Georgia Institute of Technology College of Sciences

Despite its long history, the theory of ellipticpartial differential equations in non-smooth media is abundant with openproblems. We will discuss the main achievements in the theory, recentdevelopments, surprising paradoxes related to the behavior of solutions nearthe boundary, and some fundamental questions which still remain open.

Steady fluid solutions can play a special role in characterizing the dynamics of a flow: stable states might be realized in practice, while unstable ones may act as attractors in the unsteady evolution. Unfortunately, determining stability is often a process substantially more laborious than computing steady flows; this is highlighted by the fact that, for several comparatively simple flows, stability properties have been the subject of protracted disagreement (see e.g. Dritschel et al. 2005, and references therein). In this talk, we build on some ideas of Lord Kelvin, who, over a century ago, proposed an energy-based stability argument for steady flows. In essence, Kelvin’s approach involves using the second variation of the energy to establish bounds on the growth of a perturbation. However, for numerically obtained fluid equilibria, computing the second variation of the energy explicitly is often not feasible. Whether Kelvin’s ideas could be implemented for general flows has been debated extensively (Saffman & Szeto, 1980; Dritschel, 1985; Saffman, 1992; Dritschel, 1995). We recently developed a stability approach, for families of steady flows, which constitutes a rigorous implementation of Kelvin’s argument. We build on ideas from bifurcation theory, and link turning points in a velocity-impulse diagram to exchanges of stability. We further introduce concepts from imperfection theory into these problems, enabling us to reveal hidden solution branches. Our approach detects exchanges of stability directly from families of steady flows, without resorting to more involved stability calculations. We consider several examples involving fundamental vortex and wave flows. For all flows studied, we obtain stability results in agreement with linear analysis, while additionally discovering new steady solutions, which exhibit lower symmetry. Paolo is a candidate for J Ford Fellowship at CNS. To view and/or participate in the CNS Webinar from wherever you are: evo.caltech.edu/evoNext/koala.jnlp?meeting=MeMMMu2M2iD2Di9D9nDv9e

For knots the hyperbolic geometry of the complement is known to be relatedto itsJones polynomial in various ways. We propose to study this relationship morecloselyby extending the Jones polynomial to graphs. For a planar graph we will showhow itsJones polynomial then gives rise to the hyperbolic volume of the polyhedronwhose1-skeleton is the graph. Joint with Francois Gueritaud and FrancoisCostantino.

The braid group embeds in the mapping class group, and so the symplectic representation of the mapping class group gives rise to a symplectic represenation of the braid group. The basic question Tara Brendle and I are trying to answer is: how can we describe the kernel? Hain and Morifuji have conjectured that the kernel is generated by Dehn twists. I will present some progress/evidence towards this conjecture.