Xornada "Symmetry and shape"

Ven, 16 Set 2022 12


During the 3rd week of October, specifically between the 13th and 16th, the Congress "Symmetry and shape" will be held in the Main Classroom of the Faculty of Mathematics of the University of Santiago de Compostela. The objective of this conference is to bring together experts in the study of symmetry in Differential Geometry. The conference will revolve around the study of curvature, homogeneous and symmetric spaces, the geometry of Riemann's submanifolds and other topics related to Differential Geometry and Geometric Analysis.

According to Felix Klein, geometry is the study of those properties in space that are invariant under a given transformation group. Intuitively, symmetry is the correspondence of the shape at each point in a space. An interesting problem in geometry and many physical sciences is determining the symmetries of a space from its shape.

Throughout this activity, there will be presentations and talks by the different guests (updated on 09/19/2022):

Talks

  • Christoph Böhm, Universität Münster, Germany.
  • Giovanni Catino, Politecnico di Milano, Italy.
    • Some canonical metrics on Riemannian manifolds: In this talk I will review recent results concerning the existence of some canonical Riemannian metrics on closed (compact with no boundary) smooth manifolds. The constructions of these metrics are based on Aubin's local deformations and a variant of the Yamabe problem which was first studied by Gursky.

  • Quo-Shin Chi, Washington University at St. Louis, USA.
    • Fano 3-folds and classification of constantly curved holomorphic 2-spheres of degree 6 in the complex Grassmannian G(2,5): Harmonic maps from the Riemann sphere to Grassmannians (and, more generally, to symmetric spaces) arise in the sigma-model theory in Physics. Such maps of constant curvature constitute a prototypical class of interest, for which Delisle, Hussin and Zakrzewski proposed the conjecture that the maximal degree of constantly curved holomorphic 2-spheres in the (complex) G(m,n) is m(n−m) and confirmed it when m=2 and n=4 or 5. Up to unitary equivalence, there is only one constantly curved holomorphic 2-sphere of maximal degree 4 in G(2,4) by Jin and Yu. On the other hand, up to now, the only known example in the literature of constantly curved holomorphic 2-sphere of maximal degree 6 in G(2,5) has been the first associated curve of the Veronese curve of degree 4. By exploring the rich interplay between the Riemann sphere and projectively equivalent Fano 3-folds of index 2 and degree 5, we prove, up to the ambient unitary equivalence, that the moduli space of (precisely defined) generic such 2-spheres is semialgebraic of dimension 2. All these 2-spheres are verified to have non-parallel second fundamental form except for the above known example.

  • Anna Fino, University of Torino, Italy.
  • Luis Guijarro, Universidad Autónoma de Madrid, Spain.
  • Michael Jablonski, University of Oklahoma, USA.
  • Andreas Kollross, Universität Stuttgart, Germany.
  • Adela Latorre, Universidad Politécnica de Madrid, Spain.
  • Carlos Olmos, Universidad Nacional de Córdoba, Argentina.
  • Marco Radeschi, University of Notre Dame, USA.
  • Uwe Semmelmann, Universität Stuttgart, Germany.
  • Joeri Van der Veken, University of Leuven, Belgium.

Posters

  • Local topological rigidity of 3-manifolds of hyperbolic type, Andrea Drago (Sapienza University of Rome, Italy).

We study systolic inequalities for closed, orientable, Riemannian 3-manifolds of bounded positive volume entropy. This allows us to prove that the class of atoroidal manifolds (i.e. that admit an hyperbolic metric) with uniformly bounded diameter and volume entropy is topologically rigid. In particular our main result is the following theorem:

Let X be a closed, orientable, atoroidal, Riemannian 3-manifold with Ent(X)<E and Diam(X)<D. Then there exist a function s(E,D) such that, if Y is closed, orientable, torsionless, Riemannian 3-manifold with Ent(Y)<E and dGH(X,Y)<s(E,D), then π1(X)≅π1(Y). In particular, X and Y are diffeomorphic.

  • Homogeneous spaces of G2, Cristina Draper Fontanals (Universidad de Málaga, Spain).

Pilar Benito, Cristina Draper and Alberto Elduque study the reductive homogeneous spaces obtained as quotients of the exceptional group G2 in the Draper doctoral dissertation (see [1]), from an algebraic perspective. In this poster we revisit these spaces from a more geometrical approach.

Bibliography:

  1. P. Benito, C. Draper, A. Elduque: Lie-Yamaguti algebras related to g2, J. Pure Appl. Algebra 202 (2005), 22-54.

Organizing committee

  • José Carlos Díaz Ramos, researcher linked to CITMAga, University of Santiago de Compostela
  • Miguel Domínguez Vázquez, researcher linked to CITMAga, University of Santiago de Compostela
  • Eduardo García Río, researcher linked to CITMAga, University of Santiago de Compostela
  • Victor Sanmartín López, University of Santiago de Compostela
  • M. Elena Vázquez Abal, researcher linked toCITMAga, University of Santiago de Compostela


  • Source: http://xtsunxet.usc.es/symmetry2022/committees.html