����Vlog�ٷ� Cambridge: Dr José Siqueira

Position Bye Fellow
School Physical Sciences Department of Pure Mathematics and Mathematical Statistics
Personal website
Email jvp27@cam.ac.uk

Dr Siqueira is a mathematician working on the Foundations of Mathematics, particularly Category Theory and its relationship to Logic, Geometry, and Computation. Other interests include facets of mathematical thinking and how its understanding affects education, sciences, and languages (broadly construed).

After completing an undergraduate degree in mathematics in Brazil (with one year abroad at the University of Bristol), José moved to Cambridge to undertake Part III of the Mathematical Tripos, followed by a PhD at the Department of Pure Mathematics and Mathematical Statistics. He specialises in Category Theory, a subject classed as part of the Foundations of Mathematics. His research revolves around the abstract theory of distributive laws and categorifications of nonstandard proof methods (in the sense of nonstandard analysis).

Dr Siqueira has been lecturing in Part III of the Mathematical Tripos since Lent 2022 and acted as a College Lecturer and Director of Studies in Mathematics for Gonville & Caius from 2022 to 2024. He was also a Director of Studies for St Edmund's College (where he remains a Director of Studies and Associate Tutor) and Lucy Cavendish College before joining ��Vlog�ٷ� in 2024.

2022: Doctor of Philosophy, University of Cambridge
2016: Master of Advanced Study in Pure Mathematics, University of Cambridge
2014: Bachelor of Mathematics, University of Brasília

The relationship between what we describe and the language that we use to do so is at the core of Logic as a discipline. The tools of category theory facilitate its study in two main ways: we can admit a richer semantics for our languages (by meaningfully interpreting symbols as something other than just sets and functions), and we can treat syntax and semantics on equal grounds, as both are just instances of categories. One of Dr Siqueira's research interests is in using modern ideas in categorical logic to gain insight on complicated logical phenomena by providing adequate categorical models. An ongoing project concerns understanding and expanding the scope of nonstandard proof techniques, singling out the structure needed in order for the proof principles of nonstandard analysis to be valid inside a `topos' --- a mathematical environment that may have a different underlying logic than what you are used to, but that is nonetheless useful and consistent.

Another facet of his work concerns the abstract theory of distributive laws. Everyone is familiar with the fact that multiplication distributes over addition --- category theorists generalised this by introducing the formalism of distributive laws between monads. These are useful because they systematically allow one to combine two different sorts of `algebraic theories' to obtain a new one in a nontrivial way. Dr Siqueira's research on this topic revolves around generalisations of this idea that allow one to merge streams of data that are not necessarily algebraic in origin.