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Olivia Caramello is a mathematician known for her work in topos theory.

Olivia Caramello is an Italian mathematician. She holds a national Rita Levi-Montalcini associate professorship at the University of Insubria in Como, Italy, and the Gelfand Chair at the Institut des Hautes Etudes Scientifiques, France. She is known for her work in topos theory and for pioneering the technique of toposes as bridges. She is the author of the book "Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic bridges".

Education and early career
She got her Bachelor's Degree in Mathematics at the University of Turin and her Diploma in Piano at the Conservatorio di Cuneo at the age of 19. In 2009, she obtained her Ph.D. in Mathematics at the University of Cambridge (UK), as a Prince of Wales Student of Trinity College (6), with a thesis entitled "The duality between Grothendieck toposes and geometric theories" under the supervision of Peter Johnstone. In 2016, she obtained her Habilitation at the University Paris Diderot with a habilitation thesis entitled "Grothendieck toposes as unifying bridges in Mathematics".

She has held a Research Fellowship at Jesus College, Cambridge and post-doctoral appointments at the De Giorgi Center of the Scuola Normale Superiore di Pisa, the University Paris Diderot and the University of Milan (as holder of a Marie Curie Fellowship of the Istituto Nazionale di Alta Matematica) and the Institut des Hautes Etudes Scientifiques.

Work
Caramello has developed the theory of "toposes as bridges", which consists in methods and techniques for unifying different mathematical theories and transferring information between them by using toposes. This theory is based on the duality of sites and Grothendieck toposes, and on the notion of classifying topos of a geometric first-order theory, exploiting the diversity of possible presentations of each topos by infinitely many sites or theories. Caramello's theory involves several components : on the one hand, establishing equivalences between toposes presented in different ways; on the other hand, calculating or expressing topos invariants in terms of the various types of presentations considered, in order to produce correspondences between properties or elements of these various presentations.

She showed that this technique applied in diverse mathematical contexts generates a large number of new results, some of which are completely unexpected, as well as completely new proofs of already known results. In particular, this technique allowed Caramello to generalise and unify Fraïssé's construction in model theory and Grothendieck's Galois theory. Caramello's "bridge technique" has many applications to topos theory, category theory and other  different domains of mathematics such as topology (Stone-type dualities and Priestley-type dualities), algebra (dualities between MV-algebras and lattice-ordered groups), functional analysis (Gelfand duality and Wallman bases) and algebraic geometry (Nori motives).

The theory of "toposes as bridges" can be considered a meta-mathematical theory of the relations between different theories and her program contributes to realizing the unifying potential of the notion of topos already glimpsed by Alexander Grothendieck.

Caramello organized international conferences in topos theory, "Topos à l'IHES" (2015) and "Toposes in Como" (2018). She is an editor of the journal Logica Universalis and is running a blog and forum about toposes.

Awards and recognition
Caramello was awarded the AILA (Associazione Italiana di Logica e sue Applicazioni) Prize in 2011, a "L'Oréal-Unesco Fellowship for Women in Science" in 2014 and a "Rita Levi Montalcini" position of the Italian Ministry for Education, University and Research in 2017. Her methodology of toposes as bridges has been qualified by André Joyal as a "vast extension of Felix Klein's Erlangen Programme" and has been endorsed by Fields Medalists Alain Connes and Laurent Lafforgue. Her unification program has also featured in the popular best-selling book "How not to be wrong". Caramello is also an accomplished pianist; in 2008, she obtained the first prize at the Edith Leigh Piano Competition of Trinity College, Cambridge.

Controversy
In 2015 Caramello had a public controversy with a number of senior exponents of the category theory community, whom she accused of spreading negative ungrounded opinions on her work; her case is discussed in an academic paper.

Selected publications

 * Fraïssé's construction from a topos-theoretic perspective, Logica Universalis 8, 261-281 (2014).
 * The Morita-equivalence between MV-algebras and abelian l-groups with strong unit (with A. C. Russo), Journal of Algebra 422, 752-787 (2015).
 * Topological Galois Theory, Advances in Mathematics 291, 646-695 (2016).
 * Theories, Sites, Toposes : Relating and studying mathematical theories through topos-theoretic bridges, Oxford University Press (2017). ISBN 9780198758914;
 * Syntactic categories for Nori motives (with L. Barbieri-Viale and L. Lafforgue), Selecta Matematica 24, 3619-3648 (2018).
 * The theory of topos-theoretic bridges - a conceptual introduction, Glass Bead Journal (2016).