I’ll show how tools from model theory give a clean structural picture of the classical Lotka–Volterra predator–prey system. For generic parameters, its solution set is strongly minimal and geometrically trivial—informally, distinct non-constant solutions have no hidden algebraic relations. I’ll sketch the central ideas of the proof. 

Exponents are numerical invariants attached to regular singular formal differential equations, corresponding to the eigenvalues of the associated monodromy representation. They play a central role as local invariants of linear differential equations, being essential both for detecting the algebraicity of solutions and for classifying such equations. Informally, a complex number a is called an exponent if the equation admits a solution of the form y(t)=t^a f(t). A direct computation shows that the exponents arise as the zeros of a certain polynomial, the indicial polynomial.

However, this classical approach breaks down when the differential equation is not regular singular. In this talk, we present a construction that associates to any differential module of dimension n over a field of formal power series a complete multiset of n complex numbers, extending the notion of exponents to the general case. This multiset behaves well under standard operations such as tensor products, duals, and exact sequences. The construction is achieved through Tannakian methods, which also allow us to revisit, in the first part of the talk, the classical theory of differential modules over formal power series in a unified framework. 

The approach is sufficiently general to extend to other settings, namely to p-adic differential equations, which has been our original motivation. Time permitting, we will briefly discuss this aspect.

This is a joint work with Christopher Lazda (University of Exeter). 

In this talk, we present our recent work on the structure of differential subfields of strongly normal extensions and its applications to the study of algebraic differential equations. We revisit certain classical results of Goldman, Singer and Rosenlicht on solution fields of first order nonlinear differentials and explain how our results can be used to extend these results to higher order algebraic differential equations. We will also discuss differential algebraic dependence of solutions of a differential equation. This is a joint work with Varadharaj Ravi Srinivasan.