Alternatively have a look at the program.

## Registration

## Free boundary minimal surfaces and the Steklov eigenvalue problem (I)

If we fix a smooth compact surface $M$ with boundary, we can consider

all Riemannian metrics on $M$ with fixed boundary length. We can then

hope to find a canonical metric by maximizing a first eigenvalue. The

eigenvalue problem which turns out to lead to geometrically interesting

maximizing metrics is the Steklov eigenvalue problem; that is, the

Dirichlet-to-Neumann map on $\partial M$. There is a close connection

between this eigenvalue problem and minimal surfaces in a Euclidean ball

that are proper in the ball and meet the boundary of the ball

## Evolution Equations in Geometry (I)

Evolution equations have been used to address successfully key questions in Differential Geometry

like isoperimetric inequalities, the Poincaré conjecture, Thurston’s geometrization conjecture, or

the differentiable sphere theorem.

During this series of lectures we will give a general introduction to geometric flows, which are sort of

non-linear versions of the heat equation for a relevant geometric quantity. These equations should be

understood as a tool to canonically deform a manifold into a manifold with nicer properties, for instance,

## Geometric spectral invariants (I)

We will begin with the discovery of the first geometric spectral invariants by Hermann Weyl in 1912.

In the first lecture, we will recall Weyl's original proof using Dirichlet-Neumann bracketing, to demonstrate

"das asymptotische verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen,'' which is

now known as Weyl's law. In the second lecture, we will turn to the spectral invariant which lead to the

discovery of new geometric spectral invariants in the 1960s and 1980s: the heat trace. We will recall the

## A spectral lower bound in singular setting

A very well known result in Riemannian geometry, the Obata-Lichnerowicz theorem, relates the Ricci curvature and the spectrum of the Laplacian: for a compact Riemannian manifold of dimension n, if the Ricci tensor is bounded below by $(n-1)$, then the first non-zero eigenvalue of the Laplacian is larger than or equal to the dimension of the manifold. Equality holds if and only if the manifold is isometric to the sphere.

## Four-dimensional Cohomogeneity one Ricci flow and Nonnegative sectional curvature

We study the Ricci flow on $4$-manifolds admitting a cohomogeneity one group action, i.e. an isometric group action such that the orbit space $M/G$ is $1$-dimensional. We use this to demonstrate the first examples of 4-manifolds having nonnegative sectional curvature which under the Ricci flow, immediately acquire some negatively curved 2-planes. In particular, we show that $S^4$, $\mathbb{C}P^2$, $S^2\times S^2$ and $\mathbb{C}P^2 \# \overline{\mathbb{C}P^2}$ admit such metrics. This talk is based on joint work with Renato G. Bettiol.

## Selberg zeta function for hyperbolic orbifolds: transfer approach

To a hyperbolic surface and a finite-dimensional representation of its fundamental group, we associate a Selberg zeta function. The main goal of the talk is to show that under certain conditions, the Selberg zeta function admits a meromorphic extension to the whole complex plane. Our main tool is the use of transfer operators. This is a joint work with Anke Pohl.

## Evolution Equations in Geometry (II)

Evolution equations have been used to address successfully key questions in Differential Geometry

like isoperimetric inequalities, the Poincaré conjecture, Thurston’s geometrization conjecture, or

the differentiable sphere theorem.

During this series of lectures we will give a general introduction to geometric flows, which are sort of

non-linear versions of the heat equation for a relevant geometric quantity. These equations should be

understood as a tool to canonically deform a manifold into a manifold with nicer properties, for instance,

## The Willmore energy of non-orientable surfaces

The Willmore conjecture was proved by F.\ Marques and A.\ Neves in 2012. Since then we

know that the Clifford torus has the lowest Willmore energy among all tori in $\mathbb R^3$.

I will present results concerning immersed Klein bottles in $\mathbb R^n$ with low Willmore

energy, $n\geq 4$. For Klein bottles immersed in $\mathbb R^4$ it is known that there are three

distinct regular homotopy classes each one containing an embedding. I will explain that

one of these classes contains the embedded Klein bottle with lowest Willmore energy among

## Holonomy Groups of Locally Conformally Kähler Metrics

A Hermitian metric on a complex manifold is locally conformally Kähler

if it is conformal to a Kähler metric in the neighbourhood of each point.

In this talk will be given the classification of the possible holonomy groups

of such metrics on compact manifolds. As a by-product, we present the geometric

description of conformal classes on compact manifolds containing two non-homothetic

Kähler metrics - necessarily with respect to non-conjugate complex structures.

The talk is based on joint work with Farid Madani and Andrei Moroianu, which is