#### Geodesic Flows and Model Theory of Differential Fields

PhD manuscript of my PhD from September 2014 to June 2017 under the direction of Jean-Benoît Bost (Orsay) and Martin Hils( Münster).

PhD manuscript of my PhD from September 2014 to June 2017 under the direction of Jean-Benoît Bost (Orsay) and Martin Hils( Münster).

ArXiv version (51 pages)

** Abstract ** : Two theorems witnessing the abundance of geometrically trivial strongly minimal autonomous differential equations of arbitrary order are shown. The first one states that a generic algebraic vector field of degree \( d > 1 \) on the affine space of dimension \( n > 1 \) is strongly minimal and geometrically trivial. The second one states that if \(X_0\) is the complement of a smooth hyperplane section \( H_X \) of a smooth projective variety \( X \) of dimension \( n > 1 \) then for \(d \) sufficiently large, the system of differential equations associated with a generic vector field on \( X_0 \) with poles of order at most \( d \) along \( H_X \) is also strongly minimal and geometrically trivial.

With Rahim Moosa, ArXiv version (32 pages)

** Abstract ** : Several results on the birational geometry of algebraic vector fields in characteristic zero are obtained. In particular, (1) it is shown that if some cartesian power of an algebraic vector field admits a nontrivial rational first integral then already the second power does, (2) two-dimensional isotrivial algebraic vector fields admitting no nontrivial rational first integrals are classified up to birational equivalence, (3) a structural dichotomy is established for algebraic vector fields (of arbitrary dimension) whose finite covers admit no nontrivial factors, and (4) a necessary condition is given on the Albanese map of a smooth projective algebraic variety in order for it to admit an algebraic vector field having no nontrivial rational first integrals. Analogues in bimeromorphic geometry, for families of compact Kaehler manifolds parametrised by Moishezon varieties, are also obtained[...]

With Leo Jimenez and Anand Pillay, ArXiv version (26 pages). To appear in Advances in Mathematics, 2023.

** Abstract **: We first elaborate on the theory of relative internality in stable theories, focusing on the notion of uniform relative internality (called collapse of the groupoid in an earlier work of the second author), and relating it to orthogonality, triviality of fibrations, the strong canonical base property, differential Galois theory, and GAGA. We prove that \(DCF_0 \) does not have the strong canonical base property, correcting an earlier proof. We also prove that the theory \(CCM\) of compact complex manifolds does not have the strong CBP, and initiate a study of the definable Galois theory of projective bundles. In the rest of the paper we study definable fibrations in \(DCF_0 \), where the general fibre is internal to the constants, including differential tangent bundles, and geometric linearizations. We obtain new examples of higher rank types orthogonal to the constants.

With James Freitag, David Marker and Joel Nagloo (49 pages). To appear in International Mathematics Research Notices, 2022.

** Abstract ** : We study the structure of the solution sets in universal differential fields of certain differential equations of order two, the Poizat equations, which are particular cases of Liénard equations. We give a necessary and sufficient condition for strong minimality for equations in this class and a complete classification of the algebraic relations for solutions of strongly minimal Poizat equations. We also give an analysis of the non strongly minimal cases as well as applications concerning the Liouvillian and Pfaffian solutions of some Liénard equations.

With James Freitag and Rahim Moosa (5 pages). To appear in the Journal of Mathematical Logic, 2022.

** Abstract ** : It is shown that if \( p \) is a complete type of Lascar rank at least 2 over \( A \), in the theory of differentially closed fields of characteristic zero, then there exists a pair of realisations, \( a_1 \) and \( a_2 \), such that \( p \) has a nonalgebraic forking extension over \( A,a_1,a_2 \). Moreover, if A is contained in the field of constants then \( p \) already has a nonalgebraic forking extension over \( A,a_1 \). The results are also formulated in a more general setting.

With James Freitag and Rahim Moosa. Published in Invent. Math. 230 (2022), no. 3, 1249–1265.

** Abstract ** : Given an algebraic differential equation of order greater than one, it is shown that if there is any nontrivial algebraic relation amongst any number of distinct nonalgebraic solutions, along with their derivatives, then there is already such a relation between three solutions. In the autonomous situation when the equation is over constant parameters the assumption that the order be greater than one can be dropped, and a nontrivial algebraic relation exists already between two solutions. These theorems are deduced as an application of the following model-theoretic result: Suppose p is a stationary nonalgebraic type in the theory of differentially closed fields of characteristic zero; if any three distinct realisations of p are independent then p is minimal. If the type is over the constants then minimality (and complete disintegratedness) already follow from knowing that any two realisations are independent. An algebro-geometric formulation in terms of D-varieties is given. The same methods yield also an analogous statement about families of compact Kähler manifolds.

Published in Algebra and Number Theory 15 (2021) pp. 2449-2483

**Abstract**: In this article, we study model-theoretic properties of algebraic differential equations of order 2, defined over constant differential fields. In particular, we show that the set of solutions of a general differential equation of order 2 and of degree \( d≥3 \) in a differentially closed field is strongly minimal and disintegrated. We also give two other formulations of this result in terms of algebraic (non)-integrability and algebraic independence of the analytic solutions of a general planar algebraic vector field.

Published in Confluentes Mathematici, Tome 12 (2020) no. 2, pp. 49-78.

** Abstract **: In this article, we develop a geometric framework to study the notion of semi-minimality for the generic type of a smooth autonomous differential equation \( (X,v) \) , based on the study of rational factors of \( (X,v) \) and of algebraic foliations on \( X \) , invariant under the Lie-derivative of the vector field \( v \).
We then illustrate the effectiveness of these methods by showing that certain autonomous algebraic differential equation of order three defined over the field of real numbers --- more precisely, those associated to mixing, compact, Anosov flows of dimension three --- are generically disintegrated.

Published in Bull. de la SMF, Tome 148 (2020) 529-595

** Abstract ** : Orthogonality to the constants is property of an algebraic differential equation that originated from the model-theoretic study of differential fields and that expresses remarkable independence properties for its solutions.In this article, we study the property of orthogonality to the constants in a differential algebraic language for autonomous differential equations and describe some effective methods to establish this property. The main result is a criterion for orthogonality to the constants (and its version for families) for real absolutely irreducible D-varieties \( (X,v) \) based on the dynamical properties of the associated real analytic flow. More precisely, we show that if there exists a compact region \( K \) of \( M \), Zariski-dense in \( X \) and such that the restriction of the flow to \( K \) is topologically weakly mixing then the generic type of \( (X,v) \) is orthogonal to the constants. This criterion will be applied in a second part to study from this model-theoretic point of view the geodesic flow of compact Riemannian varieties (presented algebraically) with negative curvature.

Published in Israel Journal of Mathematics volume 230, pages 527–561(2019).

**Abstract**: We define the notion of a smooth pseudo-Riemannian algebraic variety \( (X,g) \) over a field \( k \) of characteristic 0, which is an algebraic analogue of the notion of Riemannian manifold and we study, from a model-theoretic perspective, the algebraic differential equation describing the geodesics on \( (X,g) \). When k is the field of real numbers, we prove that if the real points of \( (X,g) \) are Zariski-dense in \( X \) and if the real analytification of \( (X,g) \) is a compact Riemannian manifold with negative curvature, then the algebraic differential equation describing the geodesics on \( (X,g) \) is absolutely irreducible and its generic type is orthogonal to the constants.