EPSRC Reference: 
EP/V03619X/1 
Title: 
Model theory of Dlarge fields and connections to representation theory. 
Principal Investigator: 
Leon Sanchez, Dr O 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
University of Manchester, The 
Scheme: 
New Investigator Award 
Starts: 
01 January 2022 
Ends: 
31 December 2024 
Value (£): 
351,043

EPSRC Research Topic Classifications: 
Algebra & Geometry 
Logic & Combinatorics 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
The aim of this project is to build further connections between model theory and representation theory of algebras. This is driven by a promising line of research initiated five years ago, by the PI (together with Bell, Launois, and Moosa), that exploits the geometricstability machinery from model theory to provide a new approach to the DixmierMoeglin equivalence  a program to classify irreducible representations of noetherian algebras.
In more detail, this project aims at exploring further and unifying the model theory of tame fields with generic operators. We investigate the modeltheoretic properties of large fields equipped with generic additive operators (denoted by D) obeying certain multiplicative and commutative rules. Our results naturally lead to the notion of Dlarge field, the analogue of large fields in the Doperators setting, and we explore their role in Dfield arithmetic and Inverse DGalois questions. These developments are then deployed in the representation theory of noetherian algebras. Namely, we use the modeltheoretic machinery to characterize primitive ideals, which roughly classify irreducible representations, in purely topological and algebraic terms for a wide class of noetherian Hopf algebras.
Model theoretic algebra (or rather, the model theory of fields with operators) studies in particular the algebraic, and also many times analytic, structure of rings equipped with commuting derivations. Classical examples are rings of smooth functions and fields of meromorphic functions (in several variables), equipped with the usual differentiation operators. Most of the differential field theory can be explored in parallel to its classical algebraic counterpart. For instance, there are differential analogues of algebraically closed, real closed, and padically closed fields. Furthermore, in the spirit of Galois theory for polynomial equations, a beautiful differential Galois theory for linear differential equations has been developed and used in functional transcendence questions.
Representation theory, on the other hand, is one of the most influential fields of pure mathematics. Its development has been driven by challenging, yet very basic problems. In particular, one of the fundamental questions is to classify the irreducible representations of a given noetherian algebra (which is often quite difficult). A now standard approach to this problem is to study the kernels of irreducible representations  the socalled primitive ideals. In the case of enveloping algebras of finite dimensional complex Liealgebras, Dixmier and Moeglin proved that primitive ideals can be characterised purely algebraically and topologically. These characterisations initiated the interest in what is nowadays known as the DixmierMoeglin equivalence.
In broad terms, this project is guided by two broad visions:
(1) Derivations are simply additive operators induced by the dual numbers (a special case of a local finite algebra), we aim to unify the model theory and Galois theory of all operators with multiplicative and commutative rules induced from ANY local finite algebra (on specific classes of large fields). This includes the important case of HasseSchmidt derivations (on algebraically and real closed fields, for instance).
(2) Exploit the above modeltheoretic results (in particular, the geometricstability tools) to tackle the classification of irreducible representations of noetherian algebras. More precisely, shed a light in the BellLeung conjecture stating the all finitely generated noetherian Hopf algebras of finite GelfandKirillov dimension satisfy the DixmierMoeglin equivalence. We aim to prove this equivalence for a wide family of important cases (iterated HopfOre extensions), and its Poissonversion in full generality.

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