We introduce some of the basic objects in PCF theory like good (flat) points and exact upper bounds for sequences of functions in reduced products of regular cardinals. We will then give a proof of Shelah's 'Trichotomy' theorem.

I shall introduce local and integral diffusion processes, free boundary problems with and without memory, and discuss applications to American options and economics.

We consider product of two Cantor sets, and obtain the optimal estimates in terms of their thickness that guarantee that their product is an interval. This problem is motivated by the fact that the spectrum of the Labyrinth model, which is a two dimensional quasicrystal model, is given by the product of two Cantor sets. We also discuss the connection between our problem and the ”intersection of two Cantor sets” problem, which is a problem considered in several papers before.

Geometric apparatuses are frequently inundated with too much structural detail to be computationally tractable, while traditional topological tools often incur too much simplification of the original data to be practically useful. Persistent homology, a new branch of algebraic topology, is able to bridge the gap between geometry and topology. In this talk, I will discuss a few new developments in persistent homology. First, we introduce multiscale-multiresolution persistent homology to describe the topological fingerprints and topological transitions of nano-bio materials. Additionally, multidimensional persistence is developed for topological denoising and revealing the topology-function relationship in biomolecular data. Moreover, molecular topological fingerprints are utilized to resolve ill-posed inverse problems in cryo-EM structure determination. Finally, objective-oriented persistent homology is constructed via the variational principle and differential geometry for proactive feature extracting from big data sets, which leads to topological partial differential equations (TPDEs).