Abstracts for Young Women in Geometry

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

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

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.

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,