What part of an aircraft design process is governed by mathematics?
Maths have their place in pre-design of aircraft, but it is still a design process so creativity has a place as well. Designing a plane is an iterative process, that also includes steps like determining what the plane is used for, what the payload will be, what the range should be, is it allowed to use a runway etc.

Once these factors are determined and honed down, there is a lot of math to be applied to check if the wing can support the weight, if the engines can propel the plane, if the structures can take the static and fatigue load, if the hydraulic pumps are dimensioned correctly, if the control surfaces are dimensioned correctly...

Rockets are relatively easy in that respect. Not taking anything away from the achievement of putting men on the moon, but a rocket has a single mission type, has very little bother with aerodynamics, and does not need to be optimised for decades of safe, economical or mission ready use.

One aspect of aeroplane design where maths &amp; physics rule is aerodynamics, but what's great for aerodynamics is usually not great for structural engineering for instance. So if you would like to focus on maths only, aerodynamics would be a good place to start. Please realise that it is only one aspect of aeroplane design though.

The complete process of aeroplane design is really a lengthy process that does not fit within one answer on this site. I can recommend my goto book for design aspects, Torenbeek. It has 600 pages. Check out Appendix C. on a mathematical method for weight prediction of the wing structure for instance.