p-adic Cohomology and Applications

dms -0400272p-进元分析是由亨塞尔在上世纪之交发起的，旨在将微积分的“连续”技术应用于数论中的“离散”问题。研究了p-adi分析在算术代数几何中的应用；一个典型的问题是多项式方程组的解的计数。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处理重要特殊情况的实用算法。其中一些情况发生在密码学、纠错码和其他计算机科学领域的应用程序中。更具体地说，我们正在发展Berthelot的刚性上同调，与旧的ettale上同调理论并行，后者在理论上得到更好的理解，但在显式计算方面装备不足。一个长期目标是将Lafforgue关于函数场Langlands对应的工作扩展到p进束（晶体）。在另一个方向上，我们正在研究p进上同调的计算算法，它可能在上面提到的实际应用之上具有数学应用。

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.