“数学之美－杰出学者讲坛”系列报告

Chair：Professor Stephen Shing-Toung Yau

Abstract: There is a long history involving the algebraic geometry of mechanisms going back to the nineteenth century work of luminaries such as A. Cayley, P.L. Chebyshev and W.K. Clifford. However, with the emergence of industrial robots, topology and especially the concept of group actions in the context of Lie groups, is emerging as a powerful tool for design and analysis. Industrial robots are designed to move between points in a compact subset of the Euclidean group W ⊂ E(3). Engineering considerations usually dictate that the robot defines a map from the six-torus to E(3) and that this map takes the form of a product T(x) = e^A₁^x₁ e^A₂^x₂  ··· e^A₆^x₆ with each e^A_i^x_i defining a one-parameter compact subgroup of E(e). We will show that generically, the image of this map, a critical property in defining the robot’s capabilities, determines the A_i to within a finite set of choices, and in the non-generic situations the image determines an imbedded group isomorphic to SO(3), as well as its position in a related product form.

Short Bio: Roger Brockett is the An Wang Professor Emeritus in the School of Engineering and Applied Sciences at Harvard University. He received his BS., MS. and Ph.D. degrees from Case Western Reserve University and taught for six years in the Electrical Engineering Department at MIT before joining the Harvard faculty in 1969. He has contributed extensively to the theory of automatic control with work on stability theory, stochastic processes, dynamical systems and quantum control. He has been recognized by the IEEE, ASME and SIAM for his contributions to research and education. He is a fellow of the IEEE, SIAM, and AMS and is a member of the US Academy of Engineering.