One grand project which has generated much work is the quest for mathematical understanding of mirror symmetry (via homological mirror symmetry, or the Strominger-Yau-Zaslow/Gross-Siebert picture). Attempts to formulate and prove the conjecture have led to interesting new ideas in symplectic geometry (like the work of Paul Seidel) and attempts to confirm enumerative predictions from string theory have led to new techniques in algebraic geometry. While this grand project has been around since the early 90s (e.g. Kontsevich's ICM talk which introduced homological mirror symmetry was in 1994) it is still going strong and much progress has been made.
Another very active program in geometry was initiated by the paper of Donaldson-Thomas (see also the more recent paper of Donaldson-Segal) and is an effort to define instanton counting/Floer-theoretic invariants in the context of higher-dimensional gauge theory and exceptional geometry.
The search for constant scalar curvature Kaehler metrics was initiated by (see Donaldson's 1986 Fields Medallist lecture from the Fields Medallists' Lectures Volume or Tian's book "Canonical metrics in Kaehler geometry") and the related Donaldson-Tian-Yau conjecture on existence of Kaehler-Einstein metrics on Fano varieties was recently resolved after nearly twenty years' work by many of the world's leading geometric analysts.
The 2000 paper "Introduction to Symplectic Field Theory" by Eliashberg-Givental-Hofer certainly counts as the initiation of a grand project: the systematic study of punctured pseudoholomorphic curves in certain non-compact symplectic manifolds. The theory has many applications, and a by-product is the new foundational polyfold approach to elliptic moduli problems.
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