Georgia Institute of Technology College of Sciences

We will discuss a way of explicitly constructing ribbon knots using one-two handle canceling pairs. We will also mention how this is related to some recent work of Yasui, namely that there are infinitely many knots in (S^3, std) with negative maximal Thurston-Bennequin invariant for which Legendrian surgery yields a reducible manifold.

In this talk, we will have an overview of: the Gaming Industry, specifically on the Video Slot Machine segment; the top manufactures in the world; the game design studio Gimmie Games, who we are, what we do; what is the process of making a video slot game; what is the basic structure of the math model of a slot game; current strong math models in the market; what is the roll of a game designer in the game development process; the skill set needed to be a successful Game Designer. Only basic probability knowledge is required for this talk.

New and proposed interplanetary missions increasingly require the design of trajectories within challenging multi-body environments that stress or exceed the capabilities of the two-body design methodologies typically used over the last several decades. These current methods encounter difficulties because they often require appreciable user interaction, result in trajectories that require significant amounts of propellant, or miss potential mission-enabling options. The use of dynamical systems methods applied to three-body and multi-body models provides a pathway to obtain a fuller theoretical understanding of the problem that can then result in significant improvements to trajectory design in each of these areas. In particular, the computation of periodic Lagrange point and resonant orbits along with their associated invariant manifolds and heteroclinic connections are crucial to finding the dynamical channels that provide new or more optimal solutions. These methods are particularly effective for mission types that include multi-body tours, Earth-Moon transfers, approaches to moons, and trajectories to asteroids. The inclusion of multi-body effects early in the analysis for these applications is key to providing a more complete set of solutions that includes improved trajectories that may otherwise be missed when using two-body methods. This seminar will focus on two representative trajectory design applications that are especially challenging. The first is the design of tours using flybys of planets or moons with a particular emphasis on the Galilean moons and Europa. In this case, the exploration of the design space using the invariant manifolds of resonant and Lyapunov orbits provides information such as the resonance transitions that are required as part of the tour. The second application includes endgame scenarios, which typically involve an approach to a moon with an objective of either capturing into orbit around the moon or landing on the surface. Often, the invariant manifolds of particular orbits may be used in this case to provide a wide set of approach options for both capture and landing analyses. New methods will also be discussed that provide a foundation for rigorously analyzing the transit of trajectories through the libration point regions that is necessary for the approach and capture phase for bodies such as Europa and the Moon. These methods provide a fundamentally new method to search for the invariant manifolds of orbits and hyperbolic invariant sets associated with libration points while giving additional insight into the dynamics of the flow in these regions.

Many classical hard algorithmic problems on graphs, like coloring, clique number, or the Hamiltonian cycle problem can be sped up for planar graphs resulting in algorithms of time complexity $2^{O(\sqrt{n})}$. We study the coloring problem of unit disk intersection graphs, where the number of colors is part of the input. We conclude that, assuming the Exponential Time Hypothesis, no such speedup is possible. In fact we prove a series of lower bounds depending on further restrictions on the number of colors. Generalizations for other shapes and higher dimensions were also achieved. Joint work with E. Bonnet, D. Marx, T. Miltzow, and P Rzazewski.