Almgren's Big Regularity Paper: Q-Valued Functions Minimizing Dirichlet's Integral and the Regularity of Area-Minimizing Rectifiable Currents Up to Co ... d Scientific Monograph Series in Mathematics)

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Summary: Review of Almgren's big regularity paper
Review: The publication of the Tex formatted version of Almgren's famous big regularity paper makes the original three-volume, 1700 page paper much easier to read. Especially, since Almgren developed the whole new machinery, it takes time for the beginning readers to become familiar with the terminology and to consult essential materials in the Appendix. Readers who have suffered reading the original three hand typed ``telephone books'' will appreciate this all-in-one book.

As in the title of the book, the paper provides a method to deal with the regularity problem of mass minimizing surfaces in higher codimension. This paper not only gives the optimal upper bound of the Hausdorff dimension of the singular sets, but also provides a possible approach to understand the structure of the singular sets of mass minimizing surfaces. This deep but famous paper contains several ingredients to be understood and explored. It is worthwhile for people who are interested in geometric measure theory to spend some time on reading it.