Combinatorics Seminar - Ana Chavez Caliz
17:15-18:00
Speaker: Ana Chavez Caliz
Title: Projective self-dual polygons
Abstract: In his book "Arnold's Problems," Vladimir Arnold shares a collection of questions without answers formulated during seminars in Moscow and Paris for over 40 years. One of these problems, stated in 1994, goes as follows:
Find all projective curves projectively equivalent to their duals. The answer seems to be unknown even in RP^2.
Motivated by this question, in their paper "Self-dual polygons and self-dual curves" from 2009, D. Fuchs and S. Tabachnikov explore a discrete version of Arnold's question in 2-dimensions. If P is an n-gon with vertices A_1, A_3, ... A_{2n-1}, then its dual polygon P* has vertices B*_2, B*_4, ... B*_{2n}, where B*_i is the line connecting the points A_{i-1}, A_{i+1}. Given an integer m, a polygon P is m-self-dual if there is a projective transformation f such that f(A_i) = B_{i+m}.
In this talk, I will discuss how we can generalize Fuchs and Tabachnikov's work to polygons in higher dimensions. I will include some conjectures which are supported by computational results.