Georgia Institute of Technology College of Sciences

Dynamical systems with multiple time scales have invariant geometric objects that organize the dynamics in phase space. The slow-fast structure of the dynamical system leads to phenomena such as canards, mixed-mode oscillations, and bifurcation delay. We'll discuss two projects involving chemical oscillators. The first is the analysis of a simple chemical model that exhibits complex oscillations. Its bifurcations are studied using a geometric reduction of the system to a one-dimensional induced map. The second investigates the slow-fast mechanisms generating mixed-mode oscillations in a model of the Belousov-Zhabotinsky (BZ) reaction. A mechanism called dynamic Hopf bifurcation is responsible for shaping the dynamics of the system. This webminar will be broadcast on evo.caltech.edu (register, start EVO, webminar link is evo.caltech.edu/evoNext/koala.jnlp?meeting=MMMeMn2e2sDDDD9v9nD29M )

We examine a variety of problems in delay-differential equations. Among the new results we discuss are existence and asymptotics for multiple-delay problems, global bifurcation of periodic solutions, and analyticity (or lack thereof) in variable-delay problems. We also plan to discuss some interesting open questions in the field.

Locomotion at the micro-scale is important in biology and in industrialapplications such as targeted drug delivery and micro-fluidics. Wepresent results on the optimal shape of a rigid body locomoting in 3-DStokes flow. The actuation consists of applying a fixed moment andconstraining the body to only move along the moment axis; this models theeffect of an external magnetic torque on an object made of magneticallysusceptible material. The shape of the object is parametrized by a 3-Dcenterline with a given cross-sectional shape. No a priori assumption ismade on the centerline. We show there exists a minimizer to the infinitedimensional optimization problem in a suitable infinite class ofadmissible shapes. We develop a variational (constrained) descent methodwhich is well-posed for the continuous and discrete versions of theproblem. Sensitivities of the cost and constraints are computedvariationally via shape differential calculus. Computations areaccomplished by a boundary integral method to solve the Stokes equations,and a finite element method to obtain descent directions for theoptimization algorithm. We show examples of locomotor shapes with andwithout different fixed payload/cargo shapes.

We will discuss properties of manifolds obtained by deleting a totally geodesic ``divisor'' from hyperbolic manifold. Fundamental groups of these manifolds do not generally fit into any class of groups studied by the geometric group theory, yet the groups turn out to be relatively hyperbolic when the divisor is ``sparse'' and has ``normal crossings''.

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.

When Calderón studied his commutators, in connection with the Cauchy integral on Lipschitz curves, he ran into the bilinear Hilbert transform by dropping an average in his first commutator. He posed the question whether this new operator satisfied any L^p estimates. Lacey and Thiele showed a wide range of L^p estimates in two papers from 1997 and 1999. By dropping two averages in the second Calderón commutator one bumps into the trilinear Hilbert transform. Finding L^p estimates for this operator is still an open question. In my talk I will discuss L^p estimates for a singular integral operator motivated by Calderón's second commutator by dropping one average instead of two. I will motivate this operator from a historical perspective and give some comments on potential applications to partial differential equations motivated by recent results on the water wave problem.

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.

In this lecture the analog of Riemannian manifold will be introduced through the notion of spectral triple. The recent work on the case of a metric Cantor set, endowed with an ultrametric, will be described in detail during this lecture. An analog of the Laplace Beltrami operator for a metric Cantor set will be defined and studied.