On 3 December 2022, the inaugural event of the recently founded Grothendieck Institute will take place at 3:00 p.m. in the Aula Magna of the Mondovì campus of the Polytechnic of Turin. The event, which is sponsored by the Municipality of Mondovì, will be held in Italian and will see the participation of numerous guests, including Johanna Grothendieck (Alexander’s daughter and member of the Institute’s Board of Directors), Laurent Lafforgue (member of the Institute’s Scientific Council), Gino Zaccaria (Professor of Philosophy at Bocconi University) and Nicoletta Sabadini (Professor of Computer Science at the University of Insubria). The round table and the subsequent exchange with the public will be moderated by national TV journalist Francesca Ronchin.
The title of the event is “All’ascolto della voce delle cose. Un progetto visionario per la matematica e non solo“. The expression “all’ascolto della voce delle cose” belongs to Grothendieck himself, who said “Ce qui fait la qualité de l’inventivité et de l’imagination du chercheur, c’est la qualité de son attention, à l’écoute de la voix des choses” (“The quality of a researcher’s inventiveness and imagination is the quality of his attention, to hearing the voices of things”):
Among the topics that will be addressed in the round table, in a way which is accessible to the general public, are basic research and its applications, scientific creativity, the figure of Alexander Grothendieck as an example of the kind of relationship that can exist between scientific studies, humanistic sensitivity and social commitment, abstraction and its relationship with art, ethics in scientific research.
In order to celebrate Jean-Jacques’ work on Copernicus and his reflections on categorial and Grothendieckian mathematics, I shall give (tomorrow at 16.00 French time) a talk entitled “De Copernic à Grothendieck: la puissance du point de vue fecond“, where I shall elaborate on the importance of fruitful points of view in Science by taking as prominent examples the Copernican revolution and Grothendieck’s unifying concept of topos.
For those who wish to attend the conference online, here are the Zoom credentials:
Tomorrow I shall leave for a short mission in Rome.
I will be a Jury member for the Ph.D. thesis “Dynamical topoi” by Jacopo Garofali (supervised by Michael McQuillan) at Tor Vergata University, for which I have also acted as a referee. The thesis contains interesting applications of toposes to dynamical systems. The defense will take place in the afternoon of the 28th.
I will also visit Sergio Barbarossa’s research group at “La Sapienza” University, giving a talk at their Department on the 27th.
Tomorrow and on Sunday the “Simposio Matematico RIMSE 2022” will take place online. This two-day conference is the annual edition of the national meeting of the RIMSE (Italian Network of the Mathematicians from the Excellence Schools).
The programme of the event can be found here. I have been invited to give a Lectio Magistralis at the conference, which is scheduled for tomorrow at 15:00.
I have been invited to give a talk on this occasion. The title of my presentation is Relative toposes as a generalization of locales. The abstract is as follows:
The aim of this talk is to present a way for representing relative toposes which naturally generalizes the construction of the topos of sheaves on a locale, and which is particularly effective for describing the morphisms between relative toposes in a concrete way. Our theoretical framework is based on the language of stacks and fibred sites, and provides, amongst other things, a unified setting for investigating the relationships between Grothendieck toposes as built from sites and elementary toposes as built from triposes.
On Tuesday the 6th of September at 10 CET I will give (online) a talk at the ACAI conference 2022 (International Conference on Advances in Computing Technologies and AI), which takes place in Dubai from the 6th to the 8th of September.
The title of the talk is “On primality conditions and residue number systems“; an abstract follows.
I will present a research programme aimed at investigating generalized primality conditions through residue number systems. Sieve methods from analytic number theory have proved to be very effective tools in addressing problems concerning prime numbers, such as, most notably, the Goldbach’s conjecture. Still, these methods are mostly based on analytic estimations rather than on structural considerations about residue number systems. We propose to experimentally study, through suitable computer programs, complementary sets of solutions to systems of congruences, in order to eventually formulate theoretical conjectures about their behavior and obtain insights, in particular, on the difficult problem of effectively characterizing the natural order relation on numbers in terms of modular representations. This should lead to a unified framework in which different problems such as the Goldbach’s conjecture and the twin prime conjecture can be constructively investigated under a common roof as part of an abstract theory of generalized primality conditions.
The subject of residue number systems has fascinated me since my teenage years, when I started pondering about these issues and developing a unifying framework for investigating generalized primality conditions. This subject is actually strictly related to topos theory, as the central result in the theory, namely the Chinese Remainder Theorem, can be interpreted as some kind of sheaf condition.
I am convinced that this subject would greatly benefit from extensive experimentations on a computer aimed at formulating theoretical conjectures about the behavior of modular representations of numbers (much as in the spirit of the discovery of the quadratic reciprocity law). This is why I accepted to give a talk at this congress, which gathers some of the main experts in computing with residue number systems. Thanks again to the organizers for their invitation: I’m greatly looking forward to the conference!
Next week I will be in Domodossola giving a research course on topos theory for the 2022 edition of the International Alpine School of Mathematics and Physics.
The title of my course is Grothendieck toposes, invariants and ‘bridges’. The course will be an introduction to the theory of Grothendieck toposes, with a specific emphasis on the invariants that one can define on them. The point of view that we shall take is the one provided by the theory of toposes as ‘bridges’, which we shall present and illustrate through a selection of notable examples. We shall also discuss the application of these techniques to the investigation and discovery of dualities, equivalences and correspondences in mathematics and beyond.