Lojasiewicz-Simon inequalities and applications to geometric problems
The Lojasiewicz inequalities for analytic functions were known in the field of Algebraic Geometry from the 1960s and were used to prove the "Lojasiewicz theorem" which states that if a smooth curve in space is the negative gradient flow line of an analytic function and has an accumulation point at infinity, then the length of the curve is finite and the curve actually converges towards this limiting point. These results were the motivation for L. Simon, who generalized the Lojasiewicz inequalities for analytic functionals defined on an infinite dimensional space of smooth functions on a compact, smooth Riemannean manifold, and used these generalizations to study uniqueness of blow-ups to certain geometric PDEs. Lojasiewicz-Simon type inequalities were also recently used by F.Schulze and Colding,Minicozzi who showed uniqueness of tangent flows to Mean Curvature Flow in the case the tangent flow is a compact and non-compact hypersurface respectively. The purpose of this talk is to introduce and discuss these finite and infinite-dimensional Lojasiewicz inequalities. If time allows, we will also briefly sketch applications of these to Mean Curvature Flow.

Uniqueness of compact tangent flows to Mean Curvature Flow
One property of the flow by mean curvature is that compact, smoothly embedded hypersurfaces develop singularities in finite time. Once singularities occur, the natural procedure to understand them is to "magnify" around them, by considering the parabolically rescaled flow, and study its properties. Any limiting flow obtained by such a sequence of rescalings is called a tangent flow at the singularity, and turns out to be self-similarly shrinking. However, uniqueness of the respective shrinker, independently of the particularly chosen sequence of rescalings remained open until recently. Finally, it was given an affirmative answer by Felix Schulze in the case the shrinker is compact, and by Colding and Minicozzi in the non-compact case. In this talk we are going to discuss the proof by F. Schulze which relies on a direct application of the Lojasiewicz-Simon inequalities to Huisken's monotone Gaussian integral.

Bartnik's quasi local mass and related conjectures
I would introduce the quasi local mass defined by Bartnik, and discuss the related conjectures including the Bartnik extension problems for the static/stationary vacuum Einstein equations. We will briefly discuss some of the results by M. Anderson, M. Khuri, and P. Miao.

Type-II singularities of immersed, two-convex mean curvature flow
We will prove that any translator which arises as a type-II blow-up limit of an immersed, two-convex mean curvature flow in $\R^{n+1}$, $n\geq 3$, is rotationally symmetric. Recently, Haslhofer proved this result (also for $n=2$) under the additional assumption that the limit is non-collapsing. We remove this assumption by making use of the cylindrical and gradient estimate of Husken and Sinestrari.

Ancient solutions of the mean curvature flow
I will discuss some recent work on ancient solutions of the mean curvature flow, including a convexity estimate, which makes use of the parabolic Harnack inequality.

Biharmonic maps and their heat flow
Among other connections, biharmonic maps are a natural generalisation of harmonic maps and a model for the Willmore energy. The fact that the Euler-Lagrange equations are fourth-order (as opposed to second order for harmonic maps) presents a number of analytic challenges. In this talk, I will discuss $O(d)$-equivariant biharmonic maps and their associated heat flow. Among other results, I will present a blowup result for the biharmonic map heat flow from $B^4$ into $S^4$.