Georgia Institute of Technology College of Sciences

We will describe a recently discovered object, dual Lyapunov exponents, that has emerged as a powerful tool in the spectral analysis of  quasiperiodic operators with analytic potentials, leading to solutions of several long outstanding problems. Based on papers joint with L. Ge, J. You, and Q. Zhou

The existence of multi black hole solutions in General Relativity is one of the expectations from the final state conjecture, the analogue of soliton resolution. In this talk, I will present preliminary works in this direction via a semilinear model, the energy critical wave equation, in dimension 3. In particular, I show 1) an algorithm to construct approximate solutions to the energy critical wave equation that converge to a sum of solitons at an arbitrary polynomial rate in (t-r); 2) a robust method to solve the remaining error terms for the nonlinear equation. The methods apply to energy supercritical problems.

Thin liquid films flowing down vertical fibers spontaneously exhibit complex interfacial dynamics, leading to irregular wavy patterns and traveling liquid droplets. Such droplet dynamics are fundamental components in many engineering applications, including mass and heat exchangers for thermal desalination, as well as water vapor and particle capture. Recent experiments demonstrate that critical flow regime transitions can be triggered by varying inlet geometries and external fields. Similar interacting droplet dynamics have also been observed on hydrophobic substrates, arising from interfacial instabilities in volatile liquid films. In this talk, I will describe lubrication and weighted residual models for falling droplets. The coarsening dynamics of condensing droplets will be discussed using a lubrication model. I will also present our recent results on developing optimal boundary control and mean-field control for droplet dynamics. 

For a central simple algebra $A$ over a field $K$, there are two major invariants, viz., period and index. For a field $K$, the Brauer-$l$-dimension of $K$ for a prime number $l$, is the smallest natural number $d$ such that for every finite field extension $L/K$ and every central simple $L$-algebra $A$ (of period a power of $l$), we have that index($A$) divides period$(A)^d$.

If $K$ is a number field or a local field, then classical results from class field theory tell us that the Brauer-$l$-dimension of $K$ is 1. This invariant is expected to grow under a field extension, bounded by the transcendence degree. Some recent works in this area include that of Harbater-Hartmann-Krashen for $K$ a complete discretely valued field, in the good characteristic case. In the bad characteristic case, for such fields $K$, Parimala-Suresh have given some bounds.

Also, the u-invariant of $K$ is the maximal dimension of anisotropic quadratic forms over $K$. For example, the u-invariant of $\mathbb{C}$ is 1, for $F$ a non-real global or local field the u-invariant of $F$ is 1, 2, 4, or 8, etc.

In this talk, I will present similar bounds for the Brauer-l-dimension and the strong u-invariant of a complete non-Archimedean valued field $K$ with residue field $\kappa$.