2 pm: Luca Capogna (University of Arkansas, USA)
An Aronsson-type approach to extremal quasiconformal mappings in space.
Extremal problems for quasiconformal mappings typically involve two domains $\Omega, \Omega'\subset R^n$, (or two Riemann surfaces) for which there exists a quasiconformal mapping $f:\Omega\to\Omega'$, and ask for a quasiconformal map $u:\Omega\to\Omega'$ which minimizes one or more of the dilation functions in a given class of competitors which are usually other quasiconformal mappings with same boundary data as $f$ on a portion (or all) of the boundary of the domain $\Omega$ or in the same homotopy class as the given map $f$. In a joint work with Andrew Raich (UARK) we  study this $L^{\infty}$ variational problem from ta novel  point of view  which echoes some of G. Aronsson's work in the 60's, concerning absolute minimal extensions. We derive a non-linear, degenerate elliptic, PDE system as a formal limit of Euler Lagrange equations for $L^p$ approximations for the functional and  we prove that solving such a system is  a sufficient condition for extremality of $C^2$ quasiconformal diffeomorphisms. We also study a family of gradient flows for $L^p$ dilation and prove short time existence of solutions for smooth data.

4 pm: Giovanna Citti (University of Bologna, Italy)
Regularity of non-characteristic minimal graphs in high dimensional Heisenberg groups
Minimal surfaces in the sub-Riemannian Heisenberg group can be constructed by means of a Riemannian approximation scheme, as limit of Riemannian minimal surfaces. We study the regularity of Lipschitz, non-characteristic minimal surfaces which arise as such limits. Our main results results is $C^{\infty}$ regularity of the sub-Riemannian minimal surface.

 

 

Curvature estimates for surfaces with bounded mean curvature
In this talk I will discuss some recent results concerning estimates for the norm of the second fundamental form, |A|, for surfaces with bounded mean curvature. In particular I will show that for an embedded geodesic disk with bounded L^2 norm of |A|, |A| is bounded at interior points, provided that the W^{1,p} norm of its mean curvature is sufficiently small, p>2. This is joint work with Giuseppe Tinaglia.

 

   

   

Volume optimisation on triangulated 3-manifolds
In 1978, Thurston introduced an affine algebraic set to study hyperbolic structures on triangulated 3-manifolds. Thurston's theory can be considered as a discrete SL(2,C) Chern-Simons theory on the manifold. Recently, Feng Luo discovered a finite dimensional variational principle on triangulated 3-manifolds with the property that its critical points are related to both Thurston's algebraic set and to Haken's normal surface theory. The action functional is the volume. This is a generalisation of an earlier program by Casson and Rivin for compact 3-manifolds with torus boundary. Combining the work of Luo, Futer-Guéritaud, Segerman-Tillmann and Luo-Tillmann, this gives a new, finite dimensional variational formulation of the Poincaré conjecture and may lead to both a discrete version of the 3-dimensional Ricci flow, as well as a new proof of the conjecture.

Multiple-layer solutions to the Allen-Cahn equation on hyperbolic space (joint wok with R. Mazzeo)
In this work we study the existence of multiple-layered solutions to the elliptic Allen-Cahn equation in Hyperbolic Space. More precisely, we consider the equation \begin{equation} -\Delta_{\mathbb H} u+W'(u)=0,\label{ac}\end{equation} where $\Delta_{\mathbb H}$ is the Laplace-Beltrami operator in Hypebolic space and $W$ is a positive potential with two minima. We prove that for any given collection of non-intersecting hyperplanes in $\mathbb H$ there is a solution to \eqref{ac} that has these hyperplanes as interfaces. Our result provides a Riemannian generalization of the work of M. del Pino, M. Kowalczyk, F. Pacard and J. Wei.

How to produce Harnack inequalities via Canonical Solitons
In this talk, we first review the notion of Canonical Expanding Soliton for the Ricci Flow (which is a special kind of space-time gradient soliton in a certain limiting sense, after stretching a lot the time direction). Then we show how to apply such a new construction to derive new Harnack inequalities for the Ricci Flow. This viewpoint also gives geometric insight into the existing Harnack inequalities of Hamilton and Brendle.

Regularität elliptischer Systeme mit antisymmetrischen Potentialen
Es werden Beispiele aus einerseits der Theorie der fraktionalen polyharmonischen Abbildungen und andererseits aus der Theorie der degeneriert elliptischen $n$-harmonischen Abbildungen vorgestellt, welche in Zusammenhang mit Rivieres berühmtem Resultat von 2007 über die Regularität von kritischen Punkten von konform invarianten Variationsfunktionalen in zwei Dimensionen stehen.