p-adic Cohomology and Applications

摘要奖基兰Kedlaya DMS-0400272 p-adic分析，发起了亨泽尔在上个世纪之交，旨在使“连续”的技术，微积分承担“离散”的问题，在数论。我们研究p-自底分析在算术代数几何中的应用，一个典型的问题是计算多项式方程组的解。我们一方面致力于提高我们对这个问题的理论理解，另一方面致力于开发用于处理重要特殊情况的实用算法。其中一些例子出现在密码学、纠错码和计算机科学的其他领域。更具体地说，我们正在发展Berthelot的刚性上同调，与旧的etale上同调理论并行，后者在理论上更好地理解，但不适合显式计算。一个长期目标是将Lafforgue关于函数场Langlands对应的工作扩展到p-adic层（晶体）。在另一个方向，我们正在研究算法计算p-adic上同调，这可能有数学上的实际应用上面提到的。

Abstract for the award of Kiran Kedlaya DMS-0400272p-adic analysis, initiated by Hensel at the turn of the last century,seeks to bring the "continuous" techniques of calculus to bear on"discrete" problems in number theory. We study applications of p-adicanalysis in arithmetic algebraic geometry; a typical problem is to countsolutions of systems of polynomial equations. We work on one hand onimproving our theoretical understanding of this problem, and on the otherhand on developing practical algorithms for treating important specialcases. Some of these cases occur in applications to cryptography, errorcorrecting codes, and other areas of computer science.More specifically, we are developing Berthelot's rigid cohomology in parallel with the older theory of etale cohomology, which is better understood theoretically but ill-equipped for explicit computations.One long-term goal is to extend Lafforgue's work on the function field Langlands correspondence to p-adic sheaves (crystals). In another direction, we are investigating algorithms for computing in p-adic cohomology, which may have mathematical applications on top of the practical ones mentioned above.