PUBLIC VIVA-VOCE NOTIFICATION
Date and Time : Friday, 1 July 2022, 10:30 am
Venue : Seminar Hall (Hybrid mode)
Subject : Mathematics
Combinatorial and topological aspects of chain and planar polygon spaces
Chennai Mathematical Institute.
The planar polygon space (or the moduli space of a closed polygonal linkage) with generic side lengths is a smooth, closed manifold. In this thesis we study three aspects of these spaces.
It is known that these spaces contain the real points of moduli space of genus zero curves as a dense subset; hence are a compactification. Kapranov showed that the real points of the Deligne-Mumford-Knudson compactification can be obtained from the projective Coxeter complex of type A by blowing up along the minimal building set. In the first part of the thesis we show that the planar polygon spaces can also be obtained from the projective Coxeter complex of type A, but, by performing iterative cellular surgery along a sub-collection of the minimal building set. Interestingly, this sub-collection is encoded by a combinatorial data associated with the length vector called its genetic code.
In the second part of the thesis, we study the question of identifying aspherical polygon spaces. Using techniques from toric topology, we find some classes of length vectors for which the corresponding planar polygon spaces are aspherical.
In the last part, we study the applicability of Borsuk-Ulam theorem. In particular, we obtain a formula for the Stiefel-Whitney height in terms of genetic codes and determine for which of these spaces a generalized version of a Borsuk-Ulam theorem holds.