Georgia Institute of Technology College of Sciences

We describe sufficient conditions which guarantee that a finite set of mapping classes generate a right-angled Artin subgroup quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to Teichmuller space is a quasi-isometric embedding for both of the standard metrics. This is joint work with Chris Leininger and Johanna Mangahas.

The Southeast Geometry Seminar is a series of semiannual one-day events focusing on geometric analysis. These events are hosted in rotation by the following institutions: The University of Alabama at Birmingham;  The Georgia Institute of Technology;  Emory University;  The University of Tennessee Knoxville.  The following five speakers will give presentations on topics that include geometric analysis, and related fields, such as partial differential equations, general relativity, and geometric topology. Catherine Williams (Columbia U);  Hugh Bray (Duke U);  Simon Brendle (Stanford U);  Spyros Alexakis (U of Toronto);  Alessio Figalli (U of Texas at Austin).   There will also be an evening public lecture by plenary speaker Hugh Bray (Duke U) entitled From Black Holes and the Big Bang to Dark Energy and Dark Matter: Successes of Einstein's Theory of Relativity.

In school, we learned that fluid flow becomes simple in two limits. Over long lengthscales and at high speeds, inertia dominates and the motion can approach that of a perfect fluid with zero viscosity. On short lengthscales and at slow speeds, viscous dissipation is important. Fluid flows that correspond to the formation of a finite-time singularity in the continuum description involve both a vanishing characteristic lengthscale and a diverging velocity scale. These flows can therefore evolve into final limits that defy expectations derived from properties of their initial states. This talk focuses on 3 familiar processes that belong in this category: the formation of a splash after a liquid drop collides with a dry solid surface, the emergence of a highly-collimated sheet from the impact of a jet of densely-packed, dry grains, and the pinch-off of an underwater bubble. In all three cases, the motion is dominated by inertia but a small amount of dissipation is also present. Our works show that dissipation is important for the onset of splash, plays a minor role in the ejecta sheet formation after jet impact, but becomes irrelevant in the break-up of an underwater bubble. An important consequence of this evolution towards perfect-fluid flow is that deviations from cylindrical symmetry in the initial stages of pinch-off are not erased by the dynamics. Theory, simulation and experiment show detailed memories of initial imperfections remain encoded, eventually controlling the mode of break-up. In short, the final outcome is not controlled by a single universal singularity but instead displays an infinite variety.

Ulam's problem has to do with finding asymptotics, as $n \to +\infy$, for the length of the longest increasing subsequence of a random permutation of $\{1, .., n\}. I'll survey its history, its solutions and various extensions emphasizing progresses made at GaTech.