The Chabauty-Kim algorithm sets up a direct line from motivic (almost) linear "key calculations" to concrete Diophantine results (primarily, effective finiteness of integral points on curves (with nonabelian fundamental groups)). The "key calculations" are often no joke. The theory around the algorithm also fits well with the knots-primes analogy where on the knots-side, the "key calculations" are related to invariants in low-dimensional topology which can be well-studied via symplectic geometry.

 

We'll flesh this latter statement out and construct some enhancements (e.g. introduce derived geometry enhancements and suitable symplectic structures) on the arithmetic side which allow (nearly) analogous invariants to be constructed. Time permitting, we'll discuss some simple explicit examples where the "nearly" can be removed.