The aim of this course is to study topological aspects of some problems that arise in robotics.
We plan to cover following three broad topics:
Moduli spaces of spatial and planar polygons.
Moduli space of distinct points on the projective line.
During this semester long seminar we plan to cover some of the important results in this area that are accessible to graduate students.
For example, space of spatial polygons is generically a symplectic manifold. Moreover, its real part is planar polygon space, modulo full isometries.
The (co)homology calculations. CW decomposition of the space of planar polygons. Connections with the moduli space of points on P^1.
We will meet twice a week and discuss some of the main results and problems. Details of each meeting will be posted below.
Days: Wednesday and Friday
Time: 9:10 am to 10:25 am
Farber M. Invitation to topological robotics. European Mathematical Society; 2008.
Hausmann JC, Knutson A. Polygon spaces and grassmannians. Enseignement mathématique. 1997;43(1/2):173-98.
Hausmann JC, Knutson A. The cohomology ring of polygon spaces. Annales de l'institut Fourier 1998 Jan 1 (Vol. 48, No. 1, p. 281).
Panina G. Moduli space of planar polygonal linkage: a combinatorial description. arXiv:1209.3241.
Panina G, Zhukova A. Discrete Morse theory for moduli spaces of flexible polygons, or solitaire game on the circle. arXiv:1504.05139.
Devadoss S. Combinatorial equivalence of real moduli spaces. Notices of the AMS. 2004 Jun;51:620-8.
Yoshida M. The democratic compactification of configuration spaces of point sets on the real projective line. Kyushu Journal of Mathematics. 1996;50(2):493-512.
Background and motivation. Basic definitions and complete analysis of moduli spaces of planar 4-gons.
The smooth structure of moduli spaces. Some explicit homeomorphisms.
Polygons and Grassmannians - I
Polygons and Grassmannians - II
Moduli space of points on the real projective line - I
Moduli space of points on the real projective line - II