Georgia Institute of Technology College of Sciences

The k-disjoint paths problem takes as input a graph G and k pairs of vertices (s_1, t_1),..., (s_k, t_k) and determines if there exist internally disjoint paths P_1,..., P_k such that the endpoints of P_i are s_i and t_i for all i=1,2,...,k. While the problem is NP-complete when k is allowed to be part of the input, Robertson and Seymour showed that there exists a polynomial time algorithm for fixed values of k. The existence of such an algorithm is the major algorithmic result of the Graph Minors series. The original proof of Robertson and Seymour relies on the whole theory of graph minors, and consequently is both quite technical and involved. Recent results have dramatically simplified the proof to the point where it is now feasible to present the proof in its entirety. This seminar series will do just that, with the level of detail aimed at a graduate student level.

Vortex dynamics and solid-fluid interaction are two of the most important and most studied topics in fluid dynamics for their relevance to a wide range of applications from geophysical flows to locomotion in moving fluids. In this talk, we investigate two problems in these two areas: Part I studies the viscous evolution of point vortex equilibria; Part II studies the effects of body elasticity on the passive stability of submerged bodies.In Part I, we describe the viscous evolution of point vortex configurations that, in the absence of viscosity, are in a state of fixed or relative equilibrium. In particular, we examine four cases, three of them correspond to relative equilibria in the inviscid point vortex model and one corresponds to a fixed equilibrium. Our goal is to elucidate some of the main transient dynamical features of the flow. Using a multi-Gaussian ``core growing" type of model, we show that all four configurations immediately begin to rotate unsteadily, while the shapes of vortex configurations remain unchanged. We then examine in detail the qualitative and quantitative evolution of the structures as they evolve, and for each case show the sequence of topological bifurcations that occur both in a fixed reference frame, and in an appropriately chosen rotating reference frame. Comparisons between the cases help to reveal different features of the viscous evolution for short and intermediate time ! scales of vortex structures. The dynamical evolution of passive particles in the viscously evolving flow associated with the initial fixed equilibrium is shown and interpreted in relation to the evolving streamline patterns. In Part II, we examine the effects of body geometry and elasticity on the passive stability of motion in a perfect fluid. Our main motivation is to understand the role of body elasticity on the stability of fish swimming. The fish is modeled as an articulated body made of multiple links (assumed to be identical ellipses in 2D or identical ellipsoids in 3D) interconnected by hinge joints. It can undergo shape changes by varying the relative angles between the links. Body elasticity is accounted for via the torsional springs at the joints. The unsteadiness of the flow is modeled using the added mass effect. Equations of motion for the body-fluid system are derived using Newtonian and Lagrangian approaches for both hydrodynamically decoupled and coupled models in 2D and 3D. We specifically examine the stability associated with a relative equilibrium of the equations, traditionally referred to as the ``coast motion" (proved to be unstable for a rigid elongated body model), and f! ound that body elasticity does stabilize the system. Stable regions are identified based on linear stability analysis in the parameter space spanned by aspect ratio (body geometry) and spring constants (muscle stiffness), and the findings based on the linear analysis are verified by direct numerical simulations of the nonlinear system.

Torsion of a curve in Euclidean 3-space is a quantity which together with the curvature completely determines the curve up to a rigid motion. In this talk we use the curve shortening flow to show that the number of zero torsion points (or vertices) v a closed space curve c and the number p of the pair of parallel tangent lines of c satisfy the following sharp inequality: v + 2p > 5.

We begin this talk by discussing four different problems that arenumber theoretic or combinatorial in nature. Two of these problems remainopen and the other two have known solutions. We then explain how these seeminglyunrelated problems are connected to each other. To disclose a little more information,one of the problems with a known solution is the following: Is it possible to find anirrational number $q$ such that the infinite geometric sequence $1, q, q^{2}, \dots$has 4 terms in arithmetic progression?