Non-Archimedean Quantum k-theory and Gromov-Witten Invariants

These notes were taken during the talk given by Tony Yue YU (Caltech)

Definition (Calabi-Yau variety):
A smooth projective variety $X/\mathbb{C}$ is called Calabi-Yau if its canonical bundle $K_X$ is trivial, i.e., it has nowhere vanishing holomorphic volume form.
Examples are elliptic curves, abelian varieties, K3 surfaces, hypersurface of degree d+1 in $\mathbb{CP^d}$

Mirror symmetry is conjectural duality between Calabi-Yau varieties:
Calabi-Yau variety $X$ $\longleftrightarrow$ $\exists$ mirror variety $X^\vee$

such that geometric relations hold between $X$ and $X^\vee$: Hodge structure, Gromov-Witten invariants, Fukaya categories, derived category of coherent sheaves, SY2 torus fibration, etc.
Further study reveals that mirror symmetry is actually a duality between “maximal degenerating family” of Calabi-Yau varieties.

Classical approach to Gromov-Witten theory is via Perfect obstruction theory (by Behrend-Fantechi, Li-Tian)