Field theory is fundamental to our modern understanding of the Cosmos. From classical fields to quantum fields, we see a startling complexity of incredibly wide-ranging physical phenomena emerge from relatively simpler structures and axioms.
Often, the essence of these complicated mechanisms is captured by geometry, which among other (and somewhat bigger) things, makes it easier to see intricacy emerge from the deep structure underlying physical theories.
In the context of relatively modern refinements of the construction of field theory, an important geometric structure is that of bundles (pedantically, they are topological in nature). The geometry and dynamics of a field theory can be encoded into the bundle structure of the theory, to the point that, for instance, classical fields are now defined after bundles (as sections of fibre bundles).
My recent talk at the Canadian Undergraduate Mathematics Conference at the University of Toronto focused on trying to motivate the above developments in the context of classical field theory. Examples were drawn from Newtonian gravitation and classical electrodynamics, both being classical field theories.
Here are the slides for the presentation (the source code can be found here):
It is also my plan to add some more content to the slides in the next few weeks. For instance, there are classical field theories apart from Newtonian gravitation and classical electrodynamics — such as Klein-Gordon theory (which was the subject of my previous talk at the University of Waterloo). I will also add some more topological and geometric notions and proofs especially towards the end of the slides (in the original talk, the latter half was meant more for the purpose of intuition than rigour).
It has been very fun and illuminating to learn new mathematics and physics from people at university, the conference, and on the internet. I hope to be able to spread some of this joy back through this post! :3