We study how Gromov-Witten invariants of projective 3-folds transform under a standard flop and a small extremal transition in the algebro-geometric setting from the recent development of algebraic relative Gromov-Witten theory and its applications. This gives an algebro-geometric account of Witten's wall-crossing formula for correlation functions of the descendant nonlinear sigma model in adjacent geometric phases of a gauge linear sigma model and of the symplectic approach in an earlier work of An-Min Li and Yongbin Ruan on the same problem. A terse account from the stringy and the symplectic viewpoint is given in the appendix to complement and compare to the discussion in the main text.

The study of affine Deligne-Lusztig varieties originally arose from arithmetic geometry, but many problems on affine Deligne-Lusztig varieties are purely Lie-theoretic in nature. This survey deals with recent progress on several important problems on affine Deligne-Lusztig varieties. The emphasis is on the Lie-theoretic aspect, while some connections and applications to arithmetic geometry will also be mentioned.

Theorem 1. Let D be an irreducible bounded symmetric domain in C', n>= 2. Let F be a (nonuniform) lattice in Aut (D), ie a discrete subgroup of Aut (D) for which N:= D/F (is noncompact and) has. finite volume (wrt the locally symmetric metric induced from D). Suppose that a group F isomorphic to F (as an abstract group) acts as a discrete automorphism group on a contractible K (ihler manifold tVI. Assume that lQ/f has a finite singularity free cover M (ie M is a manifold) which is quasiprojective ie admits a compactification as a projective variety IVI and that~ I\M is of codimension at least three in~ l. Then] Q is biholomorphically equivalent to D, and f is conjugate to F in Aut (D).

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