Modern Techniques in Analysis and Differential Geometry

Speaker : Jessica JAUREZ (Universidad Nacional Autónoma de México,Mexico)

The plan is to study some basic aspects of ordinary differential equations and give a brief introduction to partial differential equations. First-order linear differential equations. Emphasis will be placed on linear equations in two variables. The solutions and phase portraits obtained in general cases will be analyzed. Second-order linear differential equations. Solutions will be obtained using the method of variation of parameters, and the geometry of the solutions of a representative equation will be illustrated. Hamiltonian differential equations. Nonlinear differential equations of this type will be analyzed using the geometry of level curves of functions in two variables. Introduction to partial differential equations. The classic equations that arise in applications will be introduced, and to solve some of them, the method of separation of variables will be introduced.

Speaker : Guillermo Antonio LOBOS VILLAGRA (Universidade Federal de São Carlos,Brazil)

This mini-course provides an accessible introduction to the global geometry of surfaces, aimed at advanced undergraduate students in mathematics. Starting from the local theory of surfaces in R^3, we will study the first and second fundamental forms, Gaussian curvature, and mean curvature through classical examples such as the sphere, torus, and catenoid. We will study classical global results including the Gauss–Bonnet theorem, Hopf’s theorem, and fundamental geometric inequalities and its topological consequences.

Speaker : Gerardo HERNÁNDEZ (Universidad Nacional Autónoma de México,Mexico)

his workshop will introduce participants to fundamental concepts and tools in geophysical fluid dynamics and numerical modeling. The aim is for students to understand the physical and mathematical foundations underlying certain mathematical models based on Partial Differential Equations (PDEs), as well as their applications to atmospheric and oceanic dynamics, and the challenges associated with their numerical solution. Emphasis will be placed on both physical intuition and the computational techniques required to explore complex phenomena. 1.Mass Conservation and the Convection Equation 2, Lorenz Equations and Chaos 3. Numerical Methods for the Transport Equation and Numerical Stability 4. Shallow-Water Equations, Linearization, and Kelvin Waves 5. Anelastic Equations, the Boussinesq Approximation, and Numerical Simulations A solid understanding of these topics is essential for students and young researchers interested in atmospheric dynamics, oceanography, or applied mathematics. Throughout the workshop, participants will acquire not only theoretical foundations, but also practical criteria for constructing and analyzing numerical models, while recognizing the intrinsic limits of predictability and the significance of approximations in scientific research.

Speaker : Jorge VELASCO (Universidad Nacional Autónoma de México,Mexico)

This course provides an overview of the classical models in epidemiology, both for directly transmitted infectious diseases and for vector-borne diseases. We will review the SIS, SIR, SEIR models, as well as the Ross–Macdonald model and their extensions to incorporate heterogeneity, age structure, risk groups, multiple strains, and related generalizations. Models for directly transmitted diseases: SIS, SIR, SEIR. Models for vector-borne diseases: Ross–Macdonald model. The basic reproduction number. Incorporating heterogeneity: age structure, risk groups. Applications and examples.

Speaker : Karina NAVARRO (Weizmann Institute of Science,Israel)

In this course, we study the continuous, symmetric and positivity of an integral kernel on a compact Lie group G in terms of its symbol. Next we talk about the Reproducing Kernel Hilbert Space (RKHS) that the previous kernel generates and finally we present estimates for entropy numbers or covering numbers of the unit ball of RKHS of functions on G. The asymptotic behavior of the bounds we obtain depends on the dimension of the group and the trace order of the symbol.

Speaker : Jorge MOZO (Universidad de Valladolid,Spain)

The objective of this project is to study analytic and algebraic problems arising from holomorphic differential equations with singular points. Participants will explore both convergent and divergent solutions and attempt to deduce properties related to the divergence of solutions. As a first step, students will be given examples of differential equations that they will solve using classical methods. They will observe that some of these equations admit solutions expressible through divergent series. In the second step, they will analyze the differences between equations with convergent solutions and those with divergent solutions. Students will discover the distinction between regular and irregular singularities. Examples of regular singularities, such as hypergeometric equations, will be used to construct new, higher-order examples and their solutions. A third objective will be to examine the types of divergence at irregular singular points. Using Newton polygons, participants will compute solutions in certain cases and will be introduced to the theory of summability. Finally, there will be a computational component. Participants will investigate why some higher-order linear differential equations have explicit solutions while others do not. Basic algorithms, such as Kovacic’s algorithm, will be introduced and applied to construct specific examples. At the end of each week, participants will prepare a poster to present to the other groups. In these posters, they will not only showcase the results they have studied but also discuss additional problems—possibly open ones—that could serve as directions for further study.

The objective of this project is to introduce the students into the more advanced techniques of the geometric analysis and their applications to the study of the global geometry of surfaces in the Euclidean space. In particular, we will intend to make accessible to the level of the students several research topics, some of them very recent, about the global behavior of the curvature and the topology of surfaces in the Euclidean space.

Speaker : Ingrid MARTÍNEZ (Universidad de El Salvador,El Salvador),Mynor MELARA (Universidad de El Salvador

This workshop aims to be an introduction to the calculus of variations, which is essentially concerned with minimizing or maximizing integral functionals, studying some of its applications, particularly in the area of ​​geometry, with emphasis on approach and solving a series of proposed problems. First we will give some preliminaries and study the concept of functionals and variations, the formulas for the first and second variations and the Euler-Lagrange equation, then we will proceed to understand and solve variational problems. As an application, we will study the anisotropic functional and the geometry of immersed surfaces in the three-dimensional Euclidean space with constant anisotropic mean curvature.