Georgia Institute of Technology College of Sciences

Part II of last week's talk.

Complex-oriented cohomology theories are a class of generalized cohomology theories with special properties with respect to orientations of complex vector bundles. Examples include all ordinary cohomology theories, complex K-theory, and (our main theory of interest) complex cobordism.In two talks on these cohomology theories, we'll construct and discuss some examples and study their properties. Our ultimate goal will be to state and understand Quillen's theorem, which at first glance describes a close relationship between complex cobordism and formal group laws. Upon closer inspection, we'll see that this is really a relationship between C-oriented cohomology theories and algebraic geometry.

The conventional point of view is that the Lagrangian is a scalar object, which through the Euler-Lagrange equations provides us with one single equation. However, there is a key integrability property of certain discrete systems called multidimensional consistency, which implies that we are dealing with infinite hierarchies of compatible equations. Wanting this property to be reflected in the Lagrangian formulation, we arrive naturally at the construction of Lagrangian multiforms, i.e., Lagrangians which are the components of a form and satisfy a closure relation. Then we can propose a new variational principle for discrete integrable systems which brings in the geometry of the space of independent variables, and from this principle derive any equation in the hierarchy.

We will present a broad overview of UTRC’s research initiative in Autonomous and Intelligent Systems (AIS) that was created to conceive, develop and mature a broad range of intelligent mobile robotic systems and capabilities to enhance and support the diverse array of businesses that comprise the United Technologies Corporation. While initial efforts have been focused on Sikorsky Aircraft unmanned rotorcraft, the initiative is now expanding to include other aerospace and commercial applications, as well. The research, conducted by a diverse team of researchers in robotics, dynamical systems, control, applied mathematics, computer vision, and computer science (in partnership with several leading universities including CMU, MIT, UPenn, and UCB) includes: • Real-time algorithms for dynamic collision avoidance in an obstacle-rich environment using probabilistic roadmaps. • Navigation with imperfect and intermittent sensors in GPS degraded environments. • Multi-vehicle missions including efficient robotic search algorithms based on ergodic theory methods. • Collaborative motion planning for multiple aerial and ground robots in large, cluttered environments, trading off mission objectives while satisfying logical/spatial/temporal constraints. • Intelligent system design methodology including architectures for autonomy, human-machine systems, and formal verification. We will conclude with research problems of interest to UTRC and discuss existing and future career and internship opportunities in the broad area of autonomy and robotics.