The Sparse Time-Frequency Representation (STFR) method is based on the assumption that many physical signals inherently contain AM-FM representations. We propose a sparse optimization method to extract the AM-FM representations of such signals. We prove the convergence of the method for periodic signals under certain assumptions and provide practical algorithms specifically for the non-periodic STFR, which extends the method to tackle problems that former STFR methods could not handle, including stability to noise and non-periodic data analysis. This is a significant improvement since many adaptive and non-adaptive signal processing methods are not fully capable of handling non-periodic signals. Moreover, we propose a new STFR algorithm to study intrawave signals with strong frequency modulation and analyze the convergence of this new algorithm for periodic signals. Such signals have previously remained a bottleneck for all signal processing methods. Furthermore, we propose a modified version of STFR that facilitates the extraction of intrawaves that have overlaping frequency content. We show that the STFR methods can be applied to the realm of dynamical systems and cardiovascular signals. In particular, we present a simplified and modified version of the STFR algorithm that is potentially useful for the diagnosis of some cardiovascular diseases. We further explain some preliminary work on the nature of Intrinsic Mode Functions (IMFs) and how they can have different representations in different phase coordinates. This analysis shows that the uncertainty principle is fundamental to all oscillating signals.

Cancer often arises through a sequence of genetic alterations.  Each of these alterations may confer a fitness advantage to the cell, resulting in a clonal expansion.  To model this process we consider a generalization of the biased voter process on a lattice which incorporates successive mutations modulating individual fitness.  We will study the rate of mutant spread and accumulation of oncogenic mutations in this process.  We then investigate the geometry and extent of premalignant fields surrounding primary tumors, and evaluate how the risk of secondary tumors arising from these fields may depend on the cancer progression pathway and tissue type.  (joint work w/K. Leder, R. Durrett, and M. Ryser).

We present a case study of the large-scale “fractal” behavior of concrete families of random processes that arise in complex systems. Among other things we will exhibit two random functions both of which are “multifractal” on large scales, but only one of which shows “intermittency.” This contradicts  the commonly-held view that “multifractality” and “intermittency” can be used interchangeably. This is based on joint work with K. Kim and Y. Xiao.