We will focus on two standard problems relating differential geometric techniques and algebraic geometry: the Hodge problem and the Torelli problem. Both conjecture the existence of algebraic sub-varieties of certain complex submanifolds of projective space under topological and analytical conditions. Both are known to be false in complete generality, but there appears to be a very interesting kernel of truth in what they suggest. This term we will study: (0.) review of basics; (1.) the parameter spaces of Hodge structures and their differential and geometric properties; (2.) the invariants associated to cycles in projective manifolds; (3.) the topology of smooth projective manifolds, and auxiliary varieties. We will discuss many of the known examples and counter-examples to these two main problems, and specific partial open problems.

Every effort will be made to keep the course accessible to anybody who is interested in learning about the subject.

Course Requirements:

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Intended Audience:

Students are expected to have a knowledge of basic differential geometry and complex analysis. Some algebraic geometry would be very useful.

Class Format:

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