# Geometry of Differential Equations, Real and Complex

Organizers:

- Ronaldo Alves Garcia (Federal University of Goiás)
- Mikhail Malakhaltsev (University of los Andes)
- Jesus Mucino Raymundo (National Autonomous University of Mexico)
- Daniel Offin (Queens University)
- Farid Tari (University of São Paulo)

Schedule:

- Tuesday, Jul 25 [McGill U., Arts Building, Room W-120]
- 11:45 Kenneth Meyer (University of Cincinnati, USA), Asymptotic Stability Estimates near an Equilibrium Point
- 12:15 Alexander Cardona (University of los Andes, Colombia), Spectral invariants and global pseudo-differential calculus on homogeneous spaces
- 14:15 Johanna Garcia Saldaña (Catholic University of the Most Holy Conception, Chile), An approach to the period function through the harmonic balance
- 14:45 Jesus Mucino Raymundo (National Autonomous University of Mexico, Morelia, Mexico), Essential singularities of complex analytic vector fields on $\mathbb{C}$
- 15:45 Regilene Delazari dos Santos Oliveira (Universidade de São Paulo, Campus de S. Carlos), Singular levels and topological invariants of Morse Bott integrable systems on surfaces
- 16:15 Nabil Kahouadji (Northeastern Illinois University, USA), Isometric Immersions of Pseudo-Spherical Surfaces via Differential Equations
- 17:00 Ana Rechtman (National Autonomous University of Mexico, Mexico City, Mexico), The trunkennes of a flow
- 17:30 Mikhail Malakhaltsev (University of los Andes, Colombia), Binary differential equations and 3-webs with singularities
- Wednesday, Jul 26 [McGill U., Arts Building, Room W-120]
- 11:15 Alessandro Portaluri (University of Torino, Italy), Index and stability of closed semi-Riemannian geodesics
- 11:45 Martha P. Dussan Angulo (Universidade de São Paulo, Brazil), Bjorling problem for timelike surfaces and solutions of homogeneous wave equation
- 13:45 Débora Lopes da Silva (Universidade Federal do Sergipe, Brazil), Codimension one partially umbilic singularities of hypersurfaces of $\mathbb{ R}^4$
- 14:15 Jean Carlos Cortissoz (University of los Andes, Colombia), On Bloch's Theorem
- 14:45 Adolfo Guillot Santiago (National Autonomous University of Mexico, Cuernavaca, Mexico), Algebraic differential equations with uniform solutions
- 15:15 Frederico Xavier (Texas Christian University, USA), On the inversion of real polynomial maps
- 16:15 Ronaldo Alves Garcia (Universidade Federal de Goiás, Brazil), Darboux curves on surfaces
- 16:45 Daniel Offin (Queens University, Canadá), Multiple periodic solutions in classical Hamiltonian systems.

- Kenneth Meyer

University of Cincinnati, USAAsymptotic Stability Estimates near an Equilibrium PointWe use the error bounds for adiabatic invariants found in the work of Chartier, Murua and Sanz-Serna to bound the solutions of a Hamiltonian system near an equilibrium over exponentially long times. Our estimates depend only on the linearized system and not on the higher order terms as in KAM theory, nor do we require any steepness or convexity conditions as in Nekhoroshev theory. We require that the equilibrium point where our estimate applies satisfy a type of formal stability called Lie stability. - Alexander Cardona

University of los Andes, ColombiaSpectral invariants and global pseudo-differential calculus on homogeneous spacesGlobal pseudo-differential calculus on compact Lie groups and homogeneous spaces gives, via representation theory, a semi-discrete description of the global analysis and spectral theory of a wide class of operators on these objects. Based on the theory developed by Ruzhansky and Turunen, during this talk we will consider spectral invariants of index type for global homogeneous pseudo-differential operators; some examples and potential applications will be addressed. - Johanna Garcia Saldaña

Catholic University of the Most Holy Conception, ChileAn approach to the period function through the harmonic balanceEach differential system with a period annulus P has associated a period function T. The geometry of T is determined by the number and properties of its critical periods, i.e. its critical points. The critical periods in T is a counterpart to the problem of limit cycles in polynomial systems, since they play a fundamental roll in the geometry of the phase portrait of the differential system. In fact, several analytic techniques have been developed for studying these problems. In this talk we will show that the shape of T can be recovered by using a quantitative approach: the harmonic balance. - Jesus Mucino Raymundo

National Autonomous University of Mexico, Morelia, MexicoEssential singularities of complex analytic vector fields on $\mathbb{C}$Let $X$ be a complex analytic vector field on $(\mathbb{C}, 0)$. The real trajectories of it are geodesics of a suitable singular flat metric. Analogously to the classical Picard's Theorem, in the vicinity of an isolated essential singularity, the local complexity of $X$ must be studied using certain ``global'' flow box maps. We describe the geometry encoded is the simplest cases, for these kind of singularities. \noindent Joint work with A. Alvarez--Parrilla. - Regilene Delazari dos Santos Oliveira

Universidade de São Paulo, Campus de S. CarlosSingular levels and topological invariants of Morse Bott integrable systems on surfacesIn this talk we shall classify (up to homeomorphisms) closed curves and eights of saddle points on orientable closed surfaces. This classification is applied to Morse Bott foliations and Morse Bott integrable systems to define a complete invariant. We also present a realization Theorem based in two transformations and one generator. These results are part of a joint work with José Martínez-Alfaro (UV, Spain) and Ingrid Sarmiento-Mesa (IBILCE-UNESP, Brazil). - Nabil Kahouadji

Northeastern Illinois University, USAIsometric Immersions of Pseudo-Spherical Surfaces via Differential EquationsPseudo-spherical surfaces are surfaces of constant negative Gaussian curvature. A way of realizing such a surface in 3d space as a surface of revolution is obtained by rotating the graph of a curve called tractrix around the z-axis (infinite funnel). There is a remarkable connection between the solutions of the sine-Gordon equation $u_{xt}=\sin u$ and pseudo-spherical surfaces, in the sense that every generic solution of this equation can be shown to give rise to a pseudo-spherical surface. Furthermore, the sine-Gordon equation has the property that the way in which the pseudo-spherical surfaces corresponding to its solutions are realized geometrically in 3d space is given in closed form through some remarkable explicit formulas. The sine-Gordon equation is but one member of a very large class of differential equations whose solutions likewise define pseudo-spherical surfaces. These were defined and classified by Chern, Tenenblat and others, and include almost all the known examples of "integrable" partial differential equations. This raises the question of whether the other equations enjoy the same remarkable property as the sine-Gordon equation when it comes to the realization of the corresponding surfaces in 3d space. We will see that the answer is no, and will provide a full classification of second-order hyperbolic and $k$th-order evolution equations. The classification results will show, among other things, that the sine-Gordon equation is quite unique in this regard amongst all integrable equations. - Ana Rechtman

National Autonomous University of Mexico, Mexico City, MexicoThe trunkennes of a flowMotivated by an invariant for knots known as the trunk, we define the trunkennes of a vector field. This quantity is invariant under mesure preserving homeomorphisms and is not proportional to helicity, as almost all known asymptotic invariants. I will explain the construction and the main results related to this invariant. This is joint work with Pierre Dehornoy. - Mikhail Malakhaltsev

University of los Andes, ColombiaBinary differential equations and 3-webs with singularitiesA $3$-web with singularities is a geometric structure locally given by three one-dimensional distributions on an open dense subset $U$ of a two-dimensional manifold $M$. A point in $U$ is called \emph{regular} if values of the distributions are pairwise transversal at this point, all the other points of $M$ are called \emph{singular}. A binary differential equation of third degree determines a $3$-web with singularities (see, for example, T. Fukui and J. J. Nuño-Ballesteros, Isolated singularities of binary differential equations of degree $n$, Publicacions Matemátiques, vol. 56, 65--89, 2012). We describe singularities of this $3$-web, and show how to find topological and differential invariants of these singularities using methods developed in the paper F.A. Arias, J.R. Arteaga, and M. Malakhaltsev, 3-webs with singularities, Lobachevskii J. of Math, 37 (1), 1--20, 2016. - Alessandro Portaluri

University of Torino, ItalyIndex and stability of closed semi-Riemannian geodesicsA celebrated result of Poincaré asserts that a closed minimizing geodesic on a orientable (Riemannian) surface is unstable when considered as an orbit of the geodesic flow. In this talk starting from this classical result, we'll discuss some recently results on the strong and linear instability of closed geodesics of any causal character on higher dimensional (maybe not oriented) Lorentzian and more general semi-Riemannian manifolds. Dropping the non-positivity assumption of the metric tensor is a quite challenging task since the Morse index is truly infinite. This is a joint work with X. Hu and R. Yang - Martha P. Dussan Angulo

Universidade de São Paulo, BrazilBjorling problem for timelike surfaces and solutions of homogeneous wave equationIn this talk we show how to construct split-holomorphic extensions of initial curves $\gamma(t)$ in the Lorentz space $\mathbb R^3_1$, using, in a natural way, the point of view of solutions of homogeneous wave equation. After extending that curve to a subset of split-complex plane, we solve explicitly the Bjorling problem for timelike surfaces in the Lorentzian spaces $\mathbb R^3_1$, $\mathbb R^4_1$ and $\mathbb R^4_2$. As consequences we construct new examples and give applications. In particular, we describe one-parameter families of timelike surfaces which are solutions of the timelike Bjorling problem. In addition, we also establish symmetry principles for the class of minimal timelike surfaces in those ambient spaces. These results are part from the published papers by the author in Journal of Mathematical Analysis and Applications, Journal of Geometry and Physics and Annali di Matematica Pura ed Applicata. We remember that the classical Bjorling problem was proposed by Bjorling in 1844 and consists of the construction of a minimal surface in $\mathbb R^3$ containing the strip in the interior. The solution was obtained by Schwarz in 1890 through of a explicit formula in terms of initial datas. After that, the Bjorling problem has been considered in other ambient spaces, including in bigger codimmension or with indefinite metrics. - Débora Lopes da Silva

Universidade Federal do Sergipe, BrazilCodimension one partially umbilic singularities of hypersurfaces of $\mathbb{ R}^4$This talk is about the mutually orthogonal one dimensional singular foliations, in oriented three dimensional manifolds $\mathbb{M}^3$, whose leaves are the integral curves of the principal curvature direction fields associated to immersions $\alpha:\mathbb{M}^3\rightarrow\mathbb{R}^4$. We focus on behavior of these foliations around singularities defined by the points, called partially umbilic, where at least two principal curvature coincide. It will be described the generic behavior of the foliations in the neighborhood of partially umbilic points of codimension one. These are the singularities which appear generically in one parameter families of hypersurfaces. We express the codimension one condition by minimally weakening the genericity condition given by R. Garcia, D. Lopes e J. Sotomayor in \textit{Partially Umbilic Singularities of Hypersurfaces of $\mathbb{R}^4$. Bulletin des Sciences Mathematiques (Paris. 1885), v. 139, p. 431-472, (2015).} - Jean Carlos Cortissoz

University of los Andes, ColombiaOn Bloch's TheoremA classical theorem of André Bloch guarantees that there is a $B>0$ such that for any holomorphic function $f:D\longrightarrow \mathbb{C}$, where $D\subset\mathbb{C}$ is the unit disk, such that $\left|f'\left(0\right)\right|=1$, there is a subdomain $D'\subset D$, so that $f$ restricted to $D'$ is one to one and $f(D')$ contains a disk of radius $B$. Computing the optimal value of $B$ is an open problem. In this talk, we will discuss a new proof of Bloch's theorem, and a possible approach to improve on the known estimates on $B$. This is joint work with Julio Montero. - Adolfo Guillot Santiago

National Autonomous University of Mexico, Cuernavaca, MexicoAlgebraic differential equations with uniform solutionsIn the complex domain, the solutions of an ordinary differential equation may present multivaluedness. We will talk about a recent result describing the ordinary differential equations (given by rational vector fields on complex algebraic surfaces) that have a solution that is not multivalued. - Frederico Xavier

Texas Christian University, USAOn the inversion of real polynomial mapsThere is some evidence to support the conjecture that a polynomial local diffeomorphism of $\mathbb R^n$ into itself, $n\geq 3$, is injective if the pre-images of all $2$-planes in $\mathbb R^n$ are homeomorphic to connected subsets of $\mathbb R^2$. In this talk, we discuss this problem and offer proofs of some related global invertibility results. The arguments involve geometric constructions that use arguments from topology and complex analysis. Part of this work is joint with S. Nollet. - Ronaldo Alves Garcia

Universidade Federal de Goiás, BrazilDarboux curves on surfacesIn 1872, Gaston Darboux defined a family of curves on surfaces in the 3-dimensional Euclidean space $\mathbb{R}^3$ which are preserved by the action of the Möbius group and share many properties with geodesics. In this talk the Darboux curves will be considered under a dynamical viewpoint and described globally in special canal surfaces, quadrics and some Darboux cyclides. It will be based, mainly on the paper by R. Garcia; R. Langevin; P. Walczak, Darboux curves on surfaces II. Bull. Braz. Math. Soc. (N.S.) 47 (2016). Some open problems will be posed. - Daniel Offin

Queens University, CanadáMultiple periodic solutions in classical Hamiltonian systems.The question of multiple periodic solutions on energy surfaces, has a long and distinguished history. We will consider the question of multiple periodic solutions on non compact energy surfaces, and find conditions which guarantee infinitely many. We use variational techniques including the mountain pass theorem, and a result of the author which guarantees that a minimizing periodic solution must be hyperbolic on its energy surface.