Between ordinary and p-adic Hodge theory

PI提出了在代数变体上不同上同调理论之间建立新的显式关系，推广了已有的Betti与de Rham上同调（Hodge理论）、etale与de Rham上同调（p进Hodge理论）之间的关系。一个特别的课题是描述非阿基米德解析空间的基本群；这将有助于更好地理解p进霍奇理论中的周期域和周期映射。其他可能的结果包括德拉姆上同调和代数k理论之间的联系，适合机器计算的显式描述，以及德拉姆上同调的变体，通过与晶体上同调的类比产生l函数的谱解释。在技术方面，这个项目可能具有相当大的变革性，将历史上并行进行但缺乏连贯综合的研究领域汇集在一起。从长远来看，这些方法可能会导致新的技术来解决数论的经典问题，如素数的分布和聚集性质。正如PI先前的调查所证明的那样，它们也可能导致数论中计算和数值方法的改进，这可能会对数论在计算机科学（特别是信息安全）中的应用领域产生影响。

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).