CMSA Colloquium: The Geography of Symplectic 4-Manifolds

The symplectic geography problem, originally posed by J. McCarthy-J. Wolfson and R. Gompf, asks which ordered pairs of nonnegative integers are realized as (\chi(X),c_1^2(X)) for a symplectic 4-manifold X, where \chi(X)=(e(X)+\sigma(X))/4. In this talk, we will address the geography of simply connected spin and nonspin symplectic 4-manifolds with small Euler characteristic, with nonnegative signature, or near the Bogomolov-Miyaoka-Yau line c_1^2(X)=9\chi_h(X). We will also discuss recent developments, including joint work with Sai-Kee Yeung on symplectic fake quadrics and related questions motivated by fake projective planes and Cartwright-Steger surfaces.