Post Date
Nov 8 2021

Combinatorial G-Conjecture for Interval Subdivisions, Communications in Algebra

There is beauty in simplicity. That said, only few things can match the striking beauty of mathematics at untangling the deepest mysteries of logic and rationale. An invention of our kilogram and a half of grey matter, math offers exquisite insights into the mysteries of numbers and shapes. One such captivating and elusive mysteries is the g-conjecture. Simply put, it is the art of deciphering meaning and logic from within patterns, that may seem too complicated at first glance. A trained mathematician will conduct a fine dissection, layer by layer, until they arrive at simpler more comprehendible bits, and will stop at nothing to make sense from there on.

Dr Imran Anwar and Dr Shaheen Nazir are two mathematicians from the Department of Mathematics at the Syed Babar Ali School of Science and Engineering, who have worked on this beautiful mystery. They have shared their findings in what has been their most recently accepted paper in the prestigious journal ‘communications in algebra’. Their paper ‘Combinatorial g-conjecture for interval subdivisions’ tried to address the well understood and arrestingly interesting problem of the g-conjecture. Simply put, the g-conjecture is a way to imagine possible numbers of faces of different dimensions of a simplicial sphere. What’s that? A simplicial sphere?! Yes – not a simple sphere, but a simplicial sphere. This open problem was originally formulated by Peter McMullen, a British mathematician, a professor emeritus of mathematics at University College London.

Think of the number of triangles can you fit on the face of a decent sized sphere. Now, break down this scenario by slicing and dicing the triangles themselves into sides and faces. If you think this is easy, increase the dimensions of the triangle and envision these shapes in 3D. Now increase the dimensions. Sounds complicated? It really isn’t! The work of Dr Imran and Dr Shaheen takes us bit by bit, side by side and face by face into the next level of imagination. Simply put, their work shows that the g-vector of the interval subdivision of a Cohen–Macaulay simplicial complex is an f-vector. Whoa! What does this mean? It simply means that they have found a logical link between increasingly complex ways of ‘triangulation of a sphere’ and the number of sides and faces of these triangles that are needed to fit neatly on to a three-dimensional object; a sphere in this case. In other words, this paper is devoted to the study of the face vectors of interval subdivisions of Cohen–Macaualy simplicial complexes. McMullen’s g–conjecture for simplicial spheres is among the most celebrated problems in this particular branch of mathematics.

We congratulate the team of Dr Imran and Dr Shaheen on having their paper published in ‘communications in algebra’. We hope to see more of such beautiful work in mathematics!

Reference to the published paper:

Imran Anwar & Shaheen Nazir (2021) Combinatorial g-conjecture for interval subdivisions, Communications in Algebra, DOI: 10.1080/00927872.2021.1993236