Georgia Institute of Technology College of Sciences

We introduce a general Minimum Linear Ordering Problem (MLOP): Given a nonnegative set function f on a finite set V, find a linear ordering on V such that the sum of the function values for all the suffixes is minimized. This problem generalizes well-known problems such as the Minimum Linear Arrangement, Min Sum Set Cover, and Multiple Intents Ranking. Extending a result of Feige, Lovasz, and Tetali (2004) on Min Sum Set Cover, we show that the greedy algorithm provides a factor 4 approximate optimal solution when the cost function f is supermodular. We also present a factor 2 rounding algorithm for MLOP with a monotone submodular cost function, while the non monotone case remains wide open. This is joint work with Satoru Iwata and Pushkar Tripathi.

The theory of Groebner bases is the foundation of many algorithms in computational algebra. A Groebner basis is a special generating set of an ideal of polynomials. In this expository talk, I will introduce Groebner bases and explain how they can be used in integer programming. In particular, for an integer program, we can associate an ideal whose Groebner basis gives a set of directions that takes any feasible solution to an optimal solution.

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.