Georgia Institute of Technology College of Sciences

Compressed sensing illustrates the possibility of acquiring and reconstructing sparse signals via underdetermined (linear) systems. It is believed that iid Gaussian measurement vectors give near optimal results, with the necessary number of measurements on the order of $s \log(n/s)$ - $n$ is ambient dimension and $s$ is the sparsity threshold. The recovery algorithm used above relies on a certain quasi-isometry property of the measurement matrix. A surprising result is that the same order of measurements gives an analogous quasi-isometry in the extreme quantization of one-bit sensing. Bylik and Lacey deliver this result as a consequence of a certain stochastic process on the sphere. We will discuss an alternative method that relies heavily on the VC-dimension of a class of subsets on the sphere.

Schistosoma is a parasitic worm that circulates between human and snail hosts. Multiple biological and ecological factors contribute to its spread and persistence in host populations. The infection is widespread in many tropical countries, and WHO has made control of schistosomiasis a priority among neglected tropical diseases.Mathematical modeling is widely used for prediction and control analysis of infectious agents. But host-parasite systems with complex life-cycles like Schistosoma, pose many challenges. The talk will outline the basic biology of Schistosoma, and the principles employed in mathematical modeling of macro parasites. We shall review conventional approaches to Schistosomiasis starting with the classical work of MacDonald, and discuss their validity and implications. Then we shall outline more detailed “stratified worm burden approach”, and show how combining mathematical and computer tools one can explore real-world systems and make reliable predictions for long term control outcomes and the problem of elimination.

I'll discuss joint work with my brother Jeff Giansiracusa in which we introduce an exterior algebra and wedge product in the idempotent setting that play for tropical linear spaces (i.e., valuated matroids) a very similar role as the usual ones do for vector spaces. In particular, by working over the Boolean semifield this gives a new perspective on matroids.

Topics: local Hausdorff dimension, local Hausdorff measure, diffusion on compact metric spaces, prospective further research.