When I tell people that cubic fourfolds are a hot topic in algebraic geometry, they're often incredulous at what sounds like a random choice of numbers -- why those and not, say, quartic threefolds? But cubic fourfolds are more interesting than hypersurfaces of other degrees and dimensions for two reasons: first, the classical question of which ones are "rational" is unexpectedly hard, lying just out of reach of both old and new techniques; second, they have unexpected connections to K3 surfaces and hyperkähler manifolds, through Hodge theory, derived categories of coherent sheaves, and beautiful geometric constructions. I'll try to give a taste of what has attracted so many people to this topic in the last 15 to 25 years.
The talk will be aimed at a general mathematical audience, including graduate students.