Possible connections and applied research resources:

Perfectoid Spaces, Arithmetic Geometry, and Quantum Theory

Quantum Geometric Langlands Correspondence

https://ncatlab.org/nlab/show/quantum+geometric+Langlands+correspondence

Grand Unification of Mathematics and Physics

http://www.math.columbia.edu/~woit/wordpress/?p=7114

A new approach to Quantum Mechanics I : Overview

Is the tone appropriate? Is the mathematics at the right level?

https://mathematicswithoutapologies.wordpress.com/2018/06/02/is-the-tone-appropriate-is-the-mathematics-at-the-right-level/

Other useful references:

http://arxiv.org/abs/1201.6343

http://math.berkeley.edu/courses/fall-2014-math-274-001-lec

http://mathoverflow.net/questions/162803/p-adic-string-theory-and-the-string-orientation-of-topological-modular-forms-tm

http://ncatlab.org/nlab/show/LanglandsOxford2014

http://ncatlab.org/nlab/show/Weil

http://ncatlab.org/nlab/show/differential

http://ncatlab.org/nlab/show/function+field+analogy

http://ncatlab.org/nlab/show/geometric+Langlands+correspondence#GerasimovLebedevOblezin0

http://ncatlab.org/nlab/show/string+orientation+of+tmf

http://sciencematters.berkeley.edu/archives/volume2/issue12/story3.php

http://www.msri.org/programs/276

http://www.msri.org/workshops/710