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.

Fractal sets arise naturally when classical notions of length, area, and smoothness are no longer adequate. A central theme in geometric measure theory is to understand how the dimension and geometry of a fractal set are reflected in its projections. Even some of the simplest questions of this type lead to surprisingly difficult problems connecting geometric measure theory with harmonic analysis, combinatorics, and number theory.

One classical example is the Favard length problem, also known as Buffon's needle problem. Given a planar set, its Favard length is the average length of its orthogonal projections. A theorem of Besicovitch implies that a purely unrectifiable set of finite length has Favard length zero. For self-similar sets obtained as limits of finite Cantor constructions, however, a much more quantitative question remains: how quickly does the Favard length of the finite approximations tend to zero? Despite the simple formulation, determining the correct decay rate remains a major open problem.

I will describe the geometry behind the classical problem, some of the main ideas that have led to progress, and what happens when linear projections are replaced by nonlinear families. This leads naturally to a nonlinear analogue of Buffon's needle problem—a Buffon circle problem—and to new questions about unions of circles, nonlinear projections, and the geometry of fractal sets. I will discuss recent results in this direction and the new phenomena that appear when lines are replaced by curves.

 We develop adaptive Chebyshev and Fourier-Chebyshev spectral collocation methods for the nonlinear prescribed mean curvature equations used to model the height of a fluid interface.  The non-linearity is treated with a Newton’s method.  Various geometries and symmetries are considered, and the resulting methods are used to solve the free boundary problems that arise from the equilibrium of a floating drop.

Motivated by applications and the classical theory of A-discriminants of Gel'fand, Kapranov, and Zelevinsky, we develop the basic theory of discriminants of multivariate reciprocal polynomials. These reciprocal discriminants parameterize reciprocal polynomials that define a singular hypersurface. We show that a reciprocal discriminant has a rich combinatorial structure. It is a reducible hypersurface whose components are dual varieties to Chebyshev subvarieties which correspond to certain layers in a toric arrangement associated to the exponents of the reciprocal polynomials. The layers which contribute correspond to certain matroidal decompositions of the exponents.

This is joint work with Trevor Karn and Simon Telen.