I’m afraid that this is quite a general question but I am hoping some experts can weigh in. I am a student generally interested in learning more about the intersection of algebraic geometry, algebraic topology/homotopy theory, and algebraic K theory. I would like to stress that this is a broad interest, and I am not exactly sure what to look at. One program in this intersection which has recently caught my eye is the homotopy theory of schemes and in particular motivic homotopy theory. Another program that has interested me is derived algebraic geometry.
I have looked at the following posts for guidance but still feel a little overwhelmed with the breadth of these areas of research.
About my background:
Of the relevant topics, I feel the most comfortable with algebraic geometry. I have some familiarity with classical varieties, schemes, and sheaf cohomology (via Hartshorne and large portions of Vakil’s notes). I am secondly most comfortable with algebraic topology (at the level of Hatcher’s book) and homotopy theory (May’s Concise). I am also roughly familiar with some topological K theory at the level of Milnor-Stasheff.
On the other hand, my understanding of stable homotopy theory and higher algebra/homotopical algebra is quite weak. I am also not familiar with the modern weaponry in algebraic geometry such as stacks and étale cohomology.
Question(s). I would like some suggestions as for:
What directions to move toward given my interests
Roadmaps and references for those directions, and in particular derived algebraic geometry and motivic homotopy theory given my background
