Pacelli Bessa
“Green functions and the Dirichlet spectrum”
Vincent Borrelli
“Fitting the Earth isometrically in a ping-pong ball”
Abstract:
In 2015, John Nash received the Abel Prize for his outstanding contributions to the theory of PDEs. One of his baffling results implies that it is possible to squeeze the surface of the Earth inside a ping-pong ball while preserving all the distances! In this talk, we will make this dream concrete by constructing the first distance preserving reduction of a sphere. We will also show numerous pictures.
Esther Cabezas-Rivas
“Ricci flow beyond non-negative curvature conditions”
Abstract:
We generalize most of the known Ricci flow invariant non-negative curvature conditions to less restrictive negative bounds that remain sufficiently controlled for a short time.
As an illustration of the contents of the talk, we prove that metrics whose curvature operator has eigenvalues greater than -1 can be evolved by the Ricci flow for some uniform time such that the eigenvalues of the curvature operator remain
greater than -C. Here the time of existence and the constant C only depend on the dimension and the degree of non-collapsedness. We obtain similar generalizations for other invariant curvature conditions, including positive biholomorphic curvature in the Kaehler case. We also get a local version of the main theorem.
As an application of our almost preservation results we deduce a variety of gap and smoothing results of independent interest, including a classification for non-collapsed manifolds with almost non-negative curvature operator and a smoothing result for singular spaces coming from sequences of manifolds with lower curvature bounds.
We also obtain a short-time existence result for the Ricci flow on open manifolds with almost non-negative curvature (without requiring upper curvature bounds).
This is a joint work with Richard Bamler (Berkeley) and Burkhard Wilking (Muenster).
Pablo Chacón
“Multisymplectic Approach to Discrete Field Theory on Cellular Complexes”
Abstract:
In the 90s of last century J. Moser, A.P Veselov, J.E. Marsden, G.W. Patrick, S. Shkoller, among other authors, proposed to discretize the Calculus of Variations in order to approximate numerically since its same root the Euler-Lagrange equations. The result was, essentially, a discretizacion of the Hamilton-Cartan formalism of the calculus of variations through the so-called variational integrators, which consist in integrating the discrete Euler-Lagrange equations by means of certain families of symplectic difeomorphisms.
Three basic references on this topic, which include a well selected bibliography, are:
-) J.E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer. 10 (2001), 357-514.
-) J.E. Marsden, G.W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Comm. Math. Phys. 199 (1998), 351-395.
-) A.C. Casimiro and C. Rodrigo, First variation formula and conservation laws in several independent discrete variables, J. Geom. Phys. 62 (2012), 61-86.
In these references, the base manifold of the corresponding fibration is the standard cellular complex in R^n, for n=1, n=2, and arbitrary n respectively, being the basic object of the two first ones a discrete counterpart of the Cartan form in the continuous case, while in the third one a variation formula of the discrete Lagrangian density expressed exclusively in terms of the geometry of the cellular complex of the base manifold is emphasized.
In this general framework, our research group at the University of Salamanca has extended to the field theory the symplectic approach of the problem of Lagrange in discrete mechanics in the sense of:
- ) P.L García, A. Fernández, and C. Rodrigo, Variational integrators for discrete Lagrange problems, J. Geom. Mech. 2 (2010), 343-374.
Such an extension is based on a previous multisymplectic formulation of the unconstrained discrete field theory on cellular complexes, that will be the topic of this talk. This is a joint work with A. Fernández (USAL), P.L García (RAC and USAL), and C. Rodrigo (Acad. Militar Portugal).
Leonor Ferrer
“Properly emdedded minimal annuli in H2 × R “
Abstract:
In this talk we ask for properly embedded minimal annuli in H2 × R which bound a pair of vertical graphs over ∂∞H2 ≡ S1. We present some compactness results for these surfaces. We also give some existence results for proper, Alexandrov-embedded, minimal annuli. Contrary to what might be expected, we show that, in general, one can not prescribe the two components of the boundary at infinity. However, we can prescribe one of the boundary data, the position of the neck and the vertical flux of the annulus.
This is a joint work with F. Martín, R. Mazzeo and M. Rodríguez.
Anna Fino
"The Symplectic Calabi-Yau problem”
Abstract:
The Calabi-Yau problem is a PDE's system whose study goes back to the celebrated Calabi conjecture. Recently, Donaldson has described how the problem can be generalized to the setting of symplectic geometry on 4-manifolds.
The aim of this talk is to show some results for certain non-Kaehler 4-manifolds which are torus fibrations. In particular we will study the symplectic Calabi-Yau problem for the Kodaira-Thurston manifold viewed as an S^1-bundle over a 3-dimensional torus.
Ilkka Holopainen
“Existence and nonexistence results for minimal graphic and $p$-harmonic functions”
Abstract:
Ana Hurtado
“Comparison theorems and parabolicity of manifolds with density”
Abstract:
We study the weighted parabolicity of a Riemannian manifold with density whose Bakry-Emery Ricci curvature is bounded from below. We give estimations in this context of the drifted Laplacian of the distance function in the weighted manifold; as a consequence, we obtain upper bounds for its capacity and establish conditions that guarantee parabolicity. We extend this analysis to the study of the geometric description of the parabolicity of submanifolds immersed in a weighted manifold and which inherits the radial density of the ambient space. Finally, we give applications of the parabolicity of such submanifolds to some classical problems, as the Bernstein problem. Joint work with V. Palmer and C. Rosales.
Luquesio Jorge
“A short view of minimal surfaces in R3”
Abstract:
After the work of Osserman in the 1960’s the theory of minimal surfaces in R3 became of great interest. In the recent development of this theory the Spain mathematicians did a very important work. We point out some of the most important results produced here and talk about some recent work and new examples. We give an idea on how to prove 3 missing points using a extension of little Picard theorem and show new examples of complete minimal surfaces whose Gauss map misses 2 points. These results are part of joint works with F. Mercuri and E. Gama.
Steen Markvorsen
“Tractors and tractrices in Riemannian manifolds”
Abstract:
We generalize the notion of planar bicycle tracks, a.k.a. one-trailer systems, to so-called tractor/tractrix systems in general Riemannian manifolds and prove explicit expressions for the length of the ensuing tractrices and for the area of the domains that are swept out by any given tractor/tractrix system. In this talk we will illustrate how these expressions are sensitive to the curvatures of the ambient Riemannian manifold, and we show explicit estimates for them based on Rauch's and Toponogov's comparison theorems. Moreover, the general length shortening property of tractor/tractrix systems is used to generate geodesics in homotopy classes of curves in the ambient manifold. The talk is based on joint work with Jesper Madsen, University of Chicago.
Peter W. Michor
“General Sobolev metrics on the manifold of all Riemannian metrics”
Abstract:
For a compact manifold $M^m$ equipped with a smooth fixed background Riemannian metric $\hat g$ we consider the space
$\operatorname{Met}_{H^s(\hat g)}(M)$ of all Riemannian metrics of Sobolev class $H^s$ for real $s>\frac m2$ with respect to $\hat g$.
The $L^2$-metric on $\operatorname{Met}_{C^\infty}(M)$ was considered in
[ Olga Gil-Medrano, Peter W. Michor: The Riemannian manifold of all Riemannian metrics. Quarterly J. Math. Oxford (2) 42 (1991), 183–202] among other papers.
Sobolev metrics of integer order on $\operatorname{Met}_{C^\infty}(M)$ were considered in
[M.Bauer, P.Harms, and P.W. Michor: Sobolev metrics on the manifold of all Riemannian metrics.
J. Differential Geom., 94(2):187--208, 2013.]
In this talk we consider variants of these Sobolev metrics which include Sobolev metrics of any positive
real (not integer) order $s$.
We derive the geodesic equations and show that they are well-posed under some conditions and induce
a locally diffeomorphic geodesic exponential mapping.
Luigi Vezzoni
“A quantitative version of a Theorem of Alexandrov”
Abstract:
