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!
The programme, which is available from the conference website, is very rich and consists of several contributions highlighting the relevance of Grothendieck’s ideas across many different fields of knowledge.
I will give my own talk, entitled “On the ‘unifying notion’ of topos“, on Friday 27th at 10 am PST (7pm CEST).
This evening, at 9pm CET, I will give an online talk on “Relative topos theory via stacks” at the University of Wisconsin Logic Seminar. Many thanks to the organizers of this Seminar, in particular to Prof. Steffen Lempp, for this invitation!
Abstract: In this talk, based on joint work with Riccardo Zanfa, we shall introduce new foundations for relative topos theory based on stacks. One of the central results in our theory is an adjunction between the category of (relatively small) toposes over the topos of sheaves on a given site (C, J) and that of C-indexed categories. This represents a wide generalization of the classical adjunction between presheaves on a topological space and bundles over it, and allows one to interpret several constructions on sheaves and stacks in a geometrical way; in particular, it leads to fibrational descriptions of direct and inverse images of sheaves and stacks, as well as to a geometric understanding of the sheafification process. It also naturally allows one to regard any Grothendieck topos as a ‘petit’ topos associated with a ‘gros’ topos, thereby providing an answer to a problem posed by Grothendieck in the seventies.
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.
Hi! Those of you who can understand Italian might be interested in an interview that I shall give for the twitch channel Meet Science next Monday at 21. Here are the poster and abstract for the event:
Nella scienza e più in generale nella vita è importante – e a volte cruciale – cercare analogie e concetti unificanti. Olivia Caramello (Università degli Studi degli Insubria, Institut des Hautes Études Scientifiques) ci spiega come in matematica e in logica il concetto di topos possa fornirci strumenti per costruire “ponti” tra diverse teorie e così unificarle. Lunedì 14 dicembre 2020 alle ore 21, su https://www.twitch.tv/meetscience.
Next week, I shall give an online course entitled The geometry of morphisms and equivalences of toposes, as part of the lecture series Topics in contemporary mathematics organized by Alexander Cruz.
I will present a number of fundamental results and constructions on the theme of sites and morphisms of toposes, some of which generalize theorems from SGA4.
In the first part of the course I shall present constructions allowing to turn any morphism of sites into a a comorphism of sites inducing the same geometric morphism (up to equivalence) and conversely; moreover, I shall introduce the notion of weak morphism of toposes and characterize the functors which induce such morphisms.
In the second part, I shall discuss continuous comorphisms of sites, present an explicit characterization for them (also in terms of relative cofinality conditions), and show that this class of comorphisms includes all fibrations as well as morphisms of fibrations. I shall also present a characterization theorem for essential geometric morphisms and locally connected morphisms in terms of continuous functors, and a topos-theoretic interpretation of (a relative version) of the comprehensive factorization of a functor.
In the third part, I shall present a theorem providing necessary and sufficient explicit conditions for a morphism of sites to induce an equivalence of toposes; this generalizes Grothendieck’s Comparison Lemma. Lastly, I shall give an overview of results characterizing important properties of geometric morphisms of toposes (such as being an inclusion, a surjection, hyperconnected, localic, local etc.) in terms of properties of morphisms or comorphisms of sites.
I am happy to announce that Joshua Wrigley has just been admitted to the Ph.D. at the University of Insubria under my supervision, ranking first in the Ph.D. competition and obtaining a Huawei studentship (as part of the recently funded project Grothendieck toposes for information and computation). Joshua comes from the University of Oxford, where he has obtained his Bachelor and Master degrees with honours.
Joshua joins the other two Ph.D. students of the group, Morgan Rogers (who came from the University of Cambridge in 2018) and Riccardo Zanfa (who came from the University of Milan in 2018). Congratulations to him!