Georgia Institute of Technology College of Sciences

Two recent extensions of optimal mass transport theory will be covered. In the first part of the talk, we will discuss measure-valued spline, which generalizes the notion of cubic spline to the space of distributions. It addresses the problem to smoothly interpolate (empirical) probability measures. Potential applications include time sequence interpolation or regression of images, histograms or aggregated datas. In the second part of the talk, we will introduce matrix-valued optimal transport. It extends the optimal transport theory to handle matrix-valued densities. Several instances are quantum states, color images, diffusion tensor images and multi-variate power spectra. The new tool is expected to have applications in these domains. We will focus on theoretical side of the stories in both parts of the talk.

Let $\nu$ denote the maximum size of a packing of edge-disjoint triangles in a graph $G$. We can clearly make $G$ triangle-free by deleting $3\nu$ edges. Tuza conjectured in 1981 that $2\nu$ edges suffice, and proved it for planar graphs. The best known general bound is $(3-\frac{3}{23})\nu$ proven by Haxell in 1997. We will discuss this proof and some related results.

If $f$ is a function supported on a truncated paraboloid, what can we say about $Ef$, the Fourier transform of f? Stein conjectured in the 1960s that for any $p>3$, $\|Ef\|_{L^p(R^3)} \lesssim \|f\|_{L^{\infty}}$.

We make a small progress toward this conjecture and show that it holds for $p> 3+3/13\approx 3.23$. In the proof, we combine polynomial partitioning techniques introduced by Guth and the two ends argument introduced by Wolff and Tao.