Doc-Course 2019: Geometric Analysis

As an activity of the Andalusian Institute of Mathematics (IAMAT) developed at the Institute of Mathematics of the University of Granada (IEMath-GR), the Institute of Mathematics of the University of Seville (IMUS) and the University of Cádiz (UCA), we offer a 10-week post-graduate intensive school about Geometric Analysis. This activity is also framed within the Research Unit in Mathematics, iMAT.

The program consists of three parts:

PART 1. SPECIFIC COURSES

  • Course 1: The isoperimetric problem (IMUS, October 1-11).
    The isoperimetric problem seeks minimizers of the perimeter among sets enclosing a given volume. While this problem is solved in Euclidean space, where the unique solutions are the round balls, the situation is very different in other spaces, where even the existence of minimizers is not guaranteed. The aim of this course is to analyze the following topics in relation to the problem:

    1. The isoperimetric profile.
    2. Existence, regularity and variational properties of minimizers.
    3. Solution to the problem in some spaces.
    4. Applications: functional and geometric inequalities.

    Taught by Antonio Cañete (US), César Rosales (UGR) and Manuel Ritoré (UGR).

  • Course 2: Constant mean curvature surfaces (IMUS, October 1-11).
    The study of surfaces attending to their mean curvature H goes back to the origin of the Calculus of Variations in times of Euler and Lagrange. Minimal surfaces (H=0) are critical points of the area functional, while surfaces with constant nonzero H (CMC) have critical area for variations that preserve volume. We will cover:

    1. Maximum Principle for elliptic PDE with applications: Alexandrov reflection method.
    2. CMC surfaces in homogeneous three-manifolds wioth isometry group of dimension 4.
    3. Stability of CMC surfaces.

    Taught by José María Espinar (UCA), Isabel Fernández (US), José Miguel Manzano (UCM) and Francisco Torralbo (UGR).

  • Course 3: Minimal surfaces (IEMath-GR, October 14-25).
    1. The Plateau problem: Solution of Douglas-Radó for minimal discs. Applications and generalizations.
    2. Complex analysis and minimal surfaces: Weierstrass representation and some recent advances on the subject.
    3. Limits of sequences of embedded minimal surfaces: uniform local área and curvature bounds, minimal laminations and introduction to Colding-Minicozzi theory.
    4. The Dirichlet problem for the minimal graph equation: divergence lines and the Jenkins-Serrin problem.

    Taught by Antonio Alarcón (UGR), Francisco J. López (UGR), Magdalena Rodríguez (UGR) and Joaquín Pérez (UGR).

  • Course 4: Other geometric PDEs (IEMath-GR, October 14-25).
    1. How to use EDPs and Riemannian Geometry to solve some problems of surface theory.
    2. Overdetermined elliptic problems: from classical theory to open conjectures. Geometric ideas and methods coming from Surface Theory.
    3. Geometric PDEs modeled by holomorphic data: classification theorems for solutions.

    Taught by Pablo Mira (UPCT), Antonio Ros (UGR) and Pieralberto Sicbaldi (UGR).

PART 2. CONFERENCE ON GEOMETRIC ANALYSIS
To be held at the University of Cádiz from October 28 to 31, this conference will gather together experts in Geometric Analysis from distinguished research institutions.

Scientific committee:

  • José M. Espinar (UCA)
  • José A. Gálvez (UGR)
  • Joaquín Pérez (UGR)
  • Harold Rosenberg (IMPA)

PART 3. SUPERVISED RESEARCH PROGRAM
PhD students attending the Doc Course will also participate in short research projects supervised by experts, in which they will learn some of the most up-to-date techniques in Geometric Analysis. This will take place at Universities of Cádiz, Granada and Seville from November 4 to December 3. On December 4 and 5, there will be an informal workshop in which the participants will present their projects.

The deadline for registration is April 30, 2019.
For detailed information about registration and grants for students, please visit this page.