# Model theory, algebra, and differential equations

Lead Research Organisation:
University of Leeds

Department Name: Pure Mathematics

### Abstract

The project concerns applying ideas from model theory, a branch of mathematical logic, to the classification of differential equations. Model theory studies structures and definable sets. A structure in this sense is roughly speaking an underlying set or universe together with some collection of distinguished sets of n-tuples from the set, for varying n. Many parts or objects of mathematics can be fruitfully studied as structures. Differential equations, on the other hand, are equations which describe the rate of change, for example, of some quantity, and they lie behind all of modern science and much of modern technology. The project involves roughly viewing the set of solutions of a differential equation as a structure in the sense of model theory, and attempting to classify or understand such equations via the tools of model theory.

### Planned Impact

The interdisciplinary and cross-disciplinary nature of the project relating abstract notions from logic to questions of interest to applied mathematicians and even physicists, represents a somewhat new kind of collaboration in the scientific arena, the success of which will impact broadlly on culture and science. A part of the project deals with equations (Painleve) which are at the forefront of nonlinear science , and whose solutions are relevant to a broad spectrum of scientific problems relevant to society, including transportation and queueing problems. It will be relevant to enhancing and expanding our mathematical toolkit to face the challenges of the 21st century. A website and blog around the project will be set up, and articles around the project, intended for a broad audience, will be written and published.

### Organisations

### Publications

ADLER H
(2014)

*GENERIC STABILITY AND STABILITY*in The Journal of Symbolic Logic
Bays M
(2018)

*UNIVERSAL COVERS OF COMMUTATIVE FINITE MORLEY RANK GROUPS*in Journal of the Institute of Mathematics of Jussieu
Benoist F
(2016)

*On function field Mordell-Lang and Manin-Mumford*in Journal of Mathematical Logic
Benoist F
(2014)

*SEMIABELIAN VARIETIES OVER SEPARABLY CLOSED FIELDS, MAXIMAL DIVISIBLE SUBGROUPS, AND EXACT SEQUENCES*in Journal of the Institute of Mathematics of Jussieu
Bertrand D
(2015)

*Galois theory, functional Lindemann-Weierstrass, and Manin maps*
Bertrand D
(2016)

*Relative Manin-Mumford for Semi-Abelian Surfaces*in Proceedings of the Edinburgh Mathematical Society
Bertrand D
(2016)

*Galois theory, functional Lindemann-Weierstrass, and Manin maps*in Pacific Journal of Mathematics
Caudrelier V
(2014)

*Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency*in Symmetry, Integrability and Geometry: Methods and Applications
Chernikov A
(2014)

*External definability and groups in NIP theories*in Journal of the London Mathematical Society
Conversano A
(2012)

*Connected components of definable groups and o -minimality I*in Advances in MathematicsDescription | Over the course of the project we solved one of the main problems raised in the proposal: the analysis of the 6 families of Painleve equations vis--a-vis transcendence properties of sets of solutions, and model-theoretic notions of strong minimality and geometric triviality. This appears in the two joint papers of Nagloo and Pillay, as well as one paper by Nagloo. Some major progress on transcendence problems for exponential function relative to families of semiabelian varieties was obtained by Bertrand and Pillay, extending earlier results published in Journal AMS. These results are being written up now. Another related development was joint work with Bertrand, Masser and Zannjier on relative Manin-Mumford for semiabelian surfaces. Over the course of the project there were other key findings. The closest in relation to the project theme was joint with Benoist and Bouscaren and involved a new proof of the celebrated Mordel-Lang conjecture for function fields in positive characteristic. Also we intitiated an in-depth study of topological dynamics and definable groups, following the lead of Newelski. Several papers were written, including two with the postdoctoral fellow on the project, Davide Penazzi. In the last 6 months of the project, due to early departure of the postdoctoral fellow, the remaining time of that position was taken up by Dr Cheng Zhang, who investigated aspects of integrable difference equations. In collaboration with Nijhoff he developed a boundary Lagrangian theory for integrable partial difference equations. |

Exploitation Route | An ongoing project between Nijhoff and Nagloo involves the geometry of higher order Painleve systems, the so-called Garnier systems. These form a yet unchartered area of research from the point of view of Model Theory and would form an exciting area in which the research of the project can be taken forward. The issues of transcendality and irreducibility of such systems form an important, but challenging area for the follow-up research, as they would reach for an insight into the understanding of novel classes of nonlinear special functions possessing properties which make them of interest for physics and the mathematical sciences of the future. |

Sectors | Digital/Communication/Information Technologies (including Software),Electronics,Manufacturing, including Industrial Biotechology |