In this lecture, we study the global qualitative behaviour of fields of principal directions for the graph of a real valued polynomial function \(f\) on the plane. We will prove that every umbilic point at infinity of the projective extension of these direction fields has a Poincar\'e-Hopf index equal to 1/2 and the topological type of a Lemon or a Monstar. As a consequence, we will provide a Poincar\'e-Hopf type formula for the graph of \(f\) pointing out that, if all umbilics are isolated, the sum of all indices of the principal directions at its umbilic points only depends upon the number of real linear factors of the homogeneous part of highest degree of \(f\).

This is a joint work with B. Guilfoyle.

In this talk, we will use a form of the maximum principle and an iteration method to show existence of solutions to the Dirichlet problem: \[\left\{\begin{array}{l}-\Delta u = \lambda\left(x\right)f\left(u\right)+h\left(x\right)\quad \mbox{in}\quad \Omega,\\u=0 \quad \mbox{on}\quad \partial \Omega,\end{array}\right.\]in bounded and thin unbounded domains. Also, we will show how these methods can be applied to show existence of nontrivial solutions to the Lane-Emden system\[\left\{\begin{array}{l}-\Delta u=v^p,\quad -\Delta v=u^p, \quad \mbox{in}\quad \Omega\\u=v=0, \quad\mbox{on}\quad \partial \Omega. \end{array}\right.\]

This is joint work with Jonatán Torres-Orozco.

Slow-fast Hamiltonian systems are characterized by a separation of the phase space into slow and fast parts typically identified by a small (or slow) parameter. This kind of Hamiltonian system is not integrable, in general; even though in one degree of freedom. In this lecture, we study an integrable model associated with a slow-fast Hamiltonian system of two degrees of freedom. Thinking of the slow parameter as a perturbative one, making suitable symmetry assumptions, and using normal form theory, we show that a slow-fast Hamiltonian system in two degrees of freedom is close to an integrable Hamiltonian model. What we gain with this model is the possibility to associate a family of Lagrangian 2-tori which is almost invariant with respect to the original slow-fast Hamiltonian system. As an important application, this family of almost invariant Lagrangian 2-tori can be used to compute approximations to the spectrum of the quantum model associated with the slow-fast Hamiltonian systems.

A line congruence in the Euclidean space of dimension \(3\) is a \(2\) parameter family of lines in \(\mathbb R^{3}\).

The first record about line congruences appeared in "Mémoire sur la Théorie des Déblais et des Remblais" (1776,1784) where Gaspard Monge seeks to solve a minimizing cost problem of transporting an amount of land from one place to another, preserving the volume.

After Monge, Ernst Eduard Kummer in ''Allgemeine Theorie der geradlinigen Strahien systeme", was the first to deal exclusively with the general theory of line congruences. Mainly due to optical applications, line congruences started to gain importance, increasing even more with the development of technology. In this lecture, we will deal with line congruence when the parameters vary in a surfaces with singularities. We will build the theory of congruence lines when the parameter space is a Frontal. As application, we will consider line congruence of cuspidal edges. We will show relations between singularities of the congruence lines and geometric properties of initial cuspidal edges.

This a joint work with Tito Medina(ICMC/USP), Igor Chagas Santos(ICMC/USP) and Maria Aparecida S. Ruas(ICMC/USP).

A line congruence is a \(2\)-dimensional arrangement of lines in \(3\)-space. It is a classical topic of projective differential geometry and has many relations with binary differential equations. In this talk we give a geometric description of the generic singularities of the focal surface of a line congruence. We show that these singularities occur along the ridge and double eigenvalue curves. A basic tool is the support function associated with an eqüiaffine vector field transversal to a surface in \(\mathbb{R}^3\).

This is a joint work with Ronaldo Garcia (UFG, Brazil).

A classical result of F. Klein states that, given a finite primitive group \(G\subseteq SL_2(\mathbb{C})\), there exists a hypergeometric equation such that any second order LODE whose differential Galois group is isomorphic to \(G\) is projectively equivalent to the pullback by a rational map of this hypergeometric equation. In this paper, we generalize this result. We show that, given a finite primitive group \(G\subseteq SL_n(\mathbb{C})\), there exist a positive integer \(d=d(G)\) and a standard equation such that any LODE whose differential Galois group is isomorphic to \(G\) is gauge equivalent, over a field extension \(F\) of degree \(d\), to an equation projectively equivalent to the pullback by a map in \(F\) of this standard equation. For \(n=3\), these standard equations can be chosen to be hypergeometric.

The theory of Frobenius manifolds has important relevance in mathematics since the application to enumerative geometry with the counting of the number of zero genus curves of degree \(d\) passing through \(3d-1\) points in the projective space \(\mathbb{C} P^2\) (M. Kontsevich).

Let \(\mathbb{A}\) be a finite-dimensional commutative associative algebra with unit over \(\mathbb{R}\). A map \(F: \mathbb{A}^n \to \mathbb{A}^n\) is called \(\mathbb{A}\)-differentiable if \(dF\) is \(\mathbb{A}\)-lineal. Let \(\Gamma_\mathbb{A}\) be the pseudogroup of local \(\mathbb{A}\)-diffeomorphisms of \(\mathbb{A}^n\). An \(\mathbb{A}\)-manifold is a manifold endowed with \(\Gamma_\mathbb{A}\)-atlas. An important example of \(\mathbb{A}\)-manifold is the total space of A. Weil's bundle of \(\mathbb{A}\)-closed points.

The theory of \(\mathbb{A}\)-manifolds (A.P. Shirokov, V.V. Vishnevskii, G.I. Kruchkovich, V.V. Shurygin) has strong relations to the foliation theory and to geometry of jet bundles and the theory of natural operations in differential geometry (I. Kolář, J. Slovák, P. W. Michor).

In our talk we will show that Frobenius manifolds in the sense of B. Dubrovin and N.J. Hitchin have the structure of an \(\mathbb{A}\)-manifold, where \(\mathbb{A}\) is a Frobenius algebra. We will introduce Weil coalgebras, the \(G\)-equivariant graduated versions of Weil algebras, and the corresponding bundle structures.

This talk deals with the averaging theory, which is a classical tool allowing us to study periodic solutions of the nonlinear differential systems.

We present some applications of averaging theory. Also, we shall show a system where the averaging method fails to detect periodic orbits.

At the International Congress of Mathematics held in Paris in 1900, D. Hilbert presented a list of 23 problems for the century. After more than one hundred years some of those problems are still unsolved. Among them is the 16th problem. This problem, in its second part, asks for the number and position of limit cycles (isolated periodic solutions) of planar polynomial differential equations. In the 70's, Y. Ilyashenko and V. I. Arnold proposed, in this context, to investigate what happens for systems infinitely close to Hamiltonians. This approach is known as the infinitesimal Hilbert 16th problem. More precisely, let \(F\in\mathbb{R}[x,y]\) a polynomial and \(\omega\) a polynomial 1-form. We consider the perturbation \(dF+\epsilon\omega=0\) of the Hamiltonian foliation \(dF=0\), where \(\epsilon\) is a small parameter. Let \(\gamma(z)\subset F^{-1}(z)\) a continuous family of regular periodic orbits of \(dF=0\). The displacement map with respect to this family \(\gamma(z)\) and with respect to the foliation \(dF+\epsilon\omega=0\) is an analytic function \(\Delta(z,\epsilon)=\epsilon^\mu M_\mu(z)+\cdots\). The first function \(M_\mu\) which does not vanish identically keeps information about limit cycles born from this perturbation. By passing to the complexification of \(F\), we can consider the orbit under monodromy of \(\gamma(z)\), which is a normal subgroup \(\mathcal{O}\) of the first homotopy group of the regular fiber \(F^{-1}(z)\). To understand the orbit under monodromy one must focus on the critical values of the function \(F\). To each critical value corresponds a vanishing cycle; the monodromy related to such cycle detects the traces of change in the topology due to the presence of the singular fiber. In non generic cases, not every cycle can be reached by the orbit under monodromy of the family of regular periodic orbits \(\gamma(z)\). Thus, to detect the cycles that are not reached one considers the quotient \(\frac{\mathcal{O}}{[\mathcal{O},\pi_1(F^{-1}(z),p_0)]}\) and defines a constant \(\kappa\), called orbit depth. It turns out that the function \(M_\mu\) is an iterated integral of length at most \(\kappa\). We stress that \(\kappa\) by its definition depends only on the topology of the regular fiber of \(F\), and on the orbit \(\mathcal{O}\) of \(\gamma(z)\), while the function \(M_\mu\) depends also on the perturbation given by \(\omega\). We will talk about this result and discuss some conjectures around this bound for non-generic polynomials.

The huge group of polynomial diffeomorphisms of \(\mathbb{C}^2\) acts on the space of polynomials \(\{ f: \mathbb{C}^2 \longrightarrow \mathbb{C} \}\) by coordinate changes. The classification of polynomials under this action is a challenging open problem. We obtain a very concrete result for degree three polynomials. An application to perturbation of singular Hamiltonian foliations on \(\mathbb{C}^2\), having \(\mathbb{C} \backslash \{ k \, \hbox{points}\}\), as generic leafs, is provided.

Joint work with John Alexander Arredondo (Colombia) and Salomón Rebollo (Chile).