Integral p_adic Hodge-theory and applications to p-adic deformation theory

我们将线性数据与几何对象相关联，如曲线，曲面或定义为整数或素数幂零的数域上代数方程的零集的三倍。人们可以问，以这种方式可以捕获多少关于原始对象的信息。线性数据由各种上同调理论给出，这是纯数学中许多领域的标准技术，包括拓扑学，复几何和算术几何。这个问题导致了所谓的p-adic变形理论;该项目解决了结晶上同调或等价地与代数圈相关的de Rham-Witt上同调类以及它们如何变形的问题。特别感兴趣的一个相关问题是识别在这些线性数据内的（p-adic）整数上定义的积分结构，这导致几何对象的更精细的不变量并编码更多的信息。这种方法在算术几何中很常见。

We associate linear data to geometric objects like curves, surfaces or threefolds defined as zero-sets of algebraic equations over the integers or over a number domain that is annihilated by a power of a prime number. One can ask how much information about the original object can be captured in this way. The linear data are given by various cohomology theories which is a standard technique in many areas in Pure Mathematics including topology, complex and arithmetic geometry. This problem leads to what is called p-adic deformation theory; the project addresses the question in the case of crystalline cohomology or equivalently de Rham-Witt cohomology classes associated to algebraic cycles and how they deform p-adically. A related problem of particular interest is to identify an integral structure defined over the (p-adic) integers inside these linear data which leads to finer invariants of the geometric objects and encode more information. This method is very common in arithmetic geometry.