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Grothendieck’s survival talks

Published August 6, 2009 by lievenlb

The Grothendieck circle is a great resource to find published as well as unpublished texts by Alexander Grothendieck.

One of the text I was unaware of is his Introduction to Functorial Algebraic Geometry, a set of notes written up by Federico Gaeta based on tape-recordings (!) of an 100-hour course given by Grothendieck in Buffalo, NY in the summer of 1973. The Grothendieck-circle page adds this funny one-line comment: “These are not based on prenotes by Grothendieck and to some extent represent Gaeta’s personal understanding of what was taught there.”.

It is a bit strange that this text is listed among Grothendieck’s unpublished texts as Gaeta writes on page 3 : “GROTHENDIECK himself does not assume any responsability for the publication of these notes”. This is just one of many ‘bracketed’ comments by Gaeta which make these notes a great read. On page 5 he adds :

“Today for many collegues, GROTHENDIECK’s Algebraic Geometry looks like one of the most abstract and unapplicable products of current mathematical thought. This prejudice caused har(‘m’ or ‘ess’, unreadable) even before the students of mathematics within the U.S. were worried about the scarcity of academic positions… . If they ever heard GROTHENDIECK deliver one of his survival talks against modern Science, research, technology, etc., … their worries might become unbearable.”

Together with Claude Chevalley and Pierre Cartier, Grothendieck was an editor of “Survivre et Vivre“, the bulletin of the ecological association of the same name which appeared at regular intervals from 1970 to 1973. Scans of all but two of these volumes can be found here. All of this has a strong 60ties feel to it, as does Gaeta’s decription of Grothendieck : “He is a very liberal man and in spite of that he allowed us to use plenty of tape recorders!” (p.5).

On page 11, Gaeta records a little Q&A exchange from one of these legendary ‘survival talks’ by Grothendieck :

Question : We understand your worries about expert knowledge,… by the way, if we try to explain to a layman what algebraic geometry is it seems to me that the title of the old book of ENRIQUES, “Geometrical theory of equations”, is still adequate. What do you think?

GROTHENDIECK : Yes, but your ‘layman’ should know what a sustem of algebraic equations is. This would cost years of study to PLATO.

Question : It should be nice to have a little faith that after two thousand years every good high school graduate can understand what an affine scheme is … What do you think?

GROTHENDIECK : …. ??

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Bourbakism & the queen bee syndrome

Published July 15, 2009 by lievenlb

Probably the smartest move I’ve made after entering math-school was to fall in love with a feminist.

Yeah well, perhaps I’ll expand a bit on this sentence another time. For now, suffice it to say that I did pick up a few words in the process, among them : the queen bee syndrome :

women who have attained senior positions do not use their power to assist struggling young women or to change the system, thereby tacitly validating it.

A recent study by the Max Planck Institute for Human Development asserts that the QBS

likely stems from women at the top who feel threatened by other women and therefore, prefer to surround themselves with men. As a result, these Queen Bees often jeapordize the promotions of other females at their companies.

Radical feminists of the late 70-ties preferred a different ‘explanation’, clearly.

Women who fought their way to the top, they said, were convinced that overcoming all obstacles along the way made them into the strong persons they became. A variant on the ‘what doesn’t kill you, makes you stronger’-mantra, quoi. These queen bees genuinely believed it to be beneficial to the next generation of young women not to offer them any shortcuts on their journey through the glass ceiling.

But, let’s return to mathematics.

By and large, the 45+generation decides about the topics that should be (or shouldn’t be) on the current math-curriculum. They also write most of the text-books and course-notes used, and inevitably, the choices they make have an impact on the new generation of math-students.

Perhaps too little thought is given to the fact that the choices we (yes, I belong to that age group) make, the topics we deem important for new students to master, are heavily influenced by our own experiences.

In the late 60ties, early 70ties, Bourbaki-style mathematics influenced the ‘modern mathematics’ revolution in schools, certainly in Belgium through the influence of George Papy.

In kintergarten, kids learned the basics of set theory. Utensils to draw Venn diagrams were as indispensable as are pocket-calculators today. In secondary school, we had a formal axiomatic approach to geometry, we learned abstract topological spaces and other advanced topics.

Our 45+generation greatly benefitted from all of this when we started doing research. We felt comfortable with the (in retrospect, over)abstraction of the EGAs and SGAs and had little difficulties in using them or generalizing them to noncommutative levels…

Bourbakism made us into stronger mathematicians. Hence, we are convinced that new students should master it if they ever want to do ‘proper’ research.

Perhaps we pay too little attention to the fact that these new students are a lot worse prepared than we were in the old days. Every revolution inevitably provokes a counter-revolution. Secondary school mathematics sank over the last two decades to a debilitating level under the pretense of ‘usability’. Tim Gowers has an interesting Ivory tower post on this.

We may deplore this evolution, we may try to reverse it. But, until we succeed, it may not be fair to freshmen to continue stubbornly as if nothing changed since our good old days.

Perhaps, Bourbakism has become our very own queen bee syndrome…