In three very interesting and suggestive papers, H. Carayol introduced new aspects of complex geometry and Hodge theory into the study of non-classical automorphic representations -- in particular, those involving the totally degenerate limits of discrete series. This talk is based on two joint projects which aim to put Carayol's work into a more general context, while hewing to his over-riding theme of producing arithmetic structures on the cohomology of non-algebraic generalizations of Shimura varieties.
We study a new type of proof system, where an unbounded prover and a polynomial time verifier interact, on inputs a string $x$ and a function $f$, so that the Verifier may learn $f(x)$. The novelty of our setting is that there no longer are ``good" or ``malicious" provers, but only rational ones. In essence, the Verifier has a budget $c$ and gives the Prover a reward $r \in [0,c]$ determined by the transcript of their interaction; the prover wishes to maximize his expected reward; and his reward is maximized only if he the verifier correctly learns $f(x)$.
We present an iterative approach to constructing pseudorandom generators, based on the repeated application of mild pseudorandom restrictions. We use this template to construct pseudorandom generators for combinatorial rectangles and read-once CNFs and a hitting set generator for width-3 branching programs that achieve near-optimal seed-length even in the low error regime.