One approach to work on singular problems in hydrodynamics is to see them as small parameter problems. In particular, the formal study of incompressible and compressible boundary layers can be stated through the application of diffeomorphisms over a rectangular domain. On this occasion, we will talk about how a particular change of variables gives us simplified limit formulas for a gaseous boundary layer with total constant energy, and the questions that remain.

In 2004, K Nishioka proved that if  y_1,..., y_n are solutions of P_I : y''= 6 y2 + x such that

tr.deg._{C(x)}(y_1,y'_1,..., y_n,y'_n) < 2n

then there exists i<j such that y_i = y_j. In this talk I will explain how the Galois Groupoid of the differential equation can be used to prove (half of) this theorem. When a second order equation has a transitive, simple, infinite dimensional Galois Groupoid then if  y_1,..., y_n are solutions such that tr.deg._{C(x)}(y_1,y'_1,..., y_n,y'_n) < 2n,

• there exist i such that tr.deg._{C(x)}(y_i,y'_i) <2, or

• there exist i<j such that  tr.deg._{C(x)}(y_i,y'_j, y_i,y'_j) =2.

Ax-Schanuel and Ax-Lindenmann type results specify the kind of algebraic relations that are preserved by some transcendental functions. For instance, if there is some algebraic relation x and y and also between j(x) and j(y), being j the modular function, then there x and y are related by a Moebius transformation with rational coefficients and j(x) and j(y) satisfy a modular polynomial. In this talk we will review recent research on the topic and speak about the link of this kind of results and the differential Galois theory for principal connections.

In this talk I will present an overview of the use of Newton's polygon in valued fields and differential valued fields to obtain formal solutions of algebraic equations and algebraic differential equations. 

In this talk I shall present various recent results joint with different collaborators, A. Dzhamay, A. Stokes, A. Ligeza and T. Kecker, which can be obtained for the Painleve and quasi-Painleve equations using the geometric approach. I shall briefly explain the analogue of the polynomial Hamiltonian structure for the Takasaki equation and general quasi-Painleve equations.

Implicit differential equations (IDEs) occur naturally in differential geometry of surfaces. For example, on a regular surface in the Euclidean or Minkowski 3-space the pairs of foliations given by the asymptotic and principal curves are solutions of IDEs that are written in the form

a(x,y) dy² + b(x,y) dxdy + c(x,y) dx²=0.

Such IDEs are often called Binary Differential Equations (BDEs).

I will present in this talk some old and new results on local invariants of BDEs and IDEs. An invariant of an IDE/BDE, introduced in [Bruce-Tari, 1998] and called the multiplicity of the IDE/BDE at its singular point, counts the maximum number of stable singularities that can appear in a local deformation of the equation. Umbilic points on a generic surface are stable, that is, they persist under small deformations of the surface and the local configuration of the lines of principal curvature also remain unchanged under the deformation. However, the BDE of these lines is not stable when deformed within the set of all BDEs. In [Fernandes-Tari, to appear] we introduce an invariant of an analytic surface M in the Euclidean/Minkowski 3-space at its umbilic points, and call it the multiplicity of the umbilic point. The multiplicity of an umbilic  counts the maximum number of stable umbilic points that can appear under small deformations of the surface at a non-stable umbilic point.

We consider analytic differential operators of order n defined on two-dimensional manifolds. Namely, linear operators locally of the form

Σ fij (∂/∂x)^i  (∂/∂y)^j

with fij analytic functions. After introducing the notion of elementary singular point for such operators, we discuss a theorem of resolution of singularities, generalizing the classical result of Bendixson-Seidenberg for vector fields in dimension two.

The Theorem this talk is concerned with was published by J. Denef and L. Lipshitz in the Mathematische Annalen (1984). It states that there exists an algorithm which solves the following problem: given a finite system of polynomial ODE with coefficients in Q[x], decide whether this system admits for solution a tuple of formal power series in K[[x]] where K is the field of the complex numbers (some other fields are considered in the paper). The proof is complicated. It is supposed to be algorithmic but, as far as I know, has never been implemented. In this talk, I will present its main ingredients.

In this talk we intend to give an overview of some of the existing methods for solving algebraic differential equations through the use of algebraic geometry techniques. More specifically, the idea is to associate an algebraic variety to the differential equation (or equations) and, using properties of this variety, infer information that allows the differential equation to be solved or simplified to be able to approach it with other techniques. In this sense, the main tool to be used will be the existence of parameterizations of the associated variety.

Following the spirit of the seminar, the talk will focus on the ideas, without trying to cover the more technical aspects; all with the help of examples.