Peter Scholze 是德国(Germany)数学家, 主要的工作领域是算术代数几何(arithmetic Algebraic geometry).

Peter Scholze 非常年轻, 他 1987 年 12 月 11 日出生在德国萨克森州(Free State of Saxony)的首府城市德累斯顿(Dresden). Peter Scholze 入读的是座落在 Berlin-Friedrichshain 的一所侧重于数学与自然科学的语言学校 Heinrich-Hertz-Gymnasium.

2004 年, Peter Scholze 第一次成为德国 IMO 国家队的队员. 当年他没解出第三题, 第六题也没有完整解答, 得分是 31, 获得一枚银牌. 第二年, 他再次披挂上阵, 发挥出色, 得满分 42. 接下来, 他又分别在 2006 年, 2007 年斩获两枚金牌.

Peter Scholze 完成本科和研究生的学习, 可谓神速. 他只用  3  semesters 完成学士, 2 semesters 得到硕士. 于是, Peter Scholze 就引起了大家的注意. 随后, 在 Bonn 大学,  在 Michael Rapoport 的指导下, 做 cohomology of Shimura varieties 和 Langlands program 相关的工作. 他在 2012 年得到博士, 论文是关于 Perfectoid spaces–该理论解决了 weight-monodromy conjecture 的一个特殊情形, 也在 p-adic Hodge theory 有重要应用.

Peter Scholze 获得博士之后不久, 25 岁那年, 成为 bonn 大学 Hausdorff 数学中心的教授. 他是德国最年轻的教授.

Steve pointed out the thing that makes EGA difficult to read is not that it is dense, but rather that it is gigantic.

Robin Hartshorne’s book algebraic geometry is an edulcorated version of Grothendieck and Dieudonné’s EGA, which changed algebraic geometry forever.

EGA was so notoriously difficult that essentially nobody outside of Grothendieck’s first circle(roughly those who attended his seminars) could (or wanted to) understand it, not even luminaries like Weil or Néron .

Things began to change with the appearance of Mumford’s mimeographed notes in the 1960’s, the celebrated Red Book, which allowed the man in the street(well, at least the streets near Harvard) to be introduced to scheme theory.

Then, in 1977, Hartshorne’s revolutionary textbook  algebraic geometry was published. With it one could really study scheme theory systematically, in a splendid textbook, chock-full of pictures, motivation, exercises and technical tools like sheaves and their cohomology.

However the book remains quite difficult and is not suitable for a first contact with algebraic geometry: its Chapter I is a sort of reminder of the classical vision but you should first acquaint yourself with that material in another book.

GTM 52 的精华是第 2, 3章, 分别介绍 Scheme 和它上面的 Cohomollogy theory.

GTM 52 有习题 464 道. 这本书的习题, 非常重要! 当然, 习题也不一定必须一个一个全部做完.

Basic Algebraic Geometry 1

The third Edition of  “Basic Algebraic Geometry” has just been published.

Shafarevich’s Basic Algebraic Geometry has been a classic and universally used introduction to the subject since its first appearance over 40 years ago. As the translator writes in a prefatory note, “For all [advanced undergraduate and beginning graduate] students, and for the many specialists in other branches of math who need a liberal education in algebraic geometry, Shafarevich’s book is a must.”

Shafarevich’s book is an attractive and accessible introduction to algebraic geometry, suitable for beginning students and nonspecialists, and the new edition is set to remain a popular introduction to the field.

The third edition, in addition to some minor corrections, now offers a new treatment of the Riemann–Roch theorem for curves, including a proof from first principles.

Basic Algebraic Geometry 2

The second volume is in two parts: Book II is a gentle cultural introduction to scheme theory, with the first aim of putting abstract algebraic varieties on a firm foundation; a second aim is to introduce Hilbert schemes and moduli spaces, that serve as parameter spaces for other geometric constructions. Book III discusses complex manifolds and their relation with algebraic varieties, Kähler geometry and Hodge theory. The final section raises an important problem in uniformising higher dimensional varieties that has been widely studied as the “Shafarevich conjecture”.

The style of Basic Algebraic Geometry 2 and its minimal prerequisites make it to a large extent independent of Basic Algebraic Geometry 1, and accessible to beginning graduate students in mathematics and in theoretical physics.