Georgia Institute of Technology College of Sciences

We already know that the Euclidean unit ball is at the center of the Banach-Mazur compactum, however its structure is still being explored to this day. In 1987, Szarek and Talagrand proved that the maximum distance $R_{\infty} ^n$ between an arbitrary $n$-dimensional normed space and $\ell _{\infty} ^n$, or equivalently the maximum distance between a symmetric convex body in $\mathbb{R} ^n$ and the $n$-dimensional unit cube is bounded above by $c n^{7/8}$. In this talk, we will discuss a related paper by A. Giannopoulos, "A note to the Banach-Mazur distance to the cube", where he proves that $R_{\infty} ^n < c n^{5/6}$.

We give derivative estimates for solutions to divergence form elliptic equations with piecewise smooth coefficients. The novelty of these estimates is that, even though they depend on the shape and on the size of the surfaces of discontinuity of the coefficients, they are independent of the distance between these surfaces.

In recent years, metamaterials have drawn a great deal of attention in the scientific community due to their unusual properties and useful applications. Metamaterials are artificial materials made of subwavelength microstructures. They are well known to exhibit exotic properties and could manipulate wave propagation in a way that is impossible by using nature materials.In this talk, I will present our recent works on membrane-type acoustic metamaterials (AMMs). First, I will talk about how to achieve near-zero density/index AMMs using membranes. We numerically show that such an AMM can be utilized to achieve angular filtering and manipulate wave-fronts. Next, I will talk about the design of an acoustic complimentary metamaterial (CMM). Such a CMM can be used to acoustically cancel out aberrating layers so that sound transmission can be greatly enhanced. This material could find usage in transcranial ultrasound beam focusing and non-destructive testing through metal layers. I will then talk about our recent work on using membrane-type AMMs for low frequency noise reduction. We integrated membranes with honeycomb structures to design simultaneously lightweight, strong, and sound-proof AMMs. Experimental results will be shown to demonstrate the effectiveness of such an AMM. Finally, I will talk about how to achieve a broad-band hyperbolic AMM using membranes.

We explain the (classical) transverse Markov Theorem which relates transverse links in the tight three sphere to classical braid closures. We review an invariant of such transverse links coming from knot Floer homology and discuss some applications which appear in the literature.