I am a PhD student of Omar León Sánchez interested in the model theory, algebra, and geometry of fields and differential fields. My most recent work has been on establishing results about the existence of model companion for various theories of fields with free operators, as well as about neo-stability theoretic properties transferred from the underlying field to the field with operators.
Here is my CV.
A contribution to the model theory of fields with free operators.
Motivated by structural properties of differential field extensions, we introduce the notion of a theory T being derivation-like with respect to another model-complete theory T₀. We prove that when T admits a model-companion T₊, then several model-theoretic properties transfer from T₀ to T₊. These properties include completeness, quantifier-elimination, stability, simplicity, and NSOP₁. We also observe that, aside from the theory of differential fields, examples of derivation-like theories are plentiful.
Generalising the uniform companion for large fields with a single derivation, we construct a theory UC𝒟 of fields of characteristic 0 with free operators – operators determined by a homomorphism from the field to its tensor product with 𝒟, a finite-dimensional ℚ-algebra – which is the model companion of any theory of a field with free operators whose associated difference field is difference large and model complete. Under the assumption that 𝒟 is a local ring, we show that simplicity is transferred from the theory of the underlying field to the theory of the field with operators, and we use this to study the model theory of bounded, PAC fields with free operators.
We prove that there exists a version of Weil descent, or Weil restriction, in the category of 𝒟-algebras. The objects of this category are k-algebras R equipped with a homomorphism e : R → R ⊗ₖ 𝒟 for some fixed field k and finite-dimensional k-algebra 𝒟. We do this under a mild assumption on the so-called associated endomorphisms. In particular, this yields the existence of the Weil descent functor in the category of difference algebras, which, to our knowledge, does not appear elsewhere.
We show that fields with free operators (in the sense of Moosa and
Scanlon's Model theory of fields with free operators in characteristic
zero
) whose operators pairwise commute can be seen as an instance of
iterative 𝒟-rings (in the sense of the same
authors' Generalised Hasse–Schmidt varieties and their jet spaces
).
Aspects of Stone duality for Boolean algebras.
Supervised by Hilary Priestley.
I have been a teaching assistant for the following courses at the University of Manchester.
Teaching assistant. Approx. 25 students. Teaching and programming assistant. Approx. 50 students. Teaching assistant. Approx. 25 students. Teaching and programming assistant. Approx. 25 students. Teaching assistant. Approx. 15 students. Teaching and programming assistant. Approx. 50 students. Led problem sessions. Approx. 25 students.
Algebraic Structures 1, 2020.
Programming with Python, 2021.
Algebraic Structures 1, 2021.
Contingencies 1, 2022.
Probability 1, 2022.
Introduction to Mathematica, 2023.
0B1: Calculus and Algebra, 2023.