Georgia Institute of Technology College of Sciences

Dynamical systems have proven to be a useful tool for the design of space missions. For instance, the use of invariant manifolds is now common to design transfer strategies. Solar Sailing is a proposed form of spacecraft propulsion, where large membrane mirrors take advantage of the solar radiation pressure to push the spacecraft. Although the acceleration produced by the radiation pressure is smaller than the one achieved by a traditional spacecraft it is continuous and unlimited. This makes some long term missions more accessible, and opens a wide new range of possible applications that cannot be achieved by a traditional spacecraft. In this presentation we will focus on the dynamics of a Solar sail in a couple of situations. We will introduce this problem focusing on a Solar sail in the Earth-Sun system. In this case, the model used will be the Restricted Three Body Problem (RTBP) plus Solar radiation pressure. The effect of the solar radiation pressure on the RTBP produces a 2D family of "artificial'' equilibria, that can be parametrised by the orientation of the sail. We will describe the dynamics around some of these "artificial'' equilibrium points. We note that, due to the solar radiation pressure, the system is Hamiltonian only for two cases: when the sail is perpendicular to the Sun - Sail line; and when the sail is aligned with the Sun - sail line (i.e., no sail effect). The main tool used to understand the dynamics is the computation of centre manifolds. The second example is the dynamics of a Solar sail close to an asteroid. Note that, in this case, the effect of the sail becomes very relevant due to the low mass of the asteroid. We will use, as a model, a Hill problem plus the effect of the Solar radiation pressure, and we will describe some aspects of the natural dynamics of the sail.

Motivated by neurally feasible computation, we study Boolean functions of an arbitrary number of input variables that can be realized by recursively applying a small set of functions with a constant number of inputs each. This restricted type of construction is neurally feasible since it uses little global coordination or control. Valiant’s construction of a majority function can be realized in this manner and, as we show, can be generalized to any uniform threshold function. We study the rate of convergence, finding that while linear convergence to the correct function can be achieved for any threshold using a fixed set of primitives, for quadratic convergence, the size of the primitives must grow as the threshold approaches 0 or 1. We also study finite realizations of this process, and show that the constructions realized are accurate outside a small interval near the target threshold, where the size of the construction at each level grows as the inverse square of the interval width. This phenomenon, that errors are higher closer to thresholds (and thresholds closer to the boundary are harder to represent), is also a well-known cognitive finding. Finally, we give a neurally feasible algorithm that uses recursive constructions to learn threshold functions. This is joint work with Christos Papadimitriou and Santosh Vempala.

Let G be a 5-connected nonplanar graph. To show the Kelmans-Seymour conjecture, we keep contracting a connected subgraph on a special vertex z until the following happens: H does not contain K_4^-, and for any subgraph T of H such that z is a vertex in T and T is K_2 or K_3, H/T is not 5-connected. In this talk, we prove a lemma using the characterization of three paths with designated ends, which will be used in the proof of the Kelmans-Seymour conjecture.

The subject of this talk is wave equations that arise from geometric considerations. Prime examples include the wave map equation and the Yang-Mills equation on the Minkowski space. On one hand, these are fundamental field theories arising in physics; on the other hand, they may be thought of as the hyperbolic analogues of the harmonic map and the elliptic Yang-Mills equations, which are interesting geometric PDEs on their own. I will discuss the recent progress on the problem of global regularity and asymptotic behavior of solutions to these PDEs.