# Research summary

My research lies in the general areas of model theory and real analytic geometry. The notion of o-minimal expansion of the field of real numbers provides a suitable setting for studying generalizations of the theory of semialgebraic sets, and I try to find new explicit examples of such expansions and study their geometric and model-theoretic properties. Many of these expansions are generated by certain solutions of differential equations, and the hope is to obtain new geometric insights from the o-minimality of these structures.

### Expanding the real field by large classes of functions

I constructed two such examples in joint work with my PhD supervisor Lou van den Dries. Both expand the structure of globally subanalytic sets and define the exponential function. Moreover, one of them (the expansion of the field of reals by multisummable series and the exponential function) also defines the Gamma function on all positive real numbers, while the other (the expansion of the field of reals by all convergent generalized power series) defines the Riemann zeta function on all real numbers greater than 1.

Each of these two examples is generated by the functions, restricted to compact sets, belonging to a certain quasianalytic class satisfying closure properties similar to (but weaker than) the closure properties satisfied by the class of all real analytic functions. Their construction relies on a combination of a form of Weierstrass preparation and blowing-ups due to Jean-Claude Tougeron, used to (locally) resolve the zerosets of functions in the respective class.

Jean-Philippe Rolin, Alex J. Wilkie and myself found a variant of normalization inspired by Edward Bierstone and Pierre Milman’s resolution of singularities algorithm. Combining this algorithm with the techniques of the previous two examples, we then showed that certain quasianalytic Denjoy-Carleman classes also generate o-minimal structures. These structures answer two open questions from the theory of o-minimal structures: they show that not every o-minimal structure admits analytic stratification, and that there is a pair of distinct o-minimal structures that are not both reducts of a common o-minimal expansion.

Working with Tobias Kaiser and Jean-Philippe Rolin, I extended the normalization algorithm to polynomially bounded Ilyashenko classes involving asymptotic expansions with arbitrary real exponents (but support of order-type ω). These classes arise in many problems in analysis: the return maps of an analytic vector field in the plane near certain types of singularities belong to such classes, and as Kaiser shows, the solutions of the Dirichlet problem in a planar domain with subanalytic boundary are also of this kind.

One of my current projects with Tobias Kaiser and my student Zeinab Galal involves generalizing the o-minimality to all quasianalytic Ilyashenko classes. To do so, we need to figure out how one-variable functions definable in $\Ranexp$ extend holomorphically to the complex plane. For more, follow the posts on Log-exp-analytic functions and quasianalytic Ilyashenko algebras.

All structures described above are model complete; but do they admit quantifier elimination (say, after adding bounded division) or even a preparation theorem? My student Dan Miller made important progress on this question in his thesis, showing that the expansion of the real field by all restricted differentially algebraic functions has a preparation theorem. Jean-Philippe Rolin and Tamara Servi extended the normalization algorithm to obtain a preparation theorem for very general quasianalytic classes, including the Denjoy-Carleman classes mentioned above.

### Pfaffian functions

In a different approach to finding examples of o-minimal structures, and building on work by Alex Wilkie as well as by Jean-Marie Lion and Jean-Philippe Rolin, I showed that any o-minimal expansion $\R$ of the field of real numbers has in turn an o-minimal expansion $\P(\R)$, called the Pfaffian closure of $\R$, that is closed under taking solutions to Pfaffian differential equations.

Working jointly with Jean-Marie Lion, I verified that the construction of the Pfaffian closure preserves analytic stratification. Refining the techniques used in the proof of the latter, we also show that the Pfaffian closure is model complete in the language of all nested Rolle leaves over $\R$, that is, in the language arising naturally from Khovanskii’s original treatment of Pfaffian systems.

In an attempt to understand the order of growth of functions definable in the Pfaffian closure of an o-minimal structure, Jean-Marie Lion, Chris Miller and myself established a variant of Borel’s conjecture: every non-oscillating solution of an ODE with coefficients definable in a polynomially bounded o-minimal structure is exponentially bounded. Combining this with the model completeness of the Pfaffian closure shows that the Pfaffian closure of a polynomially bounded o-minimal structure is exponentially bounded. In particular, all presently known o-minimal structures are exponentially bounded.

None of the above has so far shed much light on the question whether the expansion of the real field by all pfaffian functions is model complete. Working with Gareth O. Jones, we are now trying to show that the real field expanded by all nested pfaffian functions is model complete and interdefinable with the pfaffian closure of the real field.

### Generalizing o-minimality

Chris Miller has on various occasions suggested to find criteria on expansions of the field of reals (or even the additive group of reals) that are less strict than the requirement of o-minimality, but still do not have the set $\ZZ$ of all integers among their definable sets. (Once $\ZZ$ is definable in an expansion of the real field, all the sets belonging to the projective hierarchy are definable.) As a first step in this direction, Miller and I showed that an expansion $\R$ of the real additive group satisfying that every definable subset of $\RR$ is either finite or uncountable has an o-minimal open core (the open core being the reduct of $\R$ generated by all definable open sets). As a consequence, we obtain that for any subset $A$ of $\RR^n$ definable in $\R$, there is a finite partition of $\RR^n$ into finitely many definable cells such that for each of these cells $C$, either $C$ is contained in $A$, or $C$ is disjoint from $A$, or $A$ is both dense and codense in $C$. (In particular, $\ZZ$ is not definable in $\R$.)

Independently, Tom Scanlon and Alf Onshuus discovered a model-theoretic rank they call þ-rank (pronounced “thorn-rank”). As Onshuus shows in his thesis, þ-rank coincides with U-rank in stable structures; however, þ-rank also makes sense in ordered structures. For example, any o-minimal structure has þ-rank 1; in general, any þ-rank 1 ordered structure has o-minimal open core. Working with Alf Dolich, we consider an arbitrary planar differentiable vector field definable in some o-minimal expansion of the real field as an example of such a structure, by representing its flow model-theoretically as a (piecewise) ordered structure. Assuming that no cycle of the vector field is an accumulation point of limit cycles (known to be true for analytic vector fields, for instance), we show that this structure has þ-rank 2 if and only if the corresponding vector field has finitely many limit cycles.

### Trajectories of vector fields

Fernando Sanz found a polynomial vector field in $\RR^3$ for which he showed, together with Felipe Cano and Robert Moussu, that some of its trajectories are non-oscillatory with respect to globally subanalytic subsets of $\RR^3$. Rolin, Sanz and Schaefke then showed that any such trajectory actually generates an o-minimal structure. Thus, these trajectories give rise to pairwise incompatible o-minimal structures, in the sense that no two of them are both reducts of any one o-minimal structure. However, as opposed to the incompatible examples generated by the quasianalytic Denjoy-Carleman classes, the examples in this case are all generated by solutions of the same ODE.