Abstract:

In his final published mathematical work, Poincaré studied holomorphic sections of a Jacobian bundle associated to a pencil of algebraic curves.  When these sections behave "normally" (with "admissible" logarithmic singularities) along the discriminant locus, he showed that they arise from a 1-cycle on the total space.  Lefschetz's subsequent discovery, that fibering out a rational (1,1)-class on a smooth projective surface produces such a "normal function", thus verified an instance of the Hodge Conjecture in the days before the Hodge decomposition was known.

Normal functions are to families of algebraic cycles what period maps are to families of algebraic varieties:  the basic Hodge-theoretic invariant.  Informally, they are given by integrals of differential forms on non-closed chains instead of topological cycles, and satisfy inhomogeneous differential equations instead of homogeneous ones.  In this 6-lecture minicourse I will give an overview of the subject, concentrating on examples and applications.

Lectures 1-2:  On the first day I will concentrate on "classical" normal functions, beginning with families of divisors on curves and a result of Manin.  Turning to cycles of higher (co)dimension, I'll describe several invariants of normal functions, as well as the relationship to the Hodge conjecture.  I'll give particular attention to the Ceresa cycle and related examples, and their relation to moduli of curves, and to the role of normal functions in open mirror symmetry discovered by Morrison and Walcher.

Lectures 3-4:  Higher Chow cycles arise naturally from usual cycles by taking limits.  I'll give a gentle introduction to higher cycles and their Abel-Jacobi invariants (also known as regulator maps).  Letting these vary in families produces "higher" normal functions.  A key example here is given by K2 classes on curves, which are little more than pairs of rational functions, but yield spectacular applications to local mirror symmetry, Feynman integrals, and quantization.

Lectures 5-6:  The Bloch-Beilinson conjectures, which posit a relationship between special values of L-functions and Hodge-theoretic invariants of algebraic cycles, remain one of the deepest open problems in mathematics.  I will explain the statements, and a new point of view that brings normal functions into the picture.  If time allows, I'll conclude with a different application of higher normal functions to number theory:  the irrationality of zeta(3).  This connection has since matured into conjectural story linking special values of normal functions to Apéry numbers and gamma classes of Fano varieties.