We report on a new development in asymptotic Hodge theory, arising from the work of Golyshev-Zagier and Bloch-Vlasenko, and connected to the Gamma Conjectures in Fano/LG-model mirror symmetry. The talk will focus exclusively on the Hodge/period-theoretic aspects.

Given a variation of Hodge structure $M$ on a Zariski open in $P^{1}$, the periods of the limiting mixed Hodge structures at the punctures are interesting invariants of $M$. More generally, one can try to compute these asymptotic invariants for iterated extensions of $M$ by "Tate objects", which may arise, for example, from normal functions associated to algebraic cycles.

The main point of the talk will be that (with suitable assumptions on $M$) these invariants are encoded in an entire function called the motivic Gamma function, which is determined by the Picard-Fuchs operator $L$ underlying $M$. In particular, when $L$ is hypergeometric, this is easy to compute and we get a closed-form answer (and a limiting motive).

Though that is probably enough for a single talk, perhaps one more thing is worth mentioning in this abstract: in the next simplest class of cases beyond hypergeometric, the leading Taylor coefficient of the motivic Gamma at 1 is given by the special value of a normal function, and in one special case this recovers Apery's irrationality proof for zeta(3).