Georgia Institute of Technology College of Sciences

Continuation of the exposition of N. Fenichel classical paper on the persistence of Normally Hyperbolic invariant manifolds.

More than 125 years ago Osborne Reynolds launched the quantitative study of turbulent transition as he sought to understand the conditions under which fluid flowing through a pipe would be laminar or turbulent. Since laminar and turbulent flow have vastly different drag laws, this question is as important now as it was in Reynolds' day. Reynolds understood how one should define "the real critical value'' for the fluid velocity beyond which turbulence can persist indefinitely. He also appreciated the difficulty in obtaining this value. For years this critical Reynolds number, as we now call it, has been the subject of study, controversy, and uncertainty. Now, more than a century after Reynolds pioneering work, we know that the onset of turbulence in shear flows is properly understood as a statistical phase transition. How turbulence first develops in these flows is more closely related to the onset of an infectious disease than to, for example, the onset of oscillation in the flow past a body or the onset of motion in a fluid layer heated from below. Through the statistical analysis of large samples of individual decay and proliferation events, we at last have an accurate estimate of the real critical Reynolds number for the onset of turbulence in pipe flow, and with it, an understanding of the nature of transitional turbulence. This work is joint with: K. Avila, D. Moxey, M. Avila, A. de Lozar, and B. Hof.

We have recently witnessed the tremendous progress in evolutionary and regulatory genomics of eukaryotes fueled by hundreds of sequenced eukaryotic genomes, including human and dozens of animal and plant genomes and culminating in the recent release of The Encyclopedia of DNA Elements (ENCODE) project. Yet, many interesting questions about the functional and structural organization of the genomic elements and their evolution remain unsolved. Computational genomics methods have become essential in addressing these questions working with the massive genomic data. In this presentation, I will talk about two interesting open problems in computational genomics. The first problem is related to identifying and characterizing long identical multispecies elements (LIMEs), the genomic regions that were slowed down through the course of evolution to their extremes. I will discuss our recent findings of the LIMEs shared across six animal as well as six plant genomes and the computational challenges associated with expanding our results towards other species. The second problem is finding genome rearrangements for a group of genomes. I will present out latest approach approach that brings together the idea of symbolic object representation and stochastic simulation of the evolutionary graphs.

We will continue discussing co-transcriptional RNA folding, and the potential for trap models to capture these dynamics.