Geometry, Analysis and Applications Jan 25-Feb 5, 2021.

Geometry and Analysis are two of the most successful areas in the modellization of real life phenomena.

The last decades have witnessed the power of their union, a new area termed Geometric Analysis

This Master School will teach you all that you need to know to use the tools of Geometric Analysis and will serve as an introduction to research in this growing area of Mathematics. The school will include two 5-hour minicourses by the award-wining mathematician María Ángeles García Ferrero, three talks on different panoramics of the field and eigth short presentations by young researchers working on the topic.

  • Course I: Maria Ángeles García Ferrero. An introduction to Geometry from Analysis.
    Abstract: In this course we will review some introductory material in Riemannian geometry from the perspective of analysis and partial differential equations. We will deal with interesting objects in the interplay of these areas like minimal surfaces, eigenvalues or level sets.

  • Course II: Maria Ángeles García Ferrero. The Geometry of PDEs.
    Abstract: In this course we will study the theory, introduced by A. Enciso and D. Peralta-Salas, to show the existence of solutions to certain partial differential equations (PDEs) with  prescribed geometrical properties. We will review some other problems in PDEs with a tight geometric connection, for instance in the context of inverse problems.

  • Panoramics I: Leonardo Colombo. On the role of geometry and analysis in control theory.
    Abstract: A control system is a family of dynamical systems parameterized by the controls, evolving on a differentiable manifold (the state space). The first part of this talk introduces the basic concepts of geometric control theory, in particular, controllability and optimal control. Controllability is related to the ability to reach a state from any other state, using the available controls. Optimal control deals with the possibility to do it in the best possible way. These concepts will be illustrated with simple examples.  The second part of the talk focus on recent problems I am studying relating geometric control with techniques of analysis on manifolds for multi-agent systems and hybrid dynamical systems. 

  • Panoramics II: Azahara de la Torre. [CANCELLED] Some tools from conformal geometry to study non-local PDEs.
    Abstract:
    In this talk, I will explain some new tools developed in conformal geometry to solve non-local elliptic semi-linear equations. These tools, which originally arose to study geometric properties, are really useful to solve several non-local and non-linear PDE problems (through the understanding of the intrinsic geometry which is present in the PDEs).
  • Panoramics III: Fernando Sanz. On the problem of the gradient of an analytic function.
    Abstract:
    In the 70’s, Lojasiewicz and Thom promoted the investigation of geometrical properties of trajectories of a gradient vector field of a real analytic function. By means of a celebrated inequality of Lojasiewicz, bounded trajectories have finite length and accumulate to a single point. Thom conjectured then that they possess a well-defined tangent at the limit point. The problem was revived around 20 years later with the proof of Thom’s conjecture by Kurdyka, Mostowski and Parusinski and also with Moussu’s stronger Non-oscillation Conjecture: a trajectory of a gradient cannot oscillate. The answer to this last conjecture is only known in dimension two, but still open in general. In this talk, we give a panorama of the achievements concerning this problem, a subject that reveals a particular interplay between analysis of ODEs, real analytic and subanalytic geometry and qualitative theory of dynamical systems.

  • Short presentations by young researchers (in order of scheduling):
    • Manuel Lainz. The geometry of Herglotz’s principle of least action
    • Jacob Goodman. Decentralized Motion Control of Multiagent systems on Riemannian manifolds
    • Damir Ferizović. Potential Theory with Multivariate Kernels I
    • Ryan Matzke. Potential Theory with Multivariate Kernels II
    • Pablo Gómez. Comparison Theory in Riemannian Geometry
    • María Martín. Completeness and stability of spacetimes with Lorentz-Minkowski ends
    • Alexandre A. Simoes. The exponential map for second order differential equations
    • Jose Manuel Fernández Barroso. Can one hear geometric properties related with the Jacobi operator on closed Riemannian manifolds?

You can download the abstracts of these short presentations here.

Attendance to this Master School will be free of charge but participants need to register first writing and email to ciem@unican.es. All conferences and courses will take place online through a Microsoft Teams link that will be provided to the participants. No microsoft account is required to attend: only a standard internet browser.

See the schedule in this link.

Organizers: