Are there review papers, literature reviews in mathematics that describe the recent developments in various fields for a newcomer? Or is the prerequisite knowledge always provided in research monographs and textbooks? That's one can read research papers and make contributions to a field after reading some textbooks or monographs? In physics, all textbooks in my area of interest seem outdated and I should read review articles (see string wiki) that summarize more recent development. I wonder if there are review articles that describe the most important recent achievements in geometry, Analysis, algebra etc.
[Math] Review papers in mathematics
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Best Answer
In Soviet Union there was special series of "mini-books" surveying recent developments in modern mathematics, top mathematicians Arnold, Kirillov, Manin, Novikov, Sinai, Shafarevich,... wrote volumes of this series. It is really very cool series. In Russian it was called «Итоги науки и техники» Серия «Современные проблемы математики. Фундаментальные направления», green thin books in hard-covers.
Here is link to content of the series in Russian.
They were translated in English in the series Springer's "Encyclopaedia of Mathematical Sciences".
Let me point out few "student must(enjoy)-read".
First of all is Shafarevich's Basic notions of algebra. It surveys in accessible level almost all ideas of algebra from groups to K-theory, from rings to algebraic geometry.
S.P. Novikov's Topology, it is analogue of previous one for topology and geometry.
Danilov's Algebraic geometry and schemes - it is covering basic techniques and ideas of algebraic geometry in very friendly way, which I have never met before. Coverage is very wide.
Somewhat on more specific and advanced topics:
Manin, Gelfand Homological algebra - derived categories, perverse sheaves, mixed Hodge structures - all profound tools in one book.
Kirillov Geometric quantization - brief and concise survey of geometric quantization.
As some other sources of surveys I would recommend ICM (international mathematical congress) proceedings.
And Bourbaki's seminar.