Session abstracts

Session S39 - Differential Equations and Geometric Structures



Tuesday, July 13, 12:00 ~ 12:50 UTC-3

A family of parabolic tight flute surfaces

Camilo Ramírez Maluendas

Universidad Nacional de Colombia, Sede Manizales, Colombia   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

A Riemann surface $S$ is called parabolic if and only if there are no nonconstant negative subharmonic functions on $S$. From the Uniformization Theorem any Riemann surface can be thought as the quotient space of the hyperbolic plane $\mathbb{H}$ by a Fuchsian group $\Gamma$, i.e. $S=\mathbb{H}/\Gamma$. Such surfaces admit a Riemannian metric of constant curvature. So, the Riemman surface $S$ is parabolic if and only if it satisfies one of the following conditions:

1. The behaviour of the geodesic flow on the unit tangent bundle of $S$ is ergodic;

2. The Poincaré series of $\Gamma$ diverges;

3. The Fuchsian group $\Gamma$ has the Mostow rigidity property;

4. The Riemann surface $S$ has the Bowen's property;

5. Almost every geodesic ray of $S$ is recurrent.

In a recent work, A. Basmajian, H. Hakobyan and D. Šaric´ have described sufficient condition to determinate the parabolicity of certain Riemann surface of infinite-type from the Fenchel-Nielsen conditions.

In this talk, we will use the Fenchel-Nielsen conditions and build geometrically a family (moduli space) conformed by tight flute surfaces (which are topologically equivalent to the infinite-type genus zero surface whose ends space is homeomorphic to the closure $\overline{\{1/n:n\in\mathbb{N}\}}$), such that each one of these is parabolic.


[1] J. A. Arredondo and C. Ramírez Maluendas. On Infinitely generated Fuchsian groups of the Loch Ness monster, the Cantor tree and the Blooming Cantor tree. Comp. Man. 7 (2020), no. 1, 73-92.

[2] L. Ahlfors and L. Sario. Riemann surfaces. Princeton Mathematical Series, No. 26 Princeton University Press, Princeton, N.J. 1960.

[3] A. Basmajian. Hyperbolic structures for surfaces of in infinite type. Trans. Amer. Math. Soc. 336, no. 1, March 1993, 421-444.

[4] A. Basmajian, H. Hakobyan and D. Šaric´. The type problem from Riemann surfaces via Fenchel-Nielsen parameters. Preprint, available on arXiv.

Joint work with John Alexander Arredondo García (Fundación Universitaria Konrad Lorenz, Bogotá, Colombia) and Israel Morales Jiménez (Instituto de Matemáticas UNAM, Unidad Oaxaca, Oaxaca, México).

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Tuesday, July 13, 13:00 ~ 13:50 UTC-3

On the multiplicity of umbilic points

Farid Tari

University of São Paulo, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Umbilic on a regular surface $M$ in the Minkowski 3-space $\mathbb R^3_1$ are the singular points of the lines of principal curvature (these can be extended at points where the induced pseudo metric on $M$ is degenerate). They are the points where all the coefficients of the binary differential equation (BDE) of these lines vanish.

Umbilic points on a generic surface are stable, that is, they persist and with the same local configuration of the lines of principal curvature under small deformations of the surface. However, the BDE of these lines is not stable when deformed within the set of all BDEs.

An invariant of a BDE, called the multiplicity of the BDE, at its singular point was introduced in [Bruce-Tari, 1998] and counts the maximum number of well folded singularities that can appear in a local deformation the BDE. Here, we introduce an invariant of an analytic surface $M$ in the Minkowski 3-space at its umbilic points, and call it the multiplicity of the umbilic point (the concept is also valid for surfaces in the Euclidean 3-space). The multiplicity counts the maximum number of stable umbilic points that can appear under small deformations of the surface at a non-stable umbilic point. We establish its properties and compute it in various cases.

Joint work with Marco Antonio do Couto Fernandes (University of Sao Paulo, Brazil).

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Tuesday, July 13, 14:00 ~ 14:50 UTC-3

The Double Eigenvalues Curve of a Line Congruence

Marcos Craizer

Pontifícia Universidade Católica do Rio de Janeiro, Brasil   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

A line conguence 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. The curve where the principal directions coincide is called the double eigenvalues curve of the line congruence.

In this talk we classify the generic singularities of the focal surface along the double eigenvalues curve. A basic tool is the support function associated with an eq\"uiaffine vector field transversal to a surface in 3-space

Joint work with Ronaldo A. Garcia (Universidade Federal de Goiás, Brasil).

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Tuesday, July 13, 15:00 ~ 15:50 UTC-3

Curvature lines of compact surfaces of genus $k$.

Federico Sánchez-Bringas

Universidad Nacional Autónoma de México, México   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We analyze the curvature lines of a compact surface of genus $k \in \mathbb N$ embedded in $\mathbb S^3$ as the link of the real part of the Milnor fibration of a polynomial in $\mathbb C^2$. Since the polynomial is quasi-homogeneous, the link gives rise, under the natural $\mathbb C^*$-action, a family of diffeomorphic surfaces with equivalent curvature lines. We use the symmetries of the model to describe the curvature lines in the cases of genus $k=2,3$. Finally, we study a bifurcation of the case of genus $k=3$ along the umbilic points.

Joint work with Vinicio Gómez Gutiérrez (Universidad Nacional Autónoma de México).

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Wednesday, July 14, 12:00 ~ 12:50 UTC-3

Poincaré Problem for foliations on $\mathbb{CP}^2$ with a unique singularity

Claudia Reynoso Alcántara

Universidad de Guanajuato, México   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The Poincaré problem for foliations on $\mathbb{CP}^2$ is a very famous problem that asks the following: if we have a foliation on $\mathbb{CP}^2$ of degree $d$ with a rational first integral of degree $s$ then, can we bound $s$ as function of $d$? The answer in general is that it is not possible. Consider for example the foliation on $\mathbb{CP}^2$ of degree 1 given by the 1-form:

\begin{equation*} pyzdx+qxzdy-(p+q)yxdz \end{equation*}

with $p$ and $q$ positive integers, the pencil $\{\alpha x^py^q - \beta z^{p+q}\}$ of degree $p+q$ defines the rational first integral.

The main objective of this talk is to give a positive answer to this problem for the case of foliations on $\mathbb{CP}^2$ of degree $d$ with a unique singular point. To obtain the bound we use the geometry of the local singular scheme of the foliation. We will give examples of extreme cases: when de degree of the rational first integral is the maximum and minimum possible.

Joint work with Ramón Ronzón-Lavié (University of Toronto).

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Wednesday, July 14, 13:00 ~ 13:50 UTC-3

Foliated affine and projective structures

Adolfo Guillot

UNAM, Mexico   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We discuss the existence of foliated affine and projective structures along the leaves of a holomorphic foliation by curves (varying holomorphically in the transverse direction). We will see that there are many foliations that admit them. We will give an index formula that allows to show that there are also many foliations do not admit these structures. For instance, a non-singular foliation of general type on a manifold of even dimension does not admit a foliated projective structure.

Joint work with Bertrand Deroin.

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Wednesday, July 14, 14:00 ~ 14:50 UTC-3

Group bundles and group connections

David Blázquez-Sanz

Universidad Nacional de Colombia - Sede Medellín, Colombia   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

This talk is devoted to our recent findings [1, 2] about differential equations that are compatible with the group operation in a Lie group, or a Lie group bundle. Such equations are group connections. We characterize the space of group connections on a group bundle as an affine space modeled over the vector space of 1-forms with values cocycles in the Lie algebra bundle of the aforementioned group bundle. We show that group connections satisfy the Ambrose-Singer theorem and that group bundles can be seen as a particular case of associated bundles realizing group connections as associated connections. We give a construction of the Moduli space of group connections with fixed base and fiber, as an space of representations of the fundamental group of the base.

[1] B.-S, Marín, Suarez. Group bundles and group connections. arXiv:2104.04804

[2] B.-S., Marín, Ruíz. A simplified categorical approach to several Galois theories. Cahiers de topologie et géométrie différentielle catégoriques. Vol. LXI (2020) 4, 450-473.

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Wednesday, July 14, 15:00 ~ 15:50 UTC-3

Dynamics of a pollinator, plant and herbivore populations

Victor Castellanos

Universidad Juárez Autónoma de Tabasco, México   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We carried out the analysis of a three-dimensional ODE nonlinear autonomous system which is derived with the aim of describing the interaction between three populations. These take the form of two mutualistic (pollinators and plants) and a third population (herbivores) is introduced. This one is feeded by consuming plants which, in turn, damages the pollinators population too by reducing the rate of visits (to plants) behalf the pollinators. The specific type of interactions between the populations are described by two types of functional responses of type IV. The main result is the proof of the existence of an attracting limit cycle for the ODE system. This emerges from a supercritical Hopf bifurcation. Its existence is proved by using the Hopf-Andronov bifurcation theorem and its stability is proved by using the first Lyapunov coefficient. In addition of the analysis, a series of numerical simulations are carried out on the full ODE system.

Joint work with Faustino Sánchez-Gardu\~no (Universidad Nacional Autónoma de México, México) and Miguel Angel Dela-Rosa (Universidad Juárez Autónoma de Tabasco-CONACYT, México).

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Monday, July 19, 16:00 ~ 16:50 UTC-3

Rigidity and flexibility of entropies of boundary maps associated to Fuchsian groups.

Svetlana Katok

The Pennsylvania State University, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Given a closed, orientable surface of constant negative curvature and genus $g\ge 2$, we study the topological entropy and measure-theoretic entropy (with respect to a smooth invariant measure) of generalized Bowen--Series boundary maps. Each such map is defined for a particular fundamental polygon for the surface and a particular multi-parameter.

We present and sketch the proofs of two strikingly different results: topological entropy is constant in this entire family (``rigidity''), while measure-theoretic entropy varies within Teichmüller space, taking all values (``flexibility'') between zero and a maximum that is achieved on the surface which admits a regular fundamental $(8g-4)$-gon. We obtain explicit formulas for both entropies. The rigidity proof uses conjugation to maps of constant slope, while the flexibility proof---valid only for certain multi-parameters---uses the realization of geodesic flow as a special flow over the natural extension of the boundary map.

Joint work with Adam Abrams (Wrocław University of Science and Technology, Poland) and Ilie Ugarcovici (DePaul University, USA).

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Monday, July 19, 17:00 ~ 17:50 UTC-3

Maslov index and stability of periodic solutions

Daniel Offin

Queen's University , Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We give a review of the Maslov index and its variants, how it can be used in global dynamics of conservative systems to understand stability of periodic solutions in symmetric settings or low dimensional settings.

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Monday, July 19, 18:00 ~ 18:50 UTC-3

Exponential mixing implies Bernoulli

Federico Rodriguez Hertz

Penn State, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk I plan to discuss some consequences of systems having the exponential mixing property. In particular I will address our joint work with D. Dolgopyat and A. Kanigowski showing that for diffeomorphisms, exponential mixing implies Bernoulli. If time permits, some open problems will be discussed and further research as well.

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Monday, July 19, 19:00 ~ 19:50 UTC-3

On the logarithmic diffusion equation

Jean Carlos Cortissoz

Universidad de los Andes, Colombia   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we will consider the boundary value problem \[ \left\{ \begin{array}{l} \partial_t u=\partial_{xx} \log u\quad \mbox{in}\quad \left[-l,l\right]\times \left(0, \infty\right)\\ \displaystyle \partial_x u\left(\pm l, t\right)=\pm 2\gamma u^{p}\left(\pm l, t\right), \end{array} \right. \] where $\gamma$ is a constant. Let $u_0>0$ be a smooth function defined on $\left[-l,l\right]$, and which satisfies the compatibility condition $$\partial_x \log u_0\left(\pm l\right)= \pm 2\gamma u_0^{p-1}\left(\pm l\right).$$

This equation is closely related to the Ricci flow on a cylinder. We will use this relation to show some results on the behavior of solutions to this equation . For instance, depending on $p$, these solutions can be global or not, and depending on the sign of $\gamma$ there could be blow-up or blow-down (quenching) in finite or infinte time (in the case the solution is global). This is joint work with C\'esar Reyes.

Joint work with César Reyes (Universidad Manuela Beltrán).

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Regilene Oliveira

ICMC-USP, Campus São Carlos, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In the weak 16th Hilbert problem, the Poincaré-Pontryagin-Melnikov function, $M_1(h)$, is used for obtaining isolated periodic orbits bifurcating from centers up to a first-order analysis. This problem becomes more difficult when a family of centers is considered. In this work we provide a compact expression for the first-order Taylor series of the function $M_1(h,a)$ with respect to $a$, being a the multi-parameter in the unperturbed center family. More concretely, when the center family has an explicit first integral or inverse integrating factor depending on $a$. We use this new bifurcation mechanism to increase the number of limit cycles appearing up to a first-order analysis without the difficulties that higher-order studies present. We show its effectiveness by applying it to some classical examples.

Joint work with Jackson Itikawa (UNIR, Brazil) and Joan Torregrosa (UAB, Spain).

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