This post will be unusual in that it will actually be concerned with the kind of mathematics I encounter in my own research, and more unusual still in that I will reveal how these encounters affect my state of mind. Since the guiding principle of this blog, like the book, is that nothing about its author belongs in it unless it is ideal-typical, in other words exemplifies a general feature of the vocation (and this, in turn, is because there’s no reason for a reader to care about the author’s state of mind, unless the reader happens to know the author, in which case the reader should just ask in private), I begin with some background about the ideal-typical experience on which I am about to report. My guide will be Hartmut Rosa, critical theorist and professor of sociology and political science at the Friedrich-Schiller-Universität Jena, in Germany, theoretician of Beschleunigung [acceleration]:
The time structures of modernity… stand above all under the sign of acceleration.… a grave and sharpening scarcity of time has arisen in the social reality of Western societies, a crisis of time that places in question the traditional ways in which individuals and polities could secure the possibility of shaping their own existence.
He describes the predicament of a contemporary entrepreneur, but the description applies just as well to a mathematician:
He will try to maintain control over his life…and also plan scrupulously for future developments. However, the more dynamic his environment becomes, the more complexly and contingently its chains of events and horizons of possibility take shape, the more unfulfillable this intention will become.
These two quotations are taken from the preface to his book Social Acceleration, a translation of Beschleunigung. Die Veränderung der Zeitstrukturen in der Moderne, which was based on the Habilitationsschrift defended in 2004 in Jena at the (it seems to me) not particularly accelerated age of 39. His two subsequent books also have Beschleunigung in the title. I could just write “acceleration” but it seems to me that, like Durkheim’s anomie or Kierkegaard’s Angst — or Drinfel’d’s shtuka, in a different context (that is nevertheless relevant to the latter part of this post) — the German word draws attention to the singularity and novelty of the underlying concept in a way that is lost in the English equivalent. His most recent book, pictured above [World Relations in the Era of Beschleunigung], looks particularly relevant, because the sample chapter available online quotes T.S. Kuhn, and the point of this post is that I am, and therefore all the representatives of the ideal-type to which I belong are, feeling buffeted if not battered by the waves of Beschleunigung powered by not one but two, or maybe even three, singularly accelerated Kuhnian paradigm-shifts. This book has not been translated, but it is available from amazon.de in Kindle form and I would be ready to overcome my deep misgivings about Amazon and purchase it with 1-Click but my whole point is that this morning’s Beschleunigung is so violent that I can’t even spare the time for 1-Click if (mixing metaphors) I want to have any hope of hanging on by my eyelids to the last car of the new paradigm train that is rushing by at blinding speed.
I remember earlier Beschleunigung-episodes, notably the introduction of perverse sheaves with its immediate applications to geometric representation theory, or quantum groups, or the development of motivic cohomology, not to mention various stages in the Langlands program with which I was too closely involved to be able to appreciate as a spectator. Each episode had its distinctive contingent character but they shared a complex mix of affects, as initiates and bystanders alike experienced the euphoria of witnessing the novel methods providing unexpected solutions to longstanding problems or even more unexpected solutions to unexpected problems, the Angst that one might not be able to beschleunigen sich enough to keep up with the new developments, the anomie of realizing that questions remained, not only the old ones but even more new ones that arrived in the Beschleunigung‘s wake. And then there is the INERTIA of those not caught up in the Beschleunigung, as described in a paper by Bart Zantvoort entitled
ON INERTIA: RESISTANCE TO CHANGE IN INDIVIDUALS, INSTITUTIONS AND THE DEVELOPMENT OF KNOWLEDGE
that cites Hartmut Rosa and that pretty much sums up the ignominious alternative that awaits me if I fail to keep my balance amidst all this buffeting and battering.
What provoked this outburst was the simultaneous appearance this morning of two preprints on the arXiv, namely
Title: Geometrization of the local Langlands correspondence: an overview
Authors: Laurent Fargues
Categories: math.NT math.AG math.RT
Title: A canonical torsion theory for pro-p Iwahori-Hecke modules
Authors: Rachel Ollivier, Peter Schneider
Categories: math.RT math.NT
The two accelerated Kuhnian paradigm shifts to which I alluded above are Scholze’s perfectoid geometry and derived algebraic geometry, and to make sense of what’s going on in the overlapping concerns of these two (and many more) papers my fellow hobos, who would just as happily sit and watch the trains pass by, need to overcome their INERTIA and beschleunigen sich enough to stay on the far side of both paradigms. And it’s just as necessary to absorb the lessons of Vincent Lafforgue’s work on Langlands parametrization (this is where the shtukas, or chtoucas, come in), and whether or not this qualifies as a full-blown paradigm shift it certainly involves a lot of work and is likely to require a lot more as its implications hit home.
I had better get back to my personal Beschleunigung. So I’m not going to explain why someone with my ideal-typical profile would have to overcome INERTIA when the two papers cited above appear on the arXiv; instead I’ll just mention that, in response to a request to contribute to a book that is (I think) entitled What is a mathematical concept?, edited by Elisabeth de Freitas, Nathalie Sinclair, and Alf Coles, I submitted a chapter entitled
The Perfectoid Concept: Test Case for an Absent Theory
where the absent theory is not the all-too-present perfectoid geometry but rather the theory of the kind of mathematical concept it exemplifies. The chapter treats the euphoria and the Angst that accompanied this particular candidate for paradigmatic Kuhnian status, but not the anomie, except perhaps in this passage near the end:
Is a professional historian even allowed to believe that (some) value judgments are objective, that the notion of the right concept is in any way coherent? How can we make sweeping claims on behalf of perfectoid geometry when historical methodology compels us to admit that even complex numbers may someday be seen as a dead end? “Too soon to tell,” as Zhou En-Lai supposedly said when asked his opinion of the French revolution.