Georgia Institute of Technology College of Sciences

Shear flow instability is a classical problem in hydrodynamics. In particular, it is important for understanding the transition from laminar to turbulent flow.  First, I will describe some results on shear flow instability in the setting of inviscid flows in a rigid wall. Then the effects of a free surface (or water waves) and viscosity will be discussed.

In the ocean, coherent vortices account for a large portion of the ocean turbulent kinetic energy and their presence significantly affects the dynamics and the statistical properties of ocean flows, with important consequences on transport processes. Mesoscale vortices also affect the population dynamics of phyto- and zooplankton, and are associated with secondary currents responsible for localized vertical fluxes of nutrients. The fact that the nutrient fluxes have a fine spatial and temporal detail, generated by the eddy field, has important consequences on primary productivity and the horizontal velocity field induced by the eddies has been suggested to play an important role in determining plankton patchiness. Owing to their trapping properties, vortices can also act as shelters for temporarily less-favoured planktonic species. In this contribution, I will review some of the transport properties associated with coherent vortices and their impact on the dynamics of planktoni ecosystems, focusing on the simplified conceptual model provided by two-dimensional turbulence.

I will explain and prove a beautiful and useful theorem of Alon and Tarsi that uses multivariate polynomials to guarantee, under suitable hypotheses, the existence of a coloring of a graph. The proof method, sometimes called a Combinatorial Nullstellensatz, has other applications in graph theory, combinatorics and number theory.

A common subsequence of two sequences X and Y is a sequence which is a subsequence of X as well as a subsequence of Y. A Longest Common Subsequence (LCS) of X and Y is a common subsequence with maximal length. Longest Common subsequences can be represented as alignments with gaps where the aligned letter pairs corresponds to the letters in the LCS. We consider two independent i.i.d.  binary texts X and Y of length n. We show that the behavior of the the alignment corresponding to the LCS is very different depending on the number of colors.  With 2-colors, long blocks tend to be aligned with no gaps, whilst for four or more colors the opposite is true. Let Ln denote the length of the LCS of X and Y.  In general the order of the variance of Ln is not known. We explain how a biased affect of a finite pattern can influence the order of the fluctuation of Ln.

Let A be a set of n real numbers. A central problem in additive combinatorics, due to Erdos and Szemeredi, is that of showing that either the sumset A+A or the product set A.A, must have close to n^2 elements. G. Elekes, in a short and brilliant paper, showed that one can give quite good bounds for this problem by invoking the Szemeredi-Trotter incidence theorem (applied to the grid (A+A) x (A.A)). Perhaps motivated by this result, J. Solymosi posed the following problem (actually, Solymosi's original problem is slightly different from the formulation I am about to give). Show that for every real c > 0, there exists 0 < d < 1, such that the following holds for all grids A x B with |A| = |B| = n sufficiently large: If one has a family of n^c lines in general position (no three meet at a point, no two parallel), at least one of them must fail to be n^(1-d)-rich -- i.e. at least one of then meets in the grid in fewer than n^(1-d) points. In this talk I will discuss a closely related result that I and Evan Borenstein have proved, and will perhaps discuss how we think we can use it to polish off this conjecture of Solymosi.