Eigenvalue Optimization and Minimal Surfaces Abstract. I will discuss a new method, developed in collaboration with M. Karpukhin, R. Kusner, and D. Stern, for constructing minimal surfaces embedded in the 3-sphere and the 3-ball, by equivariant eigenvalue optimization. A main … Read more
Crouzeix's Conjecture, Blaschke Products, and Numerical Ranges Abstract. Crouzeix’s Conjecture is an important open problem in operator theory that posits: for every square matrix A, the numerical range of A is a two spectral-set for A. In this talk, we’ll … Read more
On the global well-posedness of the Boltzmann hierarchy Abstract. The Boltzmann hierarchy is an infinite system of linear coupled equations that are instrumental for the rigorous derivation of the Boltzmann equation from many particles. In this talk we will show … Read more
Recovery of time-dependent coefficients in hyperbolic equations on Riemannian manifolds from partial data. Abstract. In this talk we discuss inverse problems of determining time-dependent coefficients appearing in the wave equation in a compact Riemannian manifold of dimension three or higher. … Read more
Spectral lines of general-relativistic hydrogen Abstract. After reviewing basics of the spectral theory for non-relativistic quantum-mechanical Hamiltonians of hydrogenic ions, I discuss what impact the incorporation of additional effects such as special relativity, anomalous magnetic moment for the electron, and … Read more
Mixed norm decoupling for paraboloids Abstract. In this talk we discuss mixed norm decoupling estimates for the paraboloid. One motivation for considering such an estimate is a conjectured mixed norm Strichartz estimate on the torus. This is joint work with … Read more
Nodal domain count for eigenfunctions of dumbbell domains Abstract. The eigenvalues and eigenfunctions of the Laplacian of a plate determine the constant frequencies at which the plate can vibrate. The zero set of the eigenfunction then corresponds to the curves … Read more
On the Donaldson-Scaduto conjecture. Abstract. Donaldson proposed a new method to study manifolds with special holonomy groups, particularly Calabi-Yau 3-folds and G2-manifolds, utilizing a Lefschetz fibration. Motivated by collapsing G2-manifolds with K3 fibrations in the adiabatic setting, Donaldson and Scaduto … Read more
The 2024 UNC PDE Mini-school will feature a series of lectures by Gigliola Staffilani from MIT. We will have additional complementary "satellite" talks by Gonzalo Cao-Labora (MIT), Felipe Hernandez (MIT), and Xueying Yu (Oregon State University). Analysis and PDE Mini-School Website … Read more
Gravity water waves with constant vorticity at low regularity and balanced energy estimates Abstract. In this talk I will talk about the Cauchy problem of two-dimensional gravity water waves with constant vorticity. The water waves system is a nonlinear dispersive … Read more
Resonances and transmission for waves in truncated media Abstract. In a lot of toy models in quantum mechanics, you can construct “states” that live only at certain ”energies” by trapping things into specific regions, say using an infinite square well … Read more
Interior Hessian estimates for singularities of the Lagrangian mean curvature flow Abstract. In this talk, we discuss the history of the Lagrangian Mean Curvature equation beginning with the special Lagrangian equation of Harvey and Lawson. We consider the hypercritical case … Read more
