Jack Morava has some interesting ideas stemming from **stable homotopy theory** and **geometric topology** on the **Shafarevich Conjecture**.

The **Shafarevich Conjecture** states: $Gal(\bar Q / Q_{cyl})$ is free. That is, the Galois group of the algebraic closure of the rationals over the cyclotomic closure of the rationals is a free group.

References for Morava's thoughts are [here] (http://folk.uio.no/rognes/yff/morava.pdf) and [here] (http://www.math.uiuc.edu/K-theory/0654/gamma2.pdf) and [here] (http://arxiv.org/PS_cache/math/pdf/0509/0509001v2.pdf) and [here] (http://www.ruhr-uni-bochum.de/topologie/conf08/jack.pdf).

This is exciting material, but I'm having trouble coming up with a way to summarize the gist and have some questions. 

> (1)What exactly is Morava's definition of a **mixed Tate motive**?

> (2) What exactly is the connection Morava is advocating between **number theory** and **geometric topology** by invoking the appearance of the Riemann zeta function in Waldhausen's A-theory/pseudo-isotopy?

> (3) Morava states that the map from the K-theory of the integers to that of the sphere spectrum, $K(\mathbb {Z}) \to K(\mathbb {S})$, is a rational equivalence as a (partial) explanation of (2). How exactly does this work??

> (4) Where does Shafarevich fit in here?

Down-to-earth answers to these would be much appreciated!!