Universidad de Granada

I will report on new and recent work with Leandro Arosio (Universita di Roma, Tor Vergata) on chaos and other aspects of dynamics in certain highly symmetric complex manifolds. For example, we prove that for many linear algebraic groups G, chaotic automorphisms are generic among volume-preserving holomorphic automorphisms of G. I will start with some background on holomorphic dynamics in general, on chaos, and on the highly symmetric complex manifolds to which our results apply.

20 de septiembre de 2019, 11:30, Seminario 1ª planta IEMath-GR

(Joint with Kyeongsu Choi.) We study closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds, focusing on mean curvature flow for the talk. In all dimensions and codimensions, we classify ancient mean curvature flows in \(S^n\) with low area: they are steady or canonically shrinking equators. In the mean curvature flow case in \(S^3\), we classify ancient flows with more relaxed area bounds: they are steady or canonically shrinking equators or Clifford tori. In the embedded curve shortening case in \(S^2\), we completely classify ancient flows of bounded length: they are steady or canonically shrinking equators.

4 de octubre de 2019, 12:30, Seminario 1ª planta IEMath-GR

We study compactness properties of metrics of prescribed fractional \(Q\)-curvature of order \(3\) in \(\mathbb{R}^3\). We use an approach inspired from conformal geometry, regarding a metric on a subset of \(\mathbb{R}^3\) as the restriction of a metric on \(\mathbb{R}^4_+\) with vanishing fourth-order \(Q\)-curvature. In particular, in analogy with a \(4\)-dimensional result of Adimurthi, Robert and Struwe, we prove that a sequence of such metrics with uniformly bounded fractional \(Q\)-curvature can blow up on a large set (roughly, the zero set of the trace of a nonpositive biharmonic function \(\Phi\) in \(\mathbb{R}^4_+\)), and we also construct examples of such behaviour. Towards this result, an intermediate step of independent interest is the construction of general Poisson-type representation formulas (also for higher dimension).

This is a work done in collaboration with María del Mar González, Ali Hyder and Luca Martinazzi.

5 de septiembre de 2019, 10:00, Seminario 1ª planta IEMath-GR

Conferenciante: Pedro José Chocano Feito (U. Complutense).




27 de septiembre de 2019, 11:30, Seminario 1ª planta IEMath-GR

La vida apareció con la primer molécula capaz de copiarse a si misma. Llamaremos a todo ente dotado de esta capacidad un "replicador" . La biología evolutiva estudia los mecanismos, llamados fuerzas evolutivas, que moldean las formas de vida. Presentaré resultados (existencia, unicidad, regularidad y comportamiento en tiempo grande) sobre ecuaciones no locales que sirven como modelos de la evolución del fenotipo en una población de replicadores. Estas ecuaciones han sido igualmente empleadas en economía, reacciones químicas auto-catalíticas e inclusive en la evolución del lenguaje. Desde un punto de vista matemático, estas ecuaciones resultan interesantes ya que los métodos clásicos del análisis no lineal (por ejemplo el principio del máximo) no son aplicables.

14 de junio de 2019, 12:30, Seminario 2ª planta IEMath-GR