Olivia Caramello's website
Videos
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Relative toposes for artificial general intelligence,
4-hour course at CentraleSupélec (Paris-Saclay University), 2025.
Topos theory can be described as the science of invariants. The
theory of relative toposes consists in methods and techniques for
studying toposes in relation to each other. Viewing toposes as
spaces embodying information, this theory notably paves the way for
the development of new, very dynamical and structural forms of
modelling, both of ‘real’ entities and phenomena, and of (natural or
artificial) learning processes. In particular, it naturally leads to
the design of systems implementing principles of meta-learning (in
the sense of learning taking place at different levels of
abstraction constructed on top of each other). The first part of the
course provides a conceptual introduction to the theory of relative
toposes and its relevance for AI, while the second presents some
first applications of this theory in connection with the modelling
and solution of Raven progressive matrices and, more generally, of
ARC-type problems.
The slides are available
here: part 1;
part 2.
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Toposes for the working mathematician, 8-hour course at
CentraleSupélec (Paris-Saclay University), 2024.
This course provides a conceptual introduction to topos theory aimed at illustrating the sense in which Grothendieck toposes can act as unifying spaces in mathematics. It starts with a review of the necessary categorical preliminaries (for the benefit of those who are not familiar with the language of category theory), and then presents the basics of the theory of sheaves on topological spaces and on general sites. It then introduces the notion of Grothendieck topos and the theory of classifying toposes, to conclude with a survey of the key principles of the theory of toposes as ‘bridges’ and its interdisciplinary applications.
The slides are available here:
part 1;
part 2.
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Topoi e
'ponti' per la matematica e l'intelligenza artificiale,
talk for the general public given at the Mantua science festival,
2025.
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Grothendieck
toposes as unifying ‘bridges’ in Mathematics,
Colloquium talk at the Stockholm Mathematics Centre (KTH and
University of Stockholm), 2022.
The slides are available
here.
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Relative
toposes for the working mathematician,
5-hour introductory course to relative topos theory given at
"Toposes in Mondovì", 2024.
The slides are available
here.
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Deductive systems and Grothendieck topologies,
talk at the Deducteam Seminar, INRIA, ENS de Paris Saclay, 2024.
The slides are available
here.
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The 'unifying notion' of topos,
talk given at the conference in honour of Grothendieck held in May
2022 at Chapman University.
The slides are available
here.
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Introduction to relative topos theory,
research
course given at the school "Unifying
themes in Geometry", 27-30 September 2021
The slides are available
here.
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Introduction to sheaves, stacks and relative toposes,
4-hour research course given in videoconference for the school of
"Toposes online", 24-30 June 2021
Lecture 1
Lecture 2
Lecture 3
Lecture 4
The slides are available
here.
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Relative topos theory via stacks, talk at ItaCa Fest 2021
The slides are available
here.
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Unification
and morphogenesis: a topos-theoretic perspective, talk given in
videoconference from IHES for the "Seminario di Logica e Filosofia
della Scienza dell’Università di Palermo"
The slides are available
here.
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Topoi come ‘ponti’ unificanti: una morfogenesi matematica, talk
given videoconference for the "Synergia Seminar" at the Università
degli Studi di Urbino
The slides are available
here.
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The geometry of morphisms and equivalences of toposes,
6-hour research course given in videoconference for the Lecture
Series Topics in contemporary mathematics at the Universidad
Nacional de Colombia, 27-28 August 2020
Lecture 1
Lecture 2
Lecture 3
The slides are available
here.
- An
interview in Italian for Meet Science
- An invitation to topos-theoretic model theory,
research course at the Institute For Fundamental Research, Tehran
(Iran) - given through videoconference from IHES, 13, 20 and 27 May
2020:
Lecture 1
Lecture 2
Lecture 3
The slides are available
here.
- Unità e
molteplicità: l'estetica dell'oggetto in matematica,
talk (in
Italian) at the Seminario Estivo di ScienzaNuova (Merano, 20 luglio
2020): video-recording
available here. The slides
are available
here.
- Introduction to Grothendieck toposes, research
course given (in videoconference from IHES) at the conference
"Topics in Category Theory", ICMS, Edinburgh (11-13 March 2020):
Lecture n.1
Lecture n.2
Lecture n.3
Lecture n.4
The playlist is available
here.
The slides are available
here.
- Reale o immaginario?
La lezione dei topoi,
talk (in
Italian) at the Seminario Estivo di ScienzaNuova (Merano, 22 luglio
2019): video-recording
available here. The slides
are available
here.
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The idea of 'bridge' and its unifying role,
talk (in
Italian) at
TEDxLakeComo, 10 November 2018: video-recording
available here.