Between ordinary and p-adic Hodge theory

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The PI proposes to develop new explicit relationships between different cohomology theories on algebraic varieties, generalize existing relationships between Betti and de Rham cohomology (Hodge theory), and between etale and de Rham cohomology (p-adic Hodge theory). One particulra project is a description of the etale fundamental groups of nonarchimedean analytic spaces; this will lead to better understanding of period domains and period mappings in p-adic Hodge theory. Other potential results include links between de Rham cohomology and algebraic K-theory, explicit descriptions of etale cohomology suitable for machine computations, and variants of de Rham cohomology giving rise to spectral interpretations of L-functions by analogy with crystalline cohomology.On the technical side, this project is potentially quite transformative, bringing together areas of research that have historically proceeded in parallel but lacking a coherent synthesis. In the long run, these methods may lead to new techniques for attacking classical question of number theory, such as the distribution and aggregate properties of prime numbers. As evidenced by the PI's prior investigations, they are also likely to lead to improvements in computational and numerical methods in number theory, which may have impacts in areas of application of number theory to computer science (notably information security).