Georgia Institute of Technology College of Sciences

What is the minimum/maximum size of a set $A$ of integers that has the property that every integer in $\{1,2,\cdots, n\}$ can be written in at least/at most $g$ ways as a difference of elements of $A$? For the first question, we show that the limit of this minimum size divided by $\sqrt{n}$ exists and is nonzero, answering a question of Kravitz. For the second question, we give an asymptotic formula for the maximum size. We also consider the same problems but in the setting of a vector space over a finite field. During the talk we will discuss open problems and connections to coding theory and graph theory. This is joint work with Eric Schmutz.

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 $\kappa$.

We discuss an approach to estimate aggregation and adaptive estimation based upon (nearly optimal) testing of convex hypotheses. The proposed adaptive routines generalize upon now-classical Goldenshluger-Lepski adaptation schemes, and, in the situation where the observations stem from simple observation schemes (i.e., have Gaussian, discrete and Poisson distribution) and where the set of unknown signals is a finite union of convex and compact sets, attain nearly optimal performance. As an illustration, we consider application of the proposed estimates to the problem of recovery of unknown signal known to belong to a union of ellitopes in Gaussian observation scheme. The corresponding numerical routines can be implemented efficiently when the number of sets in the union is “not very large.” We illustrate “practical performance” of the method in an example of estimation in the single-index model.