A final goal is to implement these techniques on a computer, to automatically generate mathematical results by exploiting the capacity of ‘bridges’ to significantly transform the level of complexity of notions and results. Back in 2010, when I first evoked this possibility in the paper The unification of Mathematics via Topos Theory, that idea was regarded with a lot of skepticism, as something almost too good to be true. Now, the time is ripe to start making that dream into reality.
The volume of proceedings of the lecture series Lectures Grothendieckiennes given in the academic year 2017-2018 at the Ecole Normale Superieure (Paris) is now available!
This book, which celebrates Grothendieck’s mathematical heritage, is prefaced by Peter Scholze and features contributions by Pierre Cartier, Olivia Caramello, Alain Connes, Laurent Lafforgue, Colin McLarty, Gilles Pisier, Jean-Jacques Szczeciniarz and Fernando Zalamea.
Next week I shall give two (video-recorded) talks, one in Italian and another one in English, on unification and morphogenesis from a topos-theoretic perspective:
Topoi come ‘ponti’ unificanti: una morfogenesi matematicaIn questo seminario affronterò il tema dell’unificazione in matematica, e della dualità tra unità e molteplicità, dal punto di vista della teoria dei ‘ponti’ topos-teoretici. Discuterò in particolare il senso in cui il tipo di unificazione realizzato da questa teoria rappresenta una vera e propria ‘morfogenesi matematica’, intesa come insieme di dinamiche di differenziazione a partire da un’unità.
Unification and morphogenesis: a topos-theoretic perspectiveWe shall present some philosophical principles underlying the theory of toposes as unifying `bridges’ in mathematics. More specifically, after reviewing the various types of unification which occur in mathematics, we shall discuss the way in which the connections established by topos-theoretic ‘bridges’ give rise to an authentic mathematical morphogenesis.
The video of my recent talk “Toposes as unifying spaces: historical aspects and prospects” at the Workshop in honor of Alexander Grothendieck’s legacy at the Universidad Nacional de Colombia is now available on YouTube:
This talk (whose slides can be downloaded here) discusses how the unifying concept of topos was introduced and conceived by Grothendieck, as well as the future prospects provided by the theory of toposes as ‘bridges’.
It is a shorter (and partially different) version of the lecture on the same subject I gave in 2018 at the ENS for the series Lectures Grothendieckiennes organized by Frédéric Jaëck (whose slides are available here):
In fact, I have recently finished writing (in French) my contribution for the Proceedings volume of that lecture series, which will be published by Spartacus: this paper can be downloaded here. An English translation is also in preparation.