the degree-one vortex under equivariant perturbations in the self-dual case.