I am recently retired from 30 years at the University of New South Wales (UNSW) Sydney. But I don’t plan on giving up on mathematics explanation and discovery any time soon — it is just too much fun, and exciting.
But to cement this new direction, I have decided to embark on an additional, quite different directions of explanation — to chart a course in mathematics exploration for the general viewer, offering you a road map to get into a wide range of interesting topics in pure mathematics that you can investigate also on your own — after some orientation on my part.
The first topic is particularly exciting — it is a series on Solving Polynomial Equations. You will all know that the standard extension of the quadratic formula to cubic equations involves complicated expressions with cube and square roots, that the quartic equation is even more complicated, and that this method breaks down, at least partially in the quintic and higher cases. Galois theory was designed partly to try to understand the obstructions to writing down formulas for zeros of higher degree polynomials in terms of radicals.
But since I don’t believe in irrational quantities except in an applied, approximate sense, these “solutions by radicals” are intrinsically suspect for me. Now I am going to show you an exciting alternative, which actually meshes closer to what physicists and engineers do to solve equations — using power series and rational extensions of them in the coefficients of the given equations.
With this rather dramatic shift in point of view, I claim that an entirely new landscape emerges, which remarkably connects with a rich hierarchy of combinatorial objects related to Catalan numbers and their generalizations. We will meet binary and ternary trees, polygonal subdivisions, Dyck paths, standard tableaux, and make lots of contact with many interesting entries in the Online Encyclopedia of Integer Sequences.
You might be surprised. Could it be that we will be able to solve the general polynomial equation with this major new point of view!?
To access this exciting series, please JOIN our Members section on my YouTube channel Wild Egg Maths. See for example this informational video:
For a minimal amount (around $5 / month) you will have a rich stream of interesting videos to watch. We are going to be delving into lots of other topics too — from graph theory to projective geometry to a new world of convexity to triangle geometry in hyperbolic geometry. There will also be quite a few advising videos on how to do research as an amateur or as a graduate student.
The videos will be informal, hands- on and will encourage you to participate. I look forward to having you join us!