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$\DeclareMathOperator\Q{\mathbf{Q}}$Jack Morava has some interesting ideas stemming from stable homotopy theory and geometric topology on the Shafarevich Conjecture. The Shafarevich Conjecture states: ...

There is an algebraic analogue of the Atiyah-Hirzebruch spectral sequence from singular cohomology to topological $K$-theory of a topological space. For a field $k$, let $X$ be smooth variety $X$ ...

Ordinary (connective) complex $K$-theory is the algebraic $K$ theory of the topological ring $\mathbb{C}$ with analytic topology. One can also study the $K$ theory of $\mathbb{C}$ with discrete ...