On the work of Peter Scholze (Q2279526)

The article surveys the mathematical work of Peter Scholze giving an exposition suitable for non-experts. So, no previous knowledge of the subjects touched by the paper is necessary but for basic knowledge of algebraic geometry. Because of the vastness of the topics of Scholze's research, it is impossible to introduce all of them in detail in a short paper, therefore only basic definitions are recalled and more advanced notions are described informally and references are given to specialized literature for details. The paper is divided into five parts. The first part surveys some basic notions of arithmetic geometry mainly related to the interplay between geometry over fields of characteristic zero and fields over characteristic \(p\), for \(p\) a prime number. The second part describes how the theory of perfectoid spaces, introduced by Peter Scholze, permits a novel way of passing between characteristic \(0\) and \(p\). The main applications of the theory of perfectoid spaces are also discussed: the proof of new important cases of the weight monodromy conjecture, obtained by Scholze himself, and the full proof of Hochster's conjecture obtained by Yves Anré using perfectoid spaces. The third part deals with the work of Scholze on \(p\)-adic Hodge theory until the most recent developments of prismatic cohomology. The fourth part is about the work of Scholze on the Langlands program. This part is the one that relies more on references to the literature as the Langlands program is such an intricate web of ideas, most of which of high technical nature, that it is impossible to summarize them in few pages. So, the author bounds the discussion in describing the main ideas of Scholze on the topic giving precise references for those who want to understand more. The last part briefly describes other topics of Scholze's research: topological Hochschild homology and the new topic of condensed mathematics.