Summer Minicourse: Polygons and Grassmannians

In the summer of 2016, I gave a short summer minicourse for graduate students on polygons, grassmannians, and linkages. These are scans of my notes for the lectures, along with some supplemental reading suggestions.

  1. Introduction: Polygons and Grassmannians (7/19/2016)
    • Random triangles and polygons in the plane. Cantarella, Needham, Shonkwiler (in preparation).
    • Hausmann, J.-C., & Knutson, A. (1997). Polygon spaces and Grassmannians. L’Enseignement Mathematique. Revue Internationale. 2e Serie, 43(1-2), 173–198.
    • Cantarella, J., Deguchi, T., & Shonkwiler, C. (2013). Probability Theory of Random Polygons from the Quaternionic Viewpoint. Communications on Pure and Applied Mathematics.
  2. The Carpenter’s Ruler Problem: Background from Tensegrity Theory (7/20/2016)
    • Roth, B., & Whiteley, W. (1981). Tensegrity frameworks. Trans. Amer. Math. Soc., 265(2), 419–446.
    • Connelly, R., Demaine, E., & Rote, G. (2003). Straightening polygonal arcs and convexifying polygonal cycles. Discrete & Computational Geometry, 30(2), 205–239.
    • Streinu, I. (2000). A combinatorial approach to planar non-colliding robot arm motion planning. Collection (pp. 443–453).
  3. The Carpenter’s Ruler Problem: Main Argument. (7/22/2016)
    • Cantarella, J., Demaine, E., Iben, H. N., Iben, H. N., O’Brien, J. F., & O’Brien, J. F. (2004). An energy-driven approach to linkage unfolding. Presented at the SCG ’04: Proceedings of the twentieth annual symposium on Computational geometry,  ACM  Request Permissions.
  4. Distance Geometry I: Gower’s Theorem constructing point clouds from distance matrices. (7/25/16)
    • Gower, J. C. (1982). Euclidean distance geometry. The Mathematical Scientist, 7(1), 1–14.
    • Liberti, L., Lavor, C., Maculan, N., & Mucherino, A. (2014). Euclidean Distance Geometry and Applications. SIAM Rev., 56(1), 3–69.
  5. Distance Geometry II: More on Gower’s Theorem, the Cayley-Menger Determinant. (7/26/16)
    • A web proof of the Cayley-Menger formula.
    • Liberti, L., & Lavor, C. (2016). Six mathematical gems from the history of distance geometry. International Transactions in Operational Research, 23(5), 897–920.
  6. Distance Geometry III: Tay’s Proof of Laman’s Theorem and the 3T2 decomposition (7/29/16)