I am primarily inclined towards topology, geometric group theory and graph theory. My past research projects lie in the domain of algebraic topology and category theory. I wish to further diversify my research within topology and geometry.
I am currently learning about coxeter groups from Mike Davis' book "The geometry and topology of coxeter groups" while taking occasional detours to relevant research papers or other references, advised by Jean Lafont.
In Fall 2021, I completed my Honours Thesis, on the Bott Periodicity Theorem, supervised by Martin Frankland at the University of Regina, Canada. You could read more about my thesis by clicking on the drop-down below.
About my thesis
The Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups - 2-periodicity for the stable unitary group (the complex case) and 8-periodicity for the stable orthogonal group (the real case), discovered by Raoul Bott. The thesis covers the various ingredients that help understand the periodicity stated by the theorem, proof of the stability of the homotopy groups under consideration by adopting Bott’s approach in his paper "Periodicity Theorem for Classical Groups and Its Applications", the proof of the complex case using the K-theoretic approach from Hatcher's "Vector Bundles and K-theory", an overview of the proof of the real case using the Morse theoretic approach as outlined in Milnor's "Morse Theory" and as a consequence of the periodicity, proving that R^n is a division algebra (and the (n-1)-sphere S^n-1 is parallelizable) only for dimensions n = 1, 2, 4 and 8.
I had presented a part of the work as two separate online talks, the slides for which can be found on clicking the links below -
In the summer of 2021, I researched on the categorical aspects of graphs under Dr. Frankland's supervision.
About my summer research
There are various possible categories of graphs, depending on whether edges are directed or not, multiple edges are
allowed or not, loops are allowed or not, etc. During the summer research, we catalogued these categories of graphs and described a variety of functors between them. We then proved which functors admit adjoints and which adjoints do not exist using pointwise Kan extensions or otherwise. We also proved the existence (or non-existence) of limits and colimits in the various categories of graphs.
The research was funded by Mitacs Globalink Summer Program. The expository paper resulting from the same is currently in preparation.