Georgia Institute of Technology College of Sciences

The hyperplane conjecture of Kannan, Lovasz and Simonovits asserts that the isoperimetric constant of a logconcave measure (minimum surface to volume ratio over all subsets of measure at most half) is approximated by a halfspace to within an absolute constant factor. I will describe the motivation, implications and some developments around the conjecture and an approach to resolving it (which does not seem entirely ridiculous).

From its physical origin, the viscosity and heat conductivity in compressible fluids depend on absolute temperature through power laws. The mathematical theory on the well-posedness and regularity on this setting is widely open. I will report some recent progress made on this direction, with emphasis on the lower bound of temperature, and global existence of solutions in one or multiple dimensions. The relation between thermodynamics laws and Navier-Stokes equations will also be discussed. This talk is based on joint works with Weizhe Zhang.

A common practice in aerospace engineering has been to carry out deterministicanalysis in the design process. However, due to variations in design condition suchas material properties, physical dimensions and operating conditions; uncertainty isubiquitous to any real engineering system. Even though the use of deterministicapproaches greatly simplifies the design process since any uncertain parameter is setto a nominal value, the final design can have degraded performance if the actualparameter values are slightly different from the nominal ones.Uncertainty is important because designers are concerned about performance risk.One of the major challenges in design under uncertainty is computational efficiency,especially for expensive numerical simulations. Design under uncertainty is composedof two major parts. The first one is the propagation of uncertainties, and the otherone is the optimization method. An efficient approach for design under uncertaintyshould consider improvement in both parts.An approach for robust design based on stochastic expansions is investigated. Theresearch consists of two parts : 1) stochastic expansions for uncertainty propagationand 2) adaptive sampling for Pareto front approximation. For the first part, a strategybased on the generalized polynomial chaos (gPC) expansion method is developed. Acommon limitation in previous gPC-based approaches for robust design is the growthof the computational cost with number of uncertain parameters. In this research,the high computational cost is addressed by using sparse grids as a mean to alleviatethe curse of dimensionality. Second, in order to alleviate the computational cost ofapproximating the Pareto front, two strategies based on adaptive sampling for multi-objective problems are presented. The first one is based on the two aforementionedmethods, whereas the second one considers, in addition, two levels of fidelity of theuncertainty propagation method.The proposed approaches were tested successfully in a low Reynolds number airfoilrobust optimization with uncertain operating conditions, and the robust design of atransonic wing. The gPC based method is able to find the actual Pareto front asa Monte Carlo-based strategy, and the bi-level strategy shows further computationalefficiency.

This work studies two topics in sequence analysis. In the first part, we investigate the large deviations of the shape of the random RSK Young diagrams, associated with a random word of size n whose letters are independently drawn from an alphabet of size m=m(n). When the letters are drawn uniformly and when both n and m converge together to infinity, m not growing too fast with respect to n, the large deviations of the shape of the Young diagrams are shown to be the same as that of the spectrum of the traceless GUE. Since the length of the top row of the Young diagrams is the length of the longest (weakly) increasing subsequence of the random word, the corresponding large deviations follow. When the letters are drawn with non-uniform probability, a control of both highest probabilities will ensure that the length of the top row of the diagrams satisfies a large deviation principle. In either case, speeds and rate functions are identified. To complete this first part, non-asymptotic concentration bounds for the length of the top row of the diagrams are obtained. In the second part, we investigate the order of the r-th, 1\le r < +\infty, central moment of the length of the longest common subsequence of two independent random words of size n whose letters are identically distributed and independently drawn from a finite alphabet. When all but one of the letters are drawn with small probabilities, which depend on the size of the alphabet, the r-th central moment is shown to be of order n^{r/2}. In particular, when r=2, the order of the variance is linear.