# Books containing new results

In Endless controversy about the correctness of significant papers, Denis Serre writes:

The research community is able to point out incorrect statements, at least among those which have some importance in the development of mathematics. In time, the errors are fixed; this is the role of monographs to present a universally accepted state of the art of a topic.

That kind of a role for monographs seems a healthy one, of course, but in the past it was not so uncommon that they also contained totally new results.

So I was wondering:

Are there recent examples (say, from the last 10 years) of books/monographs containing important new results?

I'm not talking about books in which the author, here and there along the exposition, strengthens some known theorem, adds a little detail, or improves/simplifies/unifies some proofs, etc... That is quite common indeed.

I'm talking about books primarily written to communicate new results, such as, for instance, Ruelle's Thermodynamic formalism (1978).

Update

It seems from the answers and comments that the situation is more diverse than I thought, which is good, I think. Maybe it would make sense to "collect data" based on each field, because it's possible that in some areas publishing new results in books is more common than in other ones.

• What were the new results in Robinson’s 1966 book that were not in his publications from 1963-1965? That example may be more ambiguous. Feb 5 at 9:57
• I believe the monographs of Aguiar and Mahajan contain a lot of new material. Feb 5 at 10:16
• @TheAmplitwist: Yes, there is a certain point at which a very long research paper becomes a "monograph" and then it might as well be considered a "book." Incidentally, another recent monograph with a similar theme to the work of Aguiar-Mahajan you mentioned is "Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry" by Margolis, Saliola, and Steinberg. Feb 5 at 16:02
• Memoirs of AMS and Springer LNM are books primarily publishing new results (LNM also publishes proceedings of conferences). Feb 5 at 18:15
• Feb 5 at 19:04

Almgren's Big Regularity Paper appeared as a monograph in 2000, so not quite ten years ago, but still fairly recent. The results contained in it were 'new' in the sense that they had not been published before, although they had been in circulation as a preprint since 1984.

I'll add a few words about the history of the monograph. Almgren proves that an area-minimizing current, of arbitrary codimension, has a 'small' singular set: the portion of the surface where it is not smoothly embedded is a codimension two subset. (So if the current is $$n$$-dimensional, then the singular set has dimension at most $$n-2$$.) The proof is notoriously difficult, and some of the tools that Almgren developed—multivalued harmonic functions and monotonicity of frequency, for example—have found applications elsewhere.

The original preprint was 1728 pages long, which made it too long to publish in virtually any journal. Almgren was 'exploring the possibility of making it available on the web', but passed away in 1997 as a result of myelodysplasia. Two former doctoral students of Almgren, Jean Taylor and Vladimir Scheffer, posthumously published the monograph containing the result. (Jean Taylor was also Almgren's wife.) The book that eventually appeared in 2000, although shorter than the original paper, is still a hefty 970 pages.

The question seems too broad to me; it's almost like asking for a comprehensive list of long papers. For example, Aschbacher and Smith's Classification of Quasithin Groups spans two books and over a thousand pages. It came out more than 10 years ago, but the point is that it is not uncommon for a new result to require a lot of pages to prove. When that happens, it's natural for the writeup to appear in the form of a book, especially since a lot of journals have page limits. Another example mentioned in one of the comments is Higher Topos Theory by Jacob Lurie, and if his Higher Algebra is ever published, it will surely be in the form of one or more books.

Amnon Neeman's 2001 book "Triangulated categories" is the standard example of a fairly recent research monograph (in the field of homological algebra) primarily written to communicate new results.

https://press.princeton.edu/books/paperback/9780691086866/triangulated-categories-am-148-volume-148

• Let me add that my understanding (based on gossip) is that Neeman did not initially conceive his book on triangulated categories as a research paper. He had the idea to write an original research book on triangulated categories, and so he did. Feb 5 at 21:05

Also, if I may mention it, my 2010 book "Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures" was primarily written to communicate new results.

To a lesser extent, this also applies to my other two books ("Relative nonhomogeneous Koszul duality", 2021-22; and "Quadratic algebras", joint with A. Polishchuk, 2005).

https://bookstore.ams.org/view?ProductCode=ULECT/37

All the three books were indeed originally conceived as research papers, but somehow grew so long that it only made sense to publish them as books.

The Homotopy Type Theory book, from 2013 — ten years ago this summer — collaboratively written by many authors, myself included, and containing a lot of original work (some published as papers in parallel, other parts not published elsewhere at all).

In hindsight, the book is a sort of post-proceedings of the IAS special year on Univalent Foundations (2012–13), but organised into a monograph rather than a volume of papers — partly since much of the material naturally formed a single story that demanded telling as such (Part I of the book), and partly since many of us were also working on computer-formalisation of the material, so were in the mindset and culture of large-scale collaboration drawn from open-source software.

This is a physics example and I think the first edition was in 1995, but if I remember rightly there are some important results derived in Weinberg's textbook The Quantum Theory of Fields: Volume I. Although the book is marketed as being for students, it is almost a research monograph for experts and would actually be a rather difficult and confusing read for a beginner student of field theory (it is around 600 pages in length).

One of the most interesting parts of the book is on soft photons and infrared divergences. If I remember rightly he shows that that a generic infrared finite $$S$$-matrix does not exist for gauge theories, even if the relevant inclusive cross sections are all finite (this may also be proved in his 1965 paper).

He also shows that there is a soft photon theorem when the matter fields are charged particles, although again this might also be in the 1965 paper. This theorem is shown in the book to not receive any loop corrections and so is exact at tree level.

Edit: Since someone has argued that the book of Weinberg is not a research monograph with new results, I will mention instead that Jacques Tits has several books which would qualify (although I don't think any were written within the past 10 years).

• Weinberg's book is nice, but it's nothing like a research monograph. He gives his preferred version of how to treat infrared divergences, but that was already textbook material by the time his book came out, and he leaves plenty of sub-leading stuff out. The main results aren't even Weinberg's, only the simplified formalism. The real legwork was started by Bloch and Nordsieck, [Phys. Rev. 52, 54 (all the way back in1937!)] and completed by Yennie, Frautschi, and Suura [Annals of Physics 13, 379 (1961)] before Weinberg even touched the problem.
– Buzz
Feb 6 at 22:44
• Soft theorems originated in QED with the work of Bloch and Nordsieck in 1937 and were significantly developed by Low, Gell-Mann, Goldberger, Kazes, Yennie, Frautschi, and Suura between 1954 and 1961. In his 1965 paper Weinberg simplified the formalism and also generalised their soft theorems to include gravitons. Feb 8 at 19:54

I would also like to suggest the "bible" on gradient flows in metric space (and in particular in Wasserstein space) published in 2005:

On the first page of the introduction it is stated that "[This book] should be conceived [...] in part as a research book, with new results never appeared elsewhere."