What today’s number theorists need to know


Of course I’m not going to answer the question in the title.  If you want to know the answer, you can try to get it past the ferocious gatekeepers at MathOverflow; here, as always, the point is to question questions, not to answer them.

The above excerpt from Kummer’s  De numeris complexis qui unitatis radicibus et numeris integris realibus constant treats the familiar construction of units in the cyclotomic field of 5th roots of unity.  It’s a matter with which all number theorists are or should be familiar, but it’s less clear that it’s necessary today to be able to read Latin (I can’t).   It’s also arguably  unnecessary for today’s number theorists to be able to construct norm tables like this one later in Kummer’s article:

Kummer            Contemporary number theory is highly dependent on different kinds of invariant theory; for example, Vincent Lafforgue’s construction of Langlands parameters for automorphic representations over function fields is based on R.W. Richardson’s study of orbits in powers of a reductive group under the diagonal action.   But explicit invariant theory in the style of Paul Gordan’s late 19th century work is only of interest to specialists:


German seems to have gone the way of Latin, as far as number theorists (other than native German-speakers) are concerned, although it’s not so long ago that it was considered necessary to be able to read the works of Hecke and Siegel in the original language.  Although Vincent Lafforgue has provided an English-language survey of his recent work, and has devoted some serious thought to automatic bilingual publication, number theorists still do need to be able to read French, whatever Jeremy Paxman thinks.

Gordan, who apart from his work in explicit invariant theory is remembered for being Emmy Noether’s thesis advisor, reputedly exclaimed

Das ist nicht Mathematik, das ist Theologie!

when Hilbert’s Basis Theorem put an end to Gordan’s business of proving finite generation of rings of invariants.  Colin McLarty’s unpackaging of this anecdote, which was published in (the underappreciated) Circles Disturbed, should be essential reading for anyone with a serious interest in philosophy of mathematics.

Essential reading as well for number theorists, whose business of compiling norm tables has been supplanted by considerations some might qualify as theological.  Apart from the bits of mathematics number theorists need to know that have unfortunately not yet been invented — for the proof of the Tate Conjecture, for example, or Beilinson’s conjectures — there are the branches of mathematics in the process of being invented that are so massive that they will displace more mundane concerns if number theorists choose to make room for them in our finite minds.  In search of potential applications to the Langlands program I find myself tempted to dip into the vast treatise of Gaitsgory and Rozenblyum, published by installments at the bottom of this page.  It is not in Latin:


In their Preface the authors characterize their project as “A stab in the back” (I hope they mean “stab in the dark”), offer advice for “practical-minded readers,” and provide this helpful account of the contents of their book that can serve as a classification of mathematical activities more generally:

The substance of mathematical thought in this book can be roughly split into three modes of cerebral activity: (a) making constructions; (b) overcoming difficulties of homotopy-theoretic nature; (c) dealing with issues of convergence.  Mode (a) is hard to categorize or describe in general terms. This is what one calls ‘the fun part’.

Mode (b) is something much better defined: there are certain constructions that are obvious or easy for ordinary categories (e.g., define categories or functors by an explicit procedure), but require some ingenuity in the setting of higher categories. For many readers that would be the least fun part: after all it is clear that the thing should work, the only question is how to make it work without spending another 100 pages.

Mode (c) can be characterized as follows. In low-tech terms it consists of showing that certain spectral sequences converge. In a language better adapted for our needs, it consists of proving that in some given situation we can swap a limit and a colimit (the very idea of IndCoh was born from this mode of thinking). One can say that mode (c) is a sort of analysis within algebra. Some people find it fun.

Bearing in mind that the purpose of these 1000 or so pages is to lay the groundwork for a categorical understanding of the geometric Langlands program — which is undoubtedly itself only the first of an infinite chain of increasingly categorical, not to say theological Langlands programs — the authors offer encouragement in the form of a poem by Osip Mandelshtam:





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