Title: Hyperbolic programming algorithms relying on derivative cones Abstract: A hyperbolic program is a conic optimization problem in which the cone is the domain for a hyperbolic polynomial. Linear programming and semi-definite programming are the prominent examples. Directional derivatives of hyperbolic polynomials yield new hyperbolic polynomials, and with them, new cones, what we call the "derivative cones." In previous talks, we have presented algorithms built using the derivative cones, in which the original cone is first approximated by a much simpler cone, and in which, over time, the simple cone is deformed to become the original cone, all the while the resulting path of optimal solutions being traced. Although the approach possessed elegance and novelty, in certain regards the deformation cones seemed computationally sub par, especially in that the derivative directions did not evolve over time in a way that incorporated new information generated by the algorithms as the paths of optimal solutions were traced. When six months ago we turned attention to this shortcoming, we found an unexpected wealth of topics waiting to be explored. The talk will begin by carefully setting the stage, and the go on to discuss the key ideas and taunting puzzles as they stand at the time of the conference.