Olivia Caramello is an Italian

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

. She holds a national Rita Levi-Montalcini associate professorship at the

University of Insubria The University of Insubria ( it, Università degli Studi dell'Insubria) is an Italian university located in Como and Varese, with secondary locations in Busto Arsizio and Saronno. It was founded in 1998, it has been named after the area where it i ...

Como, Italy Como (, ; lmo, Còmm, label= Comasco , or ; lat, Novum Comum; rm, Com; french: Côme) is a city and ''comune'' in Lombardy, Italy. It is the administrative capital of the Province of Como. Its proximity to Lake Como and to the Alps h ...

. She is known for her work in

topos theory In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion ...

and for pioneering the technique of toposes as bridges. She authored the 2017 book ''Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic bridges''.

Education and early career

Caramello earned her bachelor's degree in mathematics at the

University of Turin The University of Turin (Italian: ''Università degli Studi di Torino'', UNITO) is a public research university in the city of Turin, in the Piedmont region of Italy. It is one of the oldest universities in Europe and continues to play an impo ...

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 , mottoeng = Literal: From here, light and sacred draughts. Non literal: From this place, we gain enlightenment and precious knowledge. , established = , other_name = The Chancellor, Masters and Schola ...

(UK), as a Prince of Wales Student of Trinity College, 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

Paris Diderot University Paris Diderot University, also known as Paris 7 (french: Université Paris Diderot), was a French university located in Paris, France. It was one of the inheritors of the historic University of Paris, which was split into 13 universities in 197 ...

with a habilitation thesis entitled "Grothendieck toposes as unifying bridges in Mathematics". Caramello 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, Paris Diderot University and the University of Milan (as holder of a Marie Curie Fellowship of the

Istituto Nazionale di Alta Matematica The Istituto Nazionale di Alta Matematica Francesco Severi, abbreviated as INdAM, is a government created non-profit research institution whose main purpose is to promote research in the field of mathematics and its applications and the diffusion ...

) and the Institut des Hautes Etudes Scientifiques.

Work

Caramello 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 In mathematics, a classifying topos for some sort of structure is a topos ''T'' such that there is a natural equivalence between geometric morphisms from a cocomplete topos ''E'' to ''T'' and the category of models for the structure in ''E''. Exam ...

of a geometric

first-order theory First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantif ...

, 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 Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry *Equivalence class (music) *''Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *'' Equival ...

between

topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion ...

es 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. 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 ''Logica Universalis'' is a peer-reviewed academic journal which covers research related to universal logic Originally the expression ''Universal logic'' was coined by analogy with the expression ''Universal algebra''. The first idea was to dev ...

'' 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. Caramello's methodology of toposes as bridges has been qualified by

André Joyal André Joyal (; born 1943) is a professor of mathematics at the Université du Québec à Montréal who works on category theory. He was a member of the School of Mathematics at the Institute for Advanced Study in 2013, where he was invited to jo ...

as a "vast extension of Felix Klein's Erlangen Programme" and has been endorsed by

Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...

ists

Alain Connes Alain Connes (; born 1 April 1947) is a French mathematician, and a theoretical physicist, known for his contributions to the study of operator algebras and noncommutative geometry. He is a professor at the , , Ohio State University and Vande ...

and

Laurent Lafforgue Laurent Lafforgue (; born 6 November 1966) is a French mathematician. He has made outstanding contributions to Langlands' program in the fields of number theory and analysis, and in particular proved the Langlands conjectures for the automorphism ...

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

* * with A. C. Russo: * * * with L. Barbieri-Viale and L. Lafforgue: *

References

{{DEFAULTSORT:Caramello, Olivia 1984 births Living people 21st-century Italian mathematicians University of Turin alumni Alumni of Trinity College, Cambridge Italian women mathematicians 21st-century Italian women