This lecture course will be centred around the celebrated Grothendieck–Riemann–Roch theorem, proven by A. Grothendieck in 1957. Along the way, we will see how it can be naturally generalized to the setting of derived algebraic geometry. Finally, we will also discuss how the derived Grothendieck-Riemann-Roch formula gives rise to formulas for the virtual fundamental class originally predicted by M. Kontsevich.
Prerequisites: a working knowledge of algebraic geometry and category theory. More advanced topics (including perfect complexes, derived schemes, and the language of higher category theory) will be reviewed as we need them.
These are preliminary notes; read with caution.